Cover
Frontmatter
Introduction and Welcome: To Know the End from the Beginning…
Chapter 1: Reasoning and Resources: Tools for Life
1.0 Introduction and Objectives
1.1 Analytical Reasoning: A Tool for Life
Project: Pascal’s Famous Triangle
1.2 Math Classics and Applied Reasoning
Project: Guards and Prisoners
Project: Cross-number Puzzle
1.3 Resources for the Present and the Future
Project: MERLOT and You
Chapter 2: Patterns and Reasoning: Fun with Math!
2.0 Introduction and Objectives
2.1 Deductive and Inductive Reasoning
Project: A Riddling Good Time
2.2 Pattern Recognition: Searching for Connections
Project: The Curious Date: 7-14-98
Project: Another Curious Date: 10-1-01
2.3 The nth Term: The Power of Generalization
Project: Sequence Creations
Project: Generalization—A Powerful Tool
2.4 Logical Thinking and Reasoning Backwards
Project: Puzzle Constructions
2.5 Magic Squares: Sum Fun!
Project: Magic Squares
Project: Magic Square Extensions
Chapter 3: Math from Other Times and Places: Historical Investigations
3.0 Introduction and Objectives
3.1 The Greeks: Students and Investigators
Project: Polygonal Numbers—A Geometric View
3.2 Pascal’s Amazing Triangle: Simple yet Profound
Project: Pascal’s Triangle
3.3 Number Systems through the Ages: Stylistic and Diverse
Project: The Abacus
Project: Numeration in Other Cultures
Chapter 4: Number Bases: Surprising Versatility
4.0 Introduction and Objectives
4.1 Understanding Other Number Bases: Places of Value
Project: The Banker’s Dilemma
4.2 The Binary System: Secret Language of Computers
Project: Chinese Trigrams and Binary Numbers
Project: The ASCII Code
Project: Course Logo: A Hidden Message
4.3 Number Base Applications: More Fun with Math!
Chapter 5: Modulus Arithmetic and Its Many Uses
5.0 Introduction and Objectives
5.1 Clock Systems: The Cycles of Life
Project: Perpetual Calendar
5.2 The Modulus in Action: Check Digits and Error Detection
Project: ISBN Numbers and Modulus Arithmetic
Project: ISMN Numbers and Modulus Arithmetic
5.3 Cryptography: The Mathematics of Privacy
Project: Cryptograms—A Popular Pastime
Project: Cryptoquotes—Another Favorite
Chapter 6: Mathematics and Music: Inseparable Partners
6.0 Introduction and Objectives
6.1 The Nature of Sound and Musical Scales: Good Vibrations
Project: Creating Good Vibrations
Project: Famous Math Quotes
6.2 The Mathematics of Stringed Instruments: Structured Harmony
Project: Guitar Analysis
6.3 Digital Music and CD’s: Applied Mathematics
Project: Digital Music and Mathematics
Chapter 7: Mathematics in Art, Architecture and Nature
7.0 Introduction and Objectives
7.1 Mathematical Perspective in Art: A Renaissance Breakthrough
Project: Mathematical Perspective
7.2 Symmetry and Tilings: Mathematical Beauty and Artistry
Project: Symmetry and Tilings
Project: The Mathematical Art of M.C. Escher
7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci
Project: The Amazing Rabbit Population
Project: Golden Rectangles in your Home
Project: Fibonacci Numbers in Nature and the Arts
Project: Logarithmic Spirals—Seashell Design
7.4 Long-term Projects
Project: Webliography Report
Project: Term Report
Project: Stock Market Investment Report
Chapter 8: Fractals: New Structures in Mathematics
8.0 Introduction and Objectives
8.1 Mathematical Fractals: A Natural Geometry
Project: Introduction to Fractals
8.2 Fractal Applications
Project: Fractals in Nature
Project: Fractal Music
Project: Fractal Art
8.3 Famous Triangles: Sierpinski Meets Pascal
Project: Triangle Connections
Chapter 9: Linear Relationships: Straight as an Arrow
9.0 Introduction and Objectives
9.1 Mathematical Relationships and Functions: “Vary-ables” in Action
Project: The Shape of Daylight
9.2 Linear Functions: A Deeper Look
9.3 Linear Models: Analytic and Predictive
Project: How Does your Lawn Grow?
Project: How Does your Garden Grow?
Chapter 10: Exponential Growth: Drama and Suspense
10.0 Introduction and Objectives
10.1 Doubling Power: An Amazing Phenomenon
10.2 The Exponential Function: A Deeper Look
Project: World Population Growth
10.3 Exponential Models and Applications
Project: U.S. Population Growth
Project: The Fastest Growing State in the U.S.?
Project: Population Growth in Florida
Chapter 11: Financial Planning: Mathematical Secrets to Acquiring Wealth
11.0 Introduction and Objectives
11.1 Compound Interest: Strategies for Wealth
Project: The Long-Term Results of Compound Interest
11.2 Savings Plans: Don’t Leave Home Without One!
Project: Intro to Savings and Investment
11.3 Stocks, Bonds and Mutual Funds
Project: Intro to Stocks, Bonds and Mutual Funds
Project: Investment Strategies
Project: A True and Amazing Story
Project: Stock Market Investment Report
11.4 Home Mortgages: Mathematical Revelations
Project: The All-Powerful Interest Rate
11.5 Credit Card Economics
Project: Credit Reports and Ratings: An Inside Look
Project: The Truth about Credit Card Debt
Appendices
Appendix A: Helps and Hints for Exercises and Projects
Appendix B: Cross-number Puzzle Terminology
Appendix C: Guide to Correlation Coefficients
Appendix D: Fractal Structures: Italian Parsley

##### Citation preview

MATH IS EVERYWHERE! EXPLORE AND DISCOVER IT! Fifth Edition Custom Edition for St. Petersburg College

James J. Rutledge Illustrations by Joyce T. Rutledge

Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Cover Art: Text illustrated by Joyce T. Rutledge All rights reserved. Permission in writing must be obtained from the publisher before any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system. All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and are used herein for identification purposes only. Pearson Learning Solutions, 501 Boylston Street, Suite 900, Boston, MA 02116 A Pearson Education Company www.pearsoned.com Printed in the United States of America 1 2 3 4 5 6 7 8 9 10 XXXX 17 16 15 14 13 12 000200010271627154 CW

ISBN 10: 1-256-67707-8 ISBN 13:978-1-256-67707-9 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

To Joyce, Justin and Nathan and To all of my students and colleagues who have inspired and encouraged me along the way J. R.

Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It!

Math Is Everywhere! Explore and Discover It! Fifth Edition

v

Chapter 1: Reasoning and Resources: Tools for Life

1

1.0 Introduction and Objectives 1.1 Analytical Reasoning: A Tool for Life Project: Pascal’s Famous Triangle 1.2 Math Classics and Applied Reasoning Project: Guards and Prisoners Project: Cross-number Puzzle 1.3 Resources for the Present and the Future Project: MERLOT and You

Chapter 2: Patterns and Reasoning: Fun with Math! 2.0 Introduction and Objectives 2.1 Deductive and Inductive Reasoning Project: A Riddling Good Time 2.2 Pattern Recognition: Searching for Connections Project: The Curious Date: 7-14-98 Project: Another Curious Date: 10-1-01 2.3 The nth Term: The Power of Generalization Project: Sequence Creations Project: Generalization—A Powerful Tool 2.4 Logical Thinking and Reasoning Backwards Project: Puzzle Constructions 2.5 Magic Squares: Sum Fun! Project: Magic Squares Project: Magic Square Extensions

Chapter 3: Math from Other Times and Places: Historical Investigations 3.0 Introduction and Objectives 3.1 The Greeks: Students and Investigators

1 3 6 8 19 20 22 26

29 29 32 46 49 62 63 65 81 82 85 99 103 116 117

121 121 123

v Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Project: Polygonal Numbers—A Geometric View 3.2 Pascal’s Amazing Triangle: Simple yet Profound Project: Pascal’s Triangle 3.3 Number Systems through the Ages: Stylistic and Diverse Project: The Abacus Project: Numeration in Other Cultures

Chapter 4: Number Bases: Surprising Versatility 4.0 Introduction and Objectives 4.1 Understanding Other Number Bases: Places of Value Project: The Banker’s Dilemma 4.2 The Binary System: Secret Language of Computers Project: Chinese Trigrams and Binary Numbers Project: The ASCII Code Project: Course Logo: A Hidden Message 4.3 Number Base Applications: More Fun with Math! Project: Hexadecimals

Chapter 5: Modulus Arithmetic and Its Many Uses

126 128 129 130 137 138

139 139 141 152 156 158 160 166 167 172

173

5.0 Introduction and Objectives 5.1 Clock Systems: The Cycles of Life Project: Perpetual Calendar 5.2 The Modulus in Action: Check Digits and Error Detection Project: ISBN Numbers and Modulus Arithmetic Project: ISMN Numbers and Modulus Arithmetic 5.3 Cryptography: The Mathematics of Privacy Project: Cryptograms—A Popular Pastime Project: Cryptoquotes—Another Favorite

173 175 180 185 190 193 195 207 207

Chapter 6: Mathematics and Music: Inseparable Partners

209

6.0 Introduction and Objectives 6.1 The Nature of Sound and Musical Scales: Good Vibrations Project: Creating Good Vibrations Project: Famous Math Quotes 6.2 The Mathematics of Stringed Instruments: Structured Harmony Project: Guitar Analysis 6.3 Digital Music and CD’s: Applied Mathematics Project: Digital Music and Mathematics

vi Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

209 211 223 224 225 227 234 236

Math Is Everywhere! Explore and Discover It!

Chapter 7: Mathematics in Art, Architecture and Nature 7.0 Introduction and Objectives 7.1 Mathematical Perspective in Art: A Renaissance Breakthrough Project: Mathematical Perspective 7.2 Symmetry and Tilings: Mathematical Beauty and Artistry Project: Tilings in your Home Project: Symmetry and Tilings Project: The Mathematical Art of M.C. Escher 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci Project: The Amazing Rabbit Population Project: Golden Rectangles in your Home Project: Fibonacci Numbers in Nature and the Arts Project: Logarithmic Spirals—Seashell Design 7.4 Long-term Projects Project: Webliography Report Project: Term Report Project: Stock Market Investment Report

Chapter 8: Fractals: New Structures in Mathematics 8.0 Introduction and Objectives 8.1 Mathematical Fractals: A Natural Geometry Project: Introduction to Fractals 8.2 Fractal Applications Project: Fractals in Nature Project: Fractal Music Project: Fractal Art 8.3 Famous Triangles: Sierpinski Meets Pascal Project: Triangle Connections

Chapter 9: Linear Relationships: Straight as an Arrow 9.0 Introduction and Objectives 9.1 Mathematical Relationships and Functions: “Vary-ables” in Action Project: The Shape of Daylight 9.2 Linear Functions: A Deeper Look 9.3 Linear Models: Analytic and Predictive Project: How Does your Lawn Grow? Project: How Does your Garden Grow?

239 239 241 245 248 253 254 255 256 260 261 262 263 266 267 268 270

273 273 275 281 284 291 292 292 293 296

299 299 301 317 318 336 346 347

vii Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It!

Chapter 10: Exponential Growth: Drama and Suspense 10.0 Introduction and Objectives 10.1 Doubling Power: An Amazing Phenomenon 10.2 The Exponential Function: A Deeper Look Project: World Population Growth 10.3 Exponential Models and Applications Project: U.S. Population Growth Project: The Fastest Growing State in the U.S.? Project: Population Growth in Florida

349 349 351 358 367 368 371 372 372

Chapter 11: Financial Planning: Mathematical Secrets to Acquiring Wealth 373 11.0 Introduction and Objectives 11.1 Compound Interest: Strategies for Wealth Project: The Long-Term Results of Compound Interest 11.2 Savings Plans: Don’t Leave Home Without One! Project: Intro to Savings and Investment 11.3 Stocks, Bonds and Mutual Funds Project: Intro to Stocks, Bonds and Mutual Funds Project: Investment Strategies Project: A True and Amazing Story Project: Stock Market Investment Report 11.4 Home Mortgages: Mathematical Revelations Project: The All-Powerful Interest Rate 11.5 Credit Card Economics Project: Credit Reports and Ratings: An Inside Look Project: The Truth about Credit Card Debt

373 375 386 389 391 392 394 395 396 397 399 409 412 414 415

********************

Appendices

421

Appendix A: Helps and Hints for Exercises and Projects Appendix B: Cross-number Puzzle Terminology Appendix C: Guide to Correlation Coefficients Appendix D: Fractal Structures: Italian Parsley

********************

viii Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

423 447 451 455

Math Is Everywhere! Explore and Discover It! Introduction and Welcome: To Know the End from the Beginning…

Introduction and Welcome: To Know the End from the Beginning… Introduction Welcome to Math Is Everywhere! Explore and Discover It!, a unique, project-based mathematics textbook and course of study for college students, particularly those who are Liberal Arts majors, and for anyone interested in mathematics and its relevance and application in today’s world. While written for entry-level college students in general, this text is especially suited for those who might be math-avoiders or math-phobic. It is also suited for those who may become elementary, middle or high school mathematics teachers; the content has much to offer in the way of mathematical ideas and resources for future educators. If you have ever felt that math was a dry and boring subject and wondered, “Why do I have to learn this?” or “When will I ever use this?”, then this textbook is for you! You’ll soon discover that mathematics has many fascinating and practical connections to music, art, architecture, nature, sports, finance, food, and much more, and I’m confident that you will thoroughly enjoy this textbook and course of study. 

In the Beginning This text was written to engage students in active learning and to reveal the beauty, practicality and pervasiveness of math in our daily lives. Math Is Everywhere! provides an effective means by which students can overcome their negative associations with math and find empowerment in their own analytical reasoning and critical thinking abilities. The text conveys a sense of enthusiasm, offers encouragement for success, and provides a wealth of opportunities for students to become engaged in mathematics in interesting and enjoyable ways.

Goals and Objectives Math Is Everywhere! is designed around the following seven goals:       

to engage students in active learning through the use of projects and activities to demonstrate the beauty, practicality and pervasiveness of math in our daily lives to lead students into a new and refreshing relationship with mathematics to alleviate student math anxiety to enable students to develop their analytical reasoning and critical thinking skills to promote technological skills to encourage students to become lifelong learners of mathematics

ix Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Introduction and Welcome: To Know the End from the Beginning…

In addition to the course goals and purposes mentioned above, there are specific content and learning objectives as well. Each textbook chapter contains a brief introduction that includes an outline of the specific course objectives for that chapter.

End Results—Evidence of Success From the outset, the materials that now comprise Math Is Everywhere! have experienced much success. Many students have used these materials and the results have been extremely positive. This curriculum was selected as a finalist in the Florida Association of Community Colleges Excellence in Curriculum Competition in 2001 and was also nominated for the Instructional Technology Council’s Outstanding Distance Learning Course Award in both 2006 and 2008. The following comments were made by some of my former students and are representative of the responses that students have had to this text and its project-based teaching/learning approach. I fully expect that you also will have a positive experience with this text and I look forward to hearing from you after you have completed this course of study. Best wishes for an exciting and adventuresome journey into the realm of mathematics, one in which you may enter into a new and enjoyable relationship with mathematics!  Sincerely, James J. Rutledge St. Petersburg College ********************

Student Comments This is the most fun I’ve ever had in Mathematics. I wonder why someone hasn’t thought of this sooner. For the first time I can say that I didn’t feel the anxiety that I would normally feel. I was calm, and didn’t even break out into a sweat! There hasn’t been a project that I didn’t enjoy. Now I find myself wondering about the Math in everything I look at. I truly enjoyed this class. I have to be honest when I say that math is definitely my weakest subject. This class gave me a different approach to math and showed me ways that I would actually use the skills I learned in the class in everyday life. More often than not, I am staring in a textbook and thinking “why do I need to know this??? I know I will never have a need for this in my life!!” With this class I did not feel that way. The textbook was very well written and user friendly.

x Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Introduction and Welcome: To Know the End from the Beginning…

xi Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Introduction and Welcome: To Know the End from the Beginning…

intriguing, something I am sure I will maintain an interest in for the rest of my life. I think that if children were taught this way they would themselves feel a closer relation to something that is so important. I absolutely loved the projects—such a much more exciting alternative to cramming for a few big tests. I found myself studying just as much if not more than with the traditional way of teaching but enjoying it. I wish all my classes were based on this theory of teaching. This was a great course! I never thought I would find math so relevant to everyday life. I have struggled with math all of my life, but this class was a lot easier to understand. It was just explained in a different way. What was great was that I actually learned and enjoyed everything about the class. Other math classes just got boring and uninteresting. So, I just want to say thank you, Mr. Rutledge, for making math fun again. You sparked a new interest for math I never knew I had. Honestly, I would have to say that this has been the most enjoyable math class I have ever taken. I feel like it opened up a whole new world of math to me, going beyond just memorizing formulas and such. It also focused on math in relation to the real world, so that I have come away with a better understanding of the major impact math has had in the world not only in science but even in art and music. I was a self proclaimed “math hater” but now I have a whole new appreciation for math. Thank You. This was the best math class that I have ever taken! I have always struggled with math and many times I have questioned the reason for needing to know much of the material in other math classes. This class was different and actually related math to something concrete in my life and I have learned so many things that I can actually use in my daily life. I have finally realized the point of doing an equation and rather than it being just “numbers”; it has shown me that by using a formula I will save time and actually figure out the answer to a problem that has some meaning to me. The text used for this course was amazing! The writing style, the layout, and most of all the price were better than any math book I have ever used in the past (and sadly I’ve taken a lot of math classes in my pursuit for my degree). I enjoyed the way things were written as to make it easier for those of us who are not math majors to understand. You related things to every day life, and that made learning the concepts so much easier. This mathematics course was probably the most interesting I have ever taken as a college student. The premise of just simply crunching numbers without ever learning about the origins or even how they may apply toward a “real-life” scenario is addressed with this course. This has been a great class and I would strongly recommend it to any math-phobic individuals who feel like they cannot grasp concepts or that math will never apply to the real world, this class disproves that logic. I enjoyed this math class very much. There were very many interesting subjects we learned about. The one that has stayed with me was the Fractal Art. I really enjoyed that subject and the pictures were great. The CD project was also interesting, but I have to say that the rabbit project was impressive. Now I know why my mother would never let me have one. Overall, I don’t

xii Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Introduction and Welcome: To Know the End from the Beginning…

think there were any assignments I didn’t like. I believe I would enjoy math more if all of the classes were as interesting, fun and comprehensive as this one. Thank you for an enjoyable semester! I loved the number puzzles and found them challenging, yet fun. I can honestly say I looked forward to doing math homework! I felt such a sense of accomplishment when I finally solved a puzzle! The projects on investments and mortgages were also valuable to me and I feel so much more knowledgeable about savings and investments. The websites offered great information that I can use long after this class is over. I can honestly say that math was my least favorite subject until this course. I enjoyed this course because it made you want to learn. I’m very good with visual things and by learning hands-on. This class helped with both and kept my interest with the jokes and M&M’s; I had a blast. I loved this class. Thank you for making it so much fun. I can honestly say that this class was the most effective math course I have ever taken. I was a bit skeptical at first, but I was truly surprised at how much I have learned. I am happy I made the decision to enroll and would encourage anyone to do the same. The most interesting aspect of this course was the “unorthodox” assignments, i.e. counting M&M’s and measuring objects in the home. I feel that these tasks definitely brought a different perspective that was so needed. Overall, I can say without hesitation that this class changed my entire attitude towards math, something I am grateful for. I feel that as an Elementary Ed major this course gave me lots of resources to look for an answer, lesson plans, insight, etc., in the future. I also think that I will be better able to explain to a class when that age-old question comes up, “Why do we have to learn this anyway?” I think I am also better equipped to appreciate the overall beauty of math and to convey that to children. I loved our text. It is by far the best mathematics book I’ve had the pleasure of using! You personalize math well and really present real life analogies for everything. So much of math seemed abstract before. You did an excellent job of bringing math into the realm of reality! Everyone should be lucky enough to get a math class like this one. If I would have had this class (or one similar to it) early on in my schooling, I would have been a lot more interested and excited in higher-level mathematics. ********************

xiii Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Introduction and Welcome: To Know the End from the Beginning…

Course Logo The logo shown below embodies some of the many topics and applications that are presented in this textbook. It also contains an encoded message that I’m sure you will appreciate later in this course of study! ******************** Meanwhile, please prepare to embark on a mathematical and educational journey that may very well change both your life and your outlook on mathematics.  ********************

Note: This logo also appears on the cover of this textbook in a multicolor format.

xiv Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.0 Introduction and Objectives

Chapter One Reasoning and Resources: Tools for Life 1.0 Introduction and Objectives The thumbnail sketches below represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

1.1 Analytical Reasoning: A Tool for Life 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1.2 Math Classics and Applied Reasoning

Farmer Fox Goat Cabbage

river

1.3 Resources for the Present and the Future MathForum

NCTM

MAA

MERLOT

1 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.0 Introduction and Objectives

Learning Objectives for Chapter 1 1. To enable students to identify and apply various problem-solving strategies. 2. To enable students to develop their analytical reasoning skills through solving problems and puzzles. 3. To enable students to develop their deductive and inductive reasoning skills in recognizing patterns of various types. 4. To promote the use of critical thinking skills through the use of puzzles and mathematical challenges. 5. To enable students to gain familiarity with Web-based mathematical resources. 6. To engage students in active learning through the use of projects and activities. 7. To alleviate student math anxiety through expositional encouragement. 8. To enable students to develop confidence in their mathematical abilities.

2 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.1 Analytical Reasoning: A Tool for Life

1.1 Analytical Reasoning: A Tool for Life Introduction One of the primary goals of this text is to foster and promote your analytical reasoning skills. Each of us has these skills and uses them in all aspects of daily life without consciously being aware of it. Determining the time at which you need to leave home to arrive at an appointment or a meeting is a simple example; calculating how much money you need to earn in a given period of time in order to pay your bills is another.

Analytical Reasoning: A Tool for Life Analytical reasoning is the primary tool used in solving problems of every kind, including math problems. While all students make use of these skills, many students have difficulty applying these skills to those dreaded “word problems” in math classes and to math problems in general. This course of study has been designed to facilitate the use of your analytical reasoning skills in a variety of settings and applications so that you gradually develop and deepen these skills as you proceed through the textbook. By the end of this course, you will be able to look back and recognize this process more clearly; for the present, please be encouraged! Solving problems and investigating mathematical relationships and connections can and will be quite enjoyable as you progress through this text. In the end, the development of your analytical reasoning skills will enable you to become more effective problem solvers in all areas of your life, not only in mathematics. (Incidentally, a number of my former students have reported that their reasoning skills improved as a result of the materials in this text.) In this section, we will outline some basic problem-solving strategies that will be helpful to you as you progress through the text. Many opportunities to apply these skills and strategies are provided throughout the text in the forms of both Opportunities to Exercise your Analytical Reasoning Skills and Discovery Moments; other opportunities are provided in the Exercises and Projects for Fun and Profit at the end of each textbook section.

Problem-solving Strategies Solving problems is something that all of us do throughout our lifetimes. It can be a very rewarding activity, especially when the situation requires some measure of creativity in order to find a solution. In order to facilitate your use of various problem-solving strategies throughout this course, I have provided an outline of the most widely used strategies below. This outline is

3 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.1 Analytical Reasoning: A Tool for Life

certainly not exhaustive, but it does include the strategies that are most commonly employed in the processes of problem solving. The strategies are loosely divided into three sections:  fundamental  algebraic  general purpose The fundamental strategies are those that are necessary and valuable in almost any situation while the algebraic strategies apply primarily to problems in which there are some unknown quantities in relationship with each other. The general purpose strategies are also valuable in many situations, especially those that are not algebraic in nature. I trust that you will find these to be helpful and instructive. fundamental strategies  clearly define the problem in your own words  construct a diagram whenever possible  clearly state the question that you will attempt to answer  identify the given facts and record them in a list or table  identify what it is that you need to know in order to answer your stated question algebraic strategies (if the problem invites an algebraic approach)  identify the important features of the problem  label the unknown features with variables  apply relevant formulas to the data, where applicable (e.g., distance = rate · time)  construct mathematical equation(s) to represent the stated facts  solve the equation(s) to obtain a solution general purpose strategies  look for patterns in the data  use trial and error, sometimes referred to as guess and check  eliminate the impossible; then only the possible remains  take the path of least resistance  use analytical reasoning to narrow the field of possibilities  gather all relevant data before jumping to a conclusion  relate the given problem to other similar problems that you have previously encountered

In the next section, Math Classics and Applied Reasoning, a number of these strategies will be illustrated as we examine some classic recreational math problems from days gone by and some new versions on which you can apply your skills. Meanwhile, the Creative Project below will allow you the opportunity to try your hand at problem solving.

4 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.1 Analytical Reasoning: A Tool for Life

Exercises and Projects for Fun and Profit Creative Project: Pascal’s Famous Triangle ********************

CALCULATOR HELP—RAISING A NUMBER TO A POWER To raise a number to a power on most scientific calculators (including the one that is accessible via your computer's Start/Programs/Accessories menu), first enter the number (also known as the base), then press the button labeled y x (sometimes labeled x y ), then enter the desired power (also known as the exponent), and press the "=" key. For example, to raise 5 to the 3rd power (i.e., 5 cubed, or 53), enter the base value of 5, press the yx button (or the x y button), enter the exponent value of 3, and press the "=" key. The result, 125, will then appear on the screen. ******************** If you have a graphing calculator, your calculator most likely uses a button labeled with the caret symbol, ^, in place of the button labeled y x on scientific calculators. To raise a number to a power, first enter the number (also known as the base), then press the button labeled ^, then enter the desired power (also known as the exponent), and press Enter. For example, to raise 5 to the 3rd power (i.e., 5 cubed, or 53), enter the base value of 5, press the ^ button, enter the exponent value of 3, and press Enter. The result, 125, will then appear on the screen. ******************** To display bases and exponents in a Word document, there are two options: 1. The simplest method is to use the caret key, ^, to indicate the use of an exponent. For example, 5 raised to the 3rd power may be typed into your Word document as 5^3. This is universally understood in mathematics (and computer programming). Note: The caret key, ^, is generally located above one of the numeric keys (e.g., the numeral 6) on your keyboard; use the Shift key to access it. 2. A more complicated method (but with a more professional appearance) is to use the Equation Editor that is usually available within Microsoft Word. 5 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.1 Analytical Reasoning: A Tool for Life

PASCAL’S FAMOUS TRIANGLE The array of numbers below shows the first six rows of Pascal’s triangle. While its historical origins are unknown, the triangle is named in honor of the French mathematician, Blaise Pascal (1623–1662), who popularized its use. 1 1 1 1 1 1

2 3

4 5

1 1 3 6

10

1 4

10

1 5

sum of entries = 1 = 2 0 sum of entries = 2 = 21 sum of entries = 4 = 2 2 sum of entries = 8 = 2 3 sum of entries = 16 = 2 4 sum of entries = 32 = 2 5

row #0 row #1 row #2 row #3 row #4 1 row #5

Note: row #3 begins with: 1, 3, …; row #4 begins with: 1, 4, … ; and so on. Each row of the triangle begins and ends with a 1. A number in the middle portion of a row is obtained by adding the entries diagonally to the left and diagonally to the right in the row immediately above. For example, in the bottom row shown above (row #5), the first 5 is the sum of the 1 that is diagonally above and to its left and the 4 that is diagonally above and to its right. Continuing with this pattern, the next row would begin with 1, and then 1+5 would produce a 6 as the second entry, 5 + 10 would produce 15 as the third entry, and so on. Hence the next row (row #6) would consist of the entries: 1

6

15

20

15

6

1

The sums of the entries in the rows of the triangle also display a pattern. A sum of entries in a row is found by adding together the entries; for example, the sum in row #3 is 1+3+3+1=8. Pascal’s triangle appears in the study of many topics in mathematics, including algebra, counting techniques, and probability.

Assignments: For each assignment, please answer the following four questions: question 1. Complete rows through: 2. Find the sum of entries in: 3. Find the sum of entries in: 4. See below

Assignment #1 Row #9

Assignment #2 Row #11

Assignment #3 Row #13

Assignment #4 Row #15

Row #10

Row #12

Row #14

Row #16

Row #20

Row #22

Row #24

Row #26

See below

See below

See below

See below

6 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.1 Analytical Reasoning: A Tool for Life

Note: You don’t necessarily have to complete the rows of the triangle through an indicated row in Questions #2 and #3; you may instead determine the sum of the entries in an indicated row based on your observation of the pattern of sums shown in the project introduction. 4. (Applies to all four assignments above.) Powers of a number are obtained by multiplying the number by itself a repeated number of times. For example:

5 2 = 5 · 5 = 25 and

5 3 = 5 · 5 · 5 = 125

By definition, a0 = 1 for all non-zero values of a. Evaluate the first five powers of 11, beginning with 0. 110 = 1 111 = 11 112 = 113 = 114 = Note: If you feel the need, please review the section entitled Calculator Help—Raising a Number to a Power that immediately precedes this project.

Notice what you obtain in comparison with Pascal’s triangle, i.e., notice how the digits in each power of 11 correspond to a row of entries in the triangle. For examples, 112 =121 and row #2 in the triangle contains the entries 1 2 1; 113 =1331 and row #3 in the triangle contains the entries 1 3 3 1. The entries in each row, when considered together as a single number, correspond to a power of 11; in fact, the row number indicates the power of 11 that the entries in the row represent. row #2: 1 2 1 row #3: 1 3 3 1 row #4: 1 4 6 4 1

112 =121 113 =1331 114 =14641

a) Does this correspondence continue in row #5 and in the following rows? b) Please explain why the correspondence continues or why the correspondence doesn’t continue based on your answer to the preceding question. ********************

7 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

1.2 Math Classics and Applied Reasoning Introduction Mathematics provides a wealth of opportunities to apply your analytical reasoning skills. In particular, the branch of mathematics known as Recreational Mathematics is a source of many creative and challenging problems that require various problem-solving techniques such as those mentioned in the preceding section of the text. In this section, we will explore a few of these math classics and then provide you with opportunities to explore additional classics in the Exercises and Projects for Fun and Profit at the end of this section. Each of these math classics has become a classic in part because the solution, even though simple in essence, is not immediately obvious. Finding the solution requires analytical reasoning, persistence and creative thinking—three hallmarks of mathematical endeavor.

Crossing the River: The Story of the Fox, the Goat and the Cabbage This story problem has been presented in various guises but perhaps the most familiar is that of the farmer who needs to cross a river with a fox, a goat and a cabbage. His objective is to safely transport them across the river under the following conditions:   

His small boat can carry only himself and one of his possessions (fox, goat or cabbage) at a time. The fox will eat the goat if they are left alone together. The goat will eat the cabbage if they are left alone together.

The question is: how can the farmer safely convey his fox, goat and cabbage across the river?

Strategy: Construct a diagram

Farmer Fox Goat Cabbage

river

8 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

Strategy: Trial and error Since the problem invites experimentation, simply choose the fox, the goat or the cabbage to accompany the farmer and see where that choice leads:   

If the farmer takes the fox, then the goat and cabbage will be left together and the goat will eat the cabbage. If the farmer takes the goat, then the fox and cabbage will be left together; this is not a problem. If the farmer takes the cabbage, then the fox and goat will be left together and the fox will eat the goat.

Consequently, the farmer must first take the goat across the river.

Fox Cabbage

river

Farmer Goat

The farmer then returns to the fox and the cabbage and must choose which to take next.

Farmer Fox Cabbage

river

Goat

Continuing with the trial and error approach, if the farmer takes the fox across the river and leaves it with the goat, the fox will eat the goat; similarly, if the farmer takes the cabbage across the river and leaves it with the goat, the goat will eat the cabbage. This presents a veritable dilemma!

9 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

***** An Opportunity to Exercise your Analytical Reasoning Skills *****  What shall the farmer do in this situation???? ************

Strategy: Eliminate the impossible; then only the possible remains. The farmer must bring either the fox or the cabbage across the river and leave it there; but since he can leave neither the fox nor the cabbage with the goat, those choices are impossible while the goat is there. The only possible remaining course of action is to bring either the fox or the cabbage across and then remove the goat by taking it back across the river with him to the original side. He can then leave the goat on the original side while he brings the remaining item (either fox or cabbage) across the river. Then the fox and the cabbage will both be together again and the farmer may make one last trip back across the river to pick up the goat and convey it to its destination as well. Success at last! 

***** Another Opportunity to Exercise your Analytical Reasoning Skills *****  The unanswered question is: After conveying the goat across the river and returning to the fox and the cabbage, which should the farmer choose to take next—the fox or the cabbage? If you’re not sure, you may find the answer to this question at the end of this section on p.21 following the Exercises and Projects for Fun and Profit. ************

The Prize Money and its Effects There are many variations of this story problem but the essence is that an additional sum of money will affect the status quo. In this version, two friends have some ten-dollar bills in their pockets and each has less than fifty dollars. Each of them has entered a contest in which the prize is \$30. The first friend declares: If I win the contest, I’ll be twice as rich as you. The second replies: Yes, but if I win the contest, I’ll be five times as rich as you. How much money does each of them have in their pockets? 10 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

Strategy: This type of problem lends itself to an algebraic approach in which you:     

identify the important features of the problem label the unknown features with variables apply relevant formulas to the data, where applicable (e.g., distance = rate · time) construct mathematical equation(s) to represent the stated facts solve the equation(s) to obtain a solution

In this case, the important facts are that the two people involved have some ten-dollar bills and each has less than five of these bills. Additional important facts are presented in the two friends’ statements. Since the ultimate goal in this problem is to determine how much money each friend has, it makes good sense to label their current amounts of money as follows: a = the amount of money that the first friend has at the beginning b = the amount of money that the second friend has at the beginning The next step is to construct simple equations that represent the statements concerning the effects of the prize money. If the first friend wins the \$30, then adding the prize money to his current amount will result in twice the current amount of the second friend. If the second friend wins, then adding the prize money to his current amount will result in five times the current amount of the first friend. Using the variables chosen above, we can write the equations: a + 30 = 2b b + 30 = 5a Finally, the objective is to solve these equations in some manner so that we may determine the values of a and b. Since this is a system of simultaneous equations, i.e., a set of equations in which the solution values for the variables must satisfy both equations, we may use the process of eliminating a variable in order to arrive at a solution. First, we’ll rearrange the equations by adding and subtracting terms in each equation to obtain: a  2b  30  5a  b  30

We may then multiply the first equation by five and add the result to the second equation in order to eliminate the variable a: 5a  10b  150  5a  b  30 By addition,  9b  180 b = 20 This means that the only value that b can have is \$20. Substituting this value back into a + 30 = 2b yields the equation: a + 30 = 40 11 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

From this we can determine that the solution value for a is \$10. The algebraic approach to this problem is quite effective and guarantees that we will obtain a result of some sort. Alternative strategy: Trial and error We could also have used the strategy of trial and error to construct a list of possible values for a and b to see which ones fit the conditions of the problem. With regard to the first friend’s statement, we have: a 10 20 30 40

b 10 20 30 40

a+30 40 50 60 70

2b 20 40 60 80

Since the value of a + 30 must equal the value of 2b, the only possible solutions are: a=10, b=20 and a=30, b=30. If we consider the other friend’s statement, we have: a 10 20 30 40

b 10 20 30 40

b+30 40 50 60 70

5a 50 100 150 200

In this case, since the value of b+30 must equal the value of 5a, the only possible solution is a=10, b=20. Consequently, since both equations have to be satisfied at the same time, the only mutual solution to the problem is a=10, b=20. Since the number of values to be tested is small, trial and error is an acceptable strategy for this problem; however, if there were many possible values for the variables, then this strategy would become quite tedious and the algebraic approach would be far more efficient. ******************** As seen above, there is often more than one way to approach a problem in an attempt to obtain a solution. In this course of study, I would encourage you to use your analytical reasoning abilities and your critical thinking skills to approach our exercises from various directions, whichever seem most suitable or convenient to you at the time. ******************** 12 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

A Math Crossword Puzzle (actually a Cross-number Puzzle) The crossword puzzle has been a standard fixture in literature for many years. In the following mathematical version, each entry both across and down is a positive whole number. The dilemma, as is usually the case with a crossword puzzle, is that many clues allow for more than one possible answer. This means that you may have to temporarily withhold your decision concerning an answer to a clue while you investigate other adjoining clues and their possible answers. As an example of this process, we’ll solve the following cross-number puzzle; the clues are provided below: 1

2 3

4

5 6

Across:

Down:

1. The square of a positive integer (i.e., a whole number multiplied by itself)

1. The cube of a positive integer (i.e., a whole number multiplied by itself twice; e.g., 9·9·9=729)

3. A prime number less than 100 (a prime is divisible only by 1 and itself) 4. A multiple of a gross (i.e., a multiple of 144: 144, 288, 432, 576, …)

2. A leap year in the 20th century (a leap year is a year that is evenly divisible by 4) 5. The number of trombones in an old popular show tune (The Music Man, 1957)

6. A number that is both a square and a cube

***** An Opportunity to Exercise your Analytical Reasoning Skills*****  Where does one begin in a situation like this? This is a seemingly perplexing dilemma. On which clue would you focus first in order to begin solving the puzzle? ************ 13 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

Strategy: Take the path of least resistance. If you chose “5 Down” or “6 Across,” then you’re on the right track—these clues appear to have only a very limited number of possible solutions. While the show tune is not contemporary, it has earned a place of recognition in our American culture and will hopefully be recognizable to you. If not, you may focus on the other clue that may have only one solution, viz., “6 Across.” Making a list of the squares and cubes less than one hundred will surely provide a solution to this clue. While taking the path of least resistance may seem like an obvious problem-solving strategy, it is often overlooked by students as they consider the various aspects of a problem.

Strategy: Use analytical reasoning to narrow the field of possibilities. Assuming that you have discovered the answers to “5 Down” (76 trombones) and “6 Across,” the next step is to decide which of the other clues are most likely to yield solutions. At this point in the puzzle, there don’t appear to be any clear choices to make. Consequently, you’ll need to resort to solving two or more clues simultaneously and then use analytical reasoning to narrow the field of potential choices. The clue for “1 Across” allows for a number of possibilities as does the clue for “1 Down.” While the number of three-digit cubes is rather limited (5), the number of three-digit squares is quite lengthy (21). This does make things complicated!

***** Another Opportunity to Exercise your Analytical Reasoning Skills *****  Since no solution is immediately forthcoming from the clues for “1 Across” and “1 Down,” where will you look next? ************ Yes, that’s it!  The clue for “2 Down” offers a valuable piece of information, viz., the leap year must begin with 1 since the year is in the twentieth century. This small scrap of information is actually quite valuable! Since “2 Down” must begin with 1, then “1 Across” must end with 1. This immediately limits the choices for “1 Across” to the following square numbers: 121, 361, 441, 841 and 961 (i.e., the squares of 11, 19, 21, 29 and 31). In turn, this limits the choices for “1 Down” to those cubes that begin with one of the digits 1, 3, 4, 8, and 9. Since the only three-digit numbers that are cubes are 125, 216, 343, 512 and 729, we can eliminate 216, 512 and 729 from consideration. This leaves only 125 and 343 as possible solutions for “1 Down.”

14 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

Strategy: Gather all relevant data before jumping to a conclusion. Before jumping to the conclusion that the answer to “1 Across” is 121 and that the answer to “1 Down” is 125, we need to be sure that we have gathered all the relevant facts concerning this mutual solution. The primary question of importance is: Are there any other possible solutions under the given circumstances? An inspection of the lists of squares and cubes given above reveals that there is another combination of a square and a cube that satisfies both pertinent clues: 361 and 343.

***** Another Opportunity to Exercise your Analytical Reasoning Skills *****  Since either of the proposed solutions seems to work equally well, we must search for yet another clue that will lead to the elimination of one of the possible solution pairs for “1 Across” and for “1 Down.” What is your next task in your pursuit of a solution to the puzzle? ************ Yes, that’s it! You need to investigate the multiples of a gross in an effort to determine the possible solutions for the clue for “4 Across. ” (Incidentally, a gross is a mathematical term for a dozen dozens, or 144.) The first several multiples of 144 are: 144, 288, 432, 576, 720 and 864.

***** Another Opportunity to Exercise your Analytical Reasoning Skills *****  Which of these multiples is the correct choice for the clue for “4 Across”? ************ You got it!  Since the only possibilities for a perfect cube are 125 and 343, and since the multiple of 144 in “4 Across” begins with the ending digit of the cubed number, the multiple of a gross must begin with either 5 or 3. An examination of the list of multiples of 144 reveals that the only possible solution is 576. At this point, you may safely enter 121, 125, and 576 as solutions. Assuming that you have already entered 64 as the solution for “6 Across”, there is only one entry that remains unsolved: “3 Across.” The clue for “2 Down” will provide a valuable piece of information in this regard. Since the leap year is in the twentieth century, its first two digits must be 1 and 9. This means that the two-digit prime number in “3 Across” must begin with 9 and once again our choices are narrowed considerably. There are only five odd numbers between 90 and 100 and only one of these is a prime: 97. 15 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

Congratulations! The puzzle has been solved! ******************** While solving this cross-number puzzle may have seemed like a lengthy and somewhat tedious process, in actuality it takes far more time to explain the analytical reasoning processes than it does to actually solve the puzzle. I hope that you enjoyed this exercise in the use of your analytical reasoning skills and problem-solving strategies. You will now have further opportunities to practice these skills and strategies in the Exercises and Projects for Fun and Profit below. Have fun! ********************

Exercises and Projects for Fun and Profit Friendly warning and assurance: The exercises and projects in this section can tend to be quite time-consuming due to their nature as puzzles; however, please don’t be alarmed—the rest of the textbook exercises do not all require this much open-ended effort!  Note: Please see Helps and Hints for Exercises on pp.423–424 in Appendix A.

Analytical Reasoning Applications 1. As I was going to St. Ives: This classic puzzle originally appeared in the Mother Goose collection of nursery rhymes and gained popularity in the ensuing years. A google.com search on this phrase will yield interesting results. There have been many variations of this puzzle down through the centuries; the one that follows has a fairy-tale flavor. As I was going to the store, I met a man with seven doors. Every door had seven knobs, Every knob had seven locks, Every lock had seven keys; Doors, knobs, locks, keys, How many were going to the store? Please identify the problem-solving strategies that you use to solve this classic and record your solution. (The strategies are enumerated on p.4 in Section 1.1 of the text.) There is more than one possible solution—please try to find all logical possibilities.

2. Nine Dots: Thinking Outside the Box: Our next math classic is a problem that you may have encountered in elementary school or elsewhere. It involves a square containing nine dots as shown below; the objective is to connect all nine dots with four straight, connected line segments 16 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

without retracing any of the line segments. In other words, you need to connect the nine dots with only four straight line segments without removing your pencil from the paper and without retracing any part of a line segment.

3. The Case of the Missing Dollar: Another classic math problem is the case of the missing dollar. A variation of the story is as follows: Three friends went out to dinner one evening for a social visit. The bill for the dinner was exactly \$30 and the three friends decided to split the bill evenly between them. Consequently, each of them paid the waiter \$10. When the waiter brought the bill to the cashier, the cashier realized that an error had been made and that the bill should have only been \$25. She gave the waiter five \$1 bills and asked him to please return the money to the guests. As the waiter returned to the table, he realized that there was no fair way to divide the money between the three guests, so he instead gave each of the guests \$1 and pocketed the remaining \$2 for himself. This means that each guest actually paid only \$9 for the meal for a total of \$27; the waiter pocketed \$2 for a combined total of \$29. So what happened to the missing \$1? Please identify the problem-solving strategies that you use to solve this classic and record your solution. (The strategies are enumerated on p.4 in Section 1.1 of the text.)

4. Exact Measures: Pouring liquids into containers of various sizes and attempting to get an exact amount of liquid into a container has long been a part of recreational mathematics. The following problem dates back several centuries at least and, as might be expected, requires some persistence and analytical reasoning in order to obtain a solution. This newer version uses a water supply in the form of a faucet with a shut-off handle. You are provided with a water supply in the form of a faucet with a shut-off handle and two empty jars, one of which holds exactly three cups of liquid while the other holds exactly five cups of liquid. Neither of the jars has any measurement markings so there is no way to partially fill them, with accuracy, directly from the faucet. Your objective is to obtain EXACTLY four cups of water in the five-cup jar by completely filling the smaller jar from the faucet when necessary and pouring water from the smaller jar into the larger until you have achieved your 17 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

objective. You may completely empty the larger jar of water by pouring it on the ground when necessary. Note: You are not allowed to partially fill a jar from the faucet. You may not use any jars or containers other than the two that are specified. Please identify the problem-solving strategies that you use to solve this classic and record your solution. (The strategies are enumerated on p.4 in Section 1.1 of the text.)

5. Exact Measures: Please follow the instructions for the preceding exercise (#4 above) and use two empty jars, one of which holds exactly four cups of liquid while the other holds exactly seven cups of liquid. Your objective is to obtain EXACTLY five cups of water in the sevencup jar.

6. Prize Money: Two friends have some ten-dollar bills in their pockets and each has less than fifty dollars. Each of them has entered a contest in which the prize is \$100. The first friend declares: If I win the contest, I’ll be three times as rich as you. The second replies: Yes, but if I win the contest, I’ll be seven times as rich as you. How much money does each of them have in their pockets? Please identify the problem-solving strategies that you use to solve this classic and record your solution. (The strategies are enumerated on p.4 in Section 1.1 of the text.)

7. Prize Money: Two friends have some twenty-dollar bills in their pockets and each has less than five hundred dollars. Each of them has entered a contest in which the prize is \$200. The first friend declares: If I win the contest, I’ll be just as rich as you. The second replies: Yes, but if I win the contest, I’ll be five times as rich as you. How much money does each of them have in their pockets? Please identify the problem-solving strategies that you use to solve this classic and record your solution. (The strategies are enumerated on p.4 in Section 1.1 of the text.)

********************

Creative Projects: Guards and Prisoners Cross-number Puzzle 18 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

GUARDS AND PRISONERS A similar problem to the fox, goat and cabbage episode involves guards and prisoners. This problem dates back to the nineteenth century and has been recast in various settings. In this particular version, there are a given number of guards and a given number of prisoners who need to cross a river. Their boat holds only two people at a time, and the number of prisoners must never be allowed to outnumber the guards on either side of the river; otherwise, the prisoners will overpower the guards and, well, the story will come to an abrupt end.  The following rules apply: 1. For the purposes of this puzzle, prisoners may be left alone on one side of the river or the other and they won’t try to escape even though they technically outnumber the guards at that point (since there are no guards present). 2. Also, it is important to note that when a person or persons arrive in the boat, they need to be counted in terms of whether the prisoners will outnumber the guards. For example, if there were one guard and one prisoner on one side of the river and if the boat arrived carrying one prisoner, at that point the two prisoners would outnumber the guard, i.e., the prisoner on the riverbank plus the prisoner in the boat would outnumber the single guard. 3. Someone (either a guard or a prisoner) needs to row the boat across the river during each crossing. Swimming and towing the boat with ropes are not permitted; only one or two people can cross the river in the boat during each crossing. Note: Using colorful markers to represent the guards and prisoners and physically moving them across a boundary line (the river) is a helpful visual strategy. In addition, drawing a diagram and constructing a table are good problem-solving strategies for recording the river crossings. Assignments: For each assignment, please do the following (please make use of the Helps and Hints for Guards and Prisoners on pp.424–429 in Appendix A): 1. Determine how many river crossings (both across and back) it will take to safely transport all of the guards and prisoners across the river. 2. List each of the river crossings that needs to be made, who is in the boat during each crossing and which direction they are traveling (across or back). 3. Identify and record the problem-solving strategies that you use to solve this classic. assignment number of guards number of prisoners 3 guards 3 prisoners Assignment #1 4 guards 2 prisoners Assignment #2 3 guards 2 prisoners Assignment #3 4 guards 3 prisoners Assignment #4

19 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

CROSS-NUMBER PUZZLE You’ll need to use several problem-solving strategies during the process of finding a solution for the following puzzle. Be persistent, and above all else, don’t give up! 1

2 4

6

3 5

7 8

9

10

11

Assignments: For each assignment, use the clues provided to solve the puzzle: Note: Numbers with leading zeros are not permissible as answers for the clues. Note: Please make use of the Helps and Hints for Cross-number Puzzle on pp.429–431 in Appendix A. It may also be helpful to read the Cross-number Puzzle Terminology on pp.449–450 in Appendix B concerning primes, factors, squares, cubes, etc. Assignment #1: Across 1. The cube of a positive integer (e.g., 9 3 =9·9·9=729) 4. A multiple of 111 (e.g., 111, 222, 333, …) 6. A number with only two prime factors (and no repeated factors), e.g., 34=2·17 8. The fourth power of a positive integer (e.g., 7 4 =7·7·7·7=2401) 11. The square of a prime less than 20 Assignment #2: Across 1. The cube of a positive integer (e.g., 9 3 =9·9·9=729) 4. A multiple of 222 (e.g., 222, 444, 666, …) 6. A multiple of a dozen 8. The cube of a positive integer 11. The cube of a positive integer

Down 1. The square of a positive integer (e.g., 9 2 =9·9=81) 2. A multiple of 9 3. A prime less than the square of 6 5. The cube of a positive integer 7. The cube of a positive integer 9. A prime number 10. An integer that is both a square and a cube

Down 1. The square of a positive integer (e.g., 9 2 =9·9=81) 2. A multiple of 9 3. Twice the value of 10 Down 5. The square of a positive integer 7. The square of a prime less than 30 9. A number divisible by 6 10. A prime number greater than 32

20 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.2 Math Classics and Applied Reasoning

Assignment #3: Across 1. The square of a positive integer (e.g., 9 2 =9·9=81) 4. The square of a positive integer 6. The fourth power of a positive integer (e.g., 7 4 =7·7·7·7=2401) 8. The cube of a positive integer 11. The square of a positive integer

Assignment #4: Across 1. The fourth power of a positive integer (e.g., 7 4 =7·7·7·7=2401) 4. The square of a positive integer (e.g., 9 2 =9·9=81) 6. A prime number between 50 and 60 8. The fourth power of a positive integer 11. The number of years in a century

Down 1. The square of a prime number 2. A number that is both a square and a cube 3. A prime number less than 30 5. A multiple of 17 7. The fourth power of a positive integer 9. One-fourth the value of 2 Down 10. A prime number between 75 and 100

Down 1. The square of a positive integer 2. A multiple of 31 3. The square of a positive integer 5. The cube of a positive integer 7. The number of degrees in a circle 9. A number that is evenly divisible by the sum of its digits 10. A prime number greater than 17

********************

Answers to Questions in this Section p.10 Crossing the River: The Story of the Fox, the Goat and the Cabbage Strategy #3: Eliminate the impossible; then only the possible remains.

***** Another Opportunity to Exercise your Analytical Reasoning Skills ***** It actually makes no difference whether the farmer takes the fox or the cabbage next. If he takes the fox across next and then brings back the goat, he can then leave the goat on the original side of the river while he takes the cabbage across the river to join the fox. If he takes the cabbage across next and then brings back the goat, he can then leave the goat on the original side of the river while he takes the fox across the river to join the cabbage. In either case, the fox and the cabbage are safe together and nothing gets inadvertently eaten… Lastly, of course, the farmer can return to get the goat and bring the goat across the river, thus safely conveying all of his possessions across the river.

21 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.3 Resources for the Present and the Future

1.3 Resources for the Present and the Future Introduction One of the goals of this course is to facilitate and promote students’ technological capabilities. In this section, we will identify some Web-based resources for mathematics and point you toward some resource collections that will prove useful in both this course and others and in your future academic careers. The number of high-quality resources has grown tremendously since the advent of the publicly accessible Internet in the mid-1990s. Due to the capabilities of online delivery of content, some of these resources offer much more than simple text-based content; animations, simulations and interactive Java applets are examples of Web-based resources that can go far beyond the traditional textbook in terms of facilitating conceptual understanding of mathematics. You will discover that there is a wealth of mathematical information on the Internet that is quite valuable with regard to your learning and mastering mathematical ideas and concepts. We will now embark on a very brief tour of the Internet and its resources for mathematics.

Intro to the Internet Even though the Internet search engines found at www.google.com and elsewhere employ some very effective and sophisticated search routines, and even though you will find relevant information for a topic of interest, the webpages returned to you on an Internet search can also contain information that is either irrelevant to your topic or of questionable quality. In response to the need for an effective means by which to find high-quality, relevant information on a given academic topic, especially at the college level, an educational organization was formed in 1997 with the purpose of creating a digital library of educational learning objects and materials that were peer-reviewed and rated for quality of content, potential effectiveness as a learning tool and ease of use. This organization became known as MERLOT, the Multimedia Educational Resource for Learning and Online Teaching. Since that time, MERLOT (www.merlot.org) has grown and developed to become one of the nation’s leading educational resources for online content. Later in this section, you may explore MERLOT and its resources and join the organization if you wish. You will find excellent resources there for a number of disciplines including Mathematics, Physics, Chemistry, Biology, History, Music, World Languages, Psychology, Teacher Education, Health Sciences, and more.

22 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.3 Resources for the Present and the Future

23 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.3 Resources for the Present and the Future  Multimedia Educational Resource for Learning and Online Teaching (MERLOT)— www.merlot.org MERLOT is an organization that is devoted to providing access to high-quality, peer-reviewed online resources for teaching and learning. It encompasses more than a dozen academic disciplines. While its primary focus is at the college level, there are a number of resources at the K–12 level as well. For more information, please visit their information page at: http://taste.merlot.org/

Conclusion Many valuable educational resources are being created and posted on the Internet for the benefit of students and readers everywhere. As you proceed through this course of study, I trust that you will benefit from the educational resources themselves and that your technology skills will improve as well.

********************

24 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.3 Resources for the Present and the Future

Exercises and Projects for Fun and Profit Internet Investigations 1. Choose one of the three mathematics organizations listed on p.23, visit its website and write a brief report (minimum of one-half page) on your reactions to the site’s materials and on what most interested you.

Creative Project: MERLOT and You

25 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 1.3 Resources for the Present and the Future

MERLOT and You

www.merlot.org

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Math Is Everywhere! Explore and Discover It! 1.3 Resources for the Present and the Future

Assignment: 1. 2. 3. 4. 5.

******************** This project is designed to introduce you to MERLOT and to give you a brief encounter with its features. I hope that you enjoy the project and that you get a sense of the value here in terms of resources and learning materials. You may also investigate a few of the MERLOT learning materials in your favorite discipline. Enjoy!  ********************

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Math Is Everywhere! Explore and Discover It! 1.3 Resources for the Present and the Future

Notes:

28 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 2.0 Introduction and Objectives

Chapter Two Patterns and Reasoning: Fun with Math! 2.0 Introduction and Objectives The thumbnail sketches below represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

2.1 Deductive and Inductive Reasoning: Top-down and Bottoms-up Logic Riddles of the Ages

Leaves no footprints, nary a track. Travels the world around and back.

Sherlock Holmes’ Analytical Reasoning

The Slippery Slope of Inductive Reasoning

2.2 Pattern Recognition: Searching for Connections Pattern Recognition

Successive Differences

??

3, 10, 17, 24, 31, … 7 7 7 7

The “Eureka!” Moment

29 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 2.0 Introduction and Objectives

2.3 The nth Term: The Power of Generalization Analyzing Relationships 7, 11, 15, 19, 23, …  4n + 3

The Missing Planet? Mercury

Venus

Earth

Mars

???

Jupiter

2.4 Logical Thinking and Reasoning Backwards: Puzzles for Fun and Profit The Case of the Missing Digits:

Word-sum Puzzles:

TE + E TH

PLAY + BALL GAME

2.5 Magic Squares: Sum Fun!

Magic Squares

16

3

2

13

5

10

11

8

9

6

7

12

4

15

14

1

30 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 2.0 Introduction and Objectives

Learning Objectives for Chapter 2 1. To enable students to develop their analytical reasoning skills through riddle solving, pattern recognition and problem solving. 2. To enable students to develop their deductive and inductive reasoning skills in recognizing patterns of various types. 3. To enable students to appreciate the power of generalization and to develop the ability to use inductive reasoning to form generalizations from sets of specific facts or data. 4. To allow students to exercise their mathematical creativity by proposing riddles and creating numerical and non-numerical sequences. 5. To promote the use of critical thinking skills through the use of puzzles and mathematical challenges. 6. To enable students to gain familiarity with Web-based mathematical resources and to gain experience and skill in assessing and evaluating these resources. 7. To engage students in active learning through the use of projects and activities. 8. To alleviate student math anxiety through expositional encouragement. 9. To enable students to develop confidence in their mathematical abilities.

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

2.1 Deductive and Inductive Reasoning: “A good puzzle should demand the exercise of our best wit and ingenuity, and although a knowledge of mathematics and of logic are often of great service in the solution of these things, yet it sometimes happens that a kind of natural cunning and sagacity is of considerable value.” —Henry Dudeney, British puzzlist

Introduction Welcome to the wonderful world of patterns and reasoning! This is a very enjoyable facet of mathematics where there are plenty of opportunities to exercise your creativity and resourcefulness and have fun in the process. I believe that there is something inherent in our human nature that delights in solving puzzles and searching for solutions; this section invites you to enter into this process and it presents some basic ideas that will carry on throughout the text.

Riddles: Ancient and Intriguing We’ll begin with a look at an interesting class of reasoning exercises known as “riddles.” Dictionaries generally define a “riddle” as: 1. a question or statement so framed as to exercise one’s ingenuity in answering it or discovering its meaning 2. a puzzling question, statement or problem, usually presented as a game or pastime 3. a puzzling or inexplicable thing or person ??? Hmm-m-m. ??? ????

Riddles are one of the oldest forms of puzzles and have a long and illustrious history. From Aristotle to Shakespeare to Tolkien, authors through the ages have used riddles to enliven their storybook characters and presentations. Riddles beckon readers to unravel the mystery contained within and use their creative thinking abilities to find a solution. Unlike the traditional math “problem” with its highly structured presentation and limited scope, riddles leave the door wide open for creativity, free-form association and lateral thinking, and invite everyone to enter in and join the fun without regard to their “math skills.” 

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

Corny, lame or inspired? You be the judge.

Here are some examples to get us started (if you should need them, hints are included at the end of this section on p.43 immediately preceding the Exercises and Projects for Fun and Profit): 1. Why did the college student name her new puppies Biology, Chemistry and Physics? 2. What is a whale’s favorite music player? 3. What did one half say to the other half as they were walking down the road together? 4. What is full of knowledge but can’t read or write? 5. What is the favorite food of an Italian lawyer who is on a diet? (You need to know some legal terminology for this one; it also helps if you’re Italian or like to eat various Italian foods….) Now, wasn’t that fun? I hope that you’ve answered, “Yes.” If you thought that any of the above were corny or lame, I take full responsibility since I created them; nonetheless, I hope that you sensed your creative thinking processes slip into gear and the accompanying surge of adrenaline as your mind began drawing on inner resources in its search for solutions. This is one of the wonderful aspects of mathematical reasoning—it energizes one’s mental systems and networks and refreshes the whole body.

Aha! Eureka! I’ve got it! While you were probably not consciously aware of it, you began to use your mathematical reasoning skills in the process of solving the riddles. You most likely:   

made some initial guesses (we formally call these “conjectures” in mathematics), tested your guesses and realized that they didn’t quite meet the conditions of the riddle (experimentation and verification), and hopefully arrived later at a solution (we call this the “Eureka!” moment if you’re in a particularly Greek mood or the “Aha!” moment otherwise; more often, you might just yell out, “I’ve got it!”).

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

******************** In any case, the discovery moment in the mathematical reasoning process is a wonderful experience—it distinctly and uniquely satisfies that inner drive to find a solution and builds an inner confidence that may then be applied to other riddles and conundrums. ********************

Bilbo’s Challenge Let’s look at some further examples of riddles and keep the creative machinery in motion. The next few are similar to those from J.R.R. Tolkien’s The Hobbit in the famous encounter between Bilbo Baggins and old Gollum (if you should need them, hints are included at the end of the section on p.43 immediately preceding the Exercises and Projects for Fun and Profit): 1. “Its bottom is hidden, Its head hard to find. Its mouth always open, As it speaks to mankind.” 2. “Leaves no footprints, Nary a track, Travels the world Around and back.” 3. “No light of its own, A guide for men; Mover of waters, Within a sailor’s ken.” 4. “A great transformer But having no form; Useful at times, At others, a storm.”

5. “I exist at war’s outset, In the midst of battle, And at the end of all conflict.”

Invigorating, isn’t it? You’ve just experienced one of the joys of mathematics—the process of searching for connections and reasons and solutions.

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

******************** This is how mathematics should always be experienced—it’s a true adventure with discoveries waiting around every bend. ******************** This course has been designed to lead you on such an adventure, one in which I trust that you will find great delight and one that will draw you into a new and exciting relationship with mathematics. Let’s proceed!

Searching for Patterns: A Sherlock Holmes Mystery Case Looking again to literature, we find that Sherlock Holmes, that most famous of detectives, often applied mathematical reasoning in solving his cases. There is a particularly appropriate example in “The Adventure of the Creeping Man” in which Holmes analyzes the mathematical pattern in the occurrences of a series of unusual events. As he delved into the details of the case, the following facts emerged: The unusual events occurred on the dates: July 2, July 11, July 20, July 29, August 7, August 16, August 26 and September 5.

***** An Opportunity to Build your Mathematical Confidence ***** brought to you by Sir Arthur Conan Doyle  Using your mathematical reasoning skills, can you determine the most likely date of the next occurrence? Please pause for a moment and give it a try—this is an opportunity to exercise your powers of observation and begin to build some confidence in your reasoning ability. ************

Congratulations! Sherlock would have been proud of you! This is exactly what Holmes induced and his keen observation led to the solution of this bizarre mystery. (Answers with which you may confirm your solutions are provided at the end of this section on p.47 immediately after the Exercises and Projects for Fun and Profit.)

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

While the above example was rather elementary, the dates in the Granada Television version of this mystery were modified to provide a bit more mathematical interest and complexity: The unusual events occurred on the dates: September 15, September 26, October 5, October 12, October 17 and October 20.

***** Another Opportunity to Build your Mathematical Confidence ***** brought to you by Granada Television  Using your reasoning skills once again, can you determine the most likely date of the next occurrence and the reason for Holmes’ sense of urgency on October 20? This pattern is slightly less obvious than the original one, but I’m sure that you are capable of meeting the challenge. Again, this pause to use your powers of observation is designed to build your confidence, so please give it a try! ************

Congratulations once again! Give yourself a pat on the back and be encouraged! (Answers with which you may confirm your solutions are provided at the end of this section on p.47 immediately after the Exercises and Projects for Fun and Profit.)

Recognizing Pattern Discrepancies In actuality, the dates in the film were shown by number as follows:

21, 15, 26, 5, 12, 17 and 20. The first date in the list, 21, was intentionally omitted from the previous example to eliminate the confusion that it introduces when attempting to determine a pattern in the dates. It appears that an inadvertent error was made at the television studio when producing this film and your next challenge is to determine what error may have been made in listing the first date.

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

***** One More Opportunity to Build your Mathematical Confidence ***** brought to you by your enthusiastic textbook author  What should the first date have been in order to properly conform to the pattern of the rest of the dates? Once more, please pause to work this out; you’ll continue to build that invaluable sense of confidence in your mathematical reasoning abilities.  ************ I trust that you successfully found the correct first date and are now experiencing that inner satisfaction that accompanies solving a puzzle. (Answers with which you may confirm your solutions are provided at the end of this section on p.47 immediately after the Exercises and Projects for Fun and Profit.)

Inductive Reasoning: Postal Deliveries In the above example, Sherlock Holmes made use of what is called inductive reasoning, a form of reasoning that uses a number of specific facts or instances to form a general rule that applies to all of the given facts. We often use this form of reasoning in our daily lives although we may not be consciously aware of the process. Consider, for example, the daily U.S. postal delivery. Most of us have probably observed the delivery patterns in our neighborhoods and have determined, on the basis of our casual and repeated observations, that the mail is delivered between certain hours of the day. In my neighborhood, for example, the mail consistently arrives between noon and 2:00 p.m. Based on my repeated observations, I feel justified in declaring that the mail always arrives between noon and 2:00 p.m. This is the inductive reasoning process in action: forming a general rule from a set of specific instances. You might think of it as “bottoms-up” logic

General principle

Fact 1

Fact 2

Fact 3

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

There is an inherent amount of risk involved in this type of reasoning; in other words, this reasoning may lead to faulty conclusions. Do you see what makes this reasoning risky? (Answer is provided at the end of this section on p.47 immediately after the Exercises and Projects for Fun and Profit.)

Deductive Reasoning: Paycheck Regularity (Thank goodness!) Despite the inherent risk involved, mathematical reasoning is a very valuable resource, one that may be used to purchase delightful results. On the other side of the reasoning coin is deductive reasoning, or “top-down” logic, the process of using an established general rule to determine specific instances. For example, if you know that your employer always issues paychecks every two weeks on Fridays, then you can look at the calendar and determine all of your specific pay dates for the coming month or year, thus making it easier to determine when you’ll have enough money to go out on that expensive date that you’ve been planning. Ah, a delightful thought! On a more mathematical note, if we begin with the principle that every prime number has only two positive whole number factors, the prime number itself and 1, then we can specifically construct a list of prime numbers. 2, 3, 5, 7, 11, 13, …

(1 is not technically considered a prime.)

Again, this is an example of deductive reasoning in which we work from a general rule to construct a set of specific instances of the rule.

General principle

Example 1

Example 2

Example 3

A Bit of Latin for Those Interested: The “ductive” Types of Reasoning Inductive and deductive reasoning both have as their etymological root the Latin verb ducto, which means “to lead.” Combining this root word with various Latin prefixes produces a number of words in our English language:

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

Latin prefix in de con ab

meaning in or into out from with away from

English word induct deduct conduct abduct

In the mathematical context,  

inductive reasoning observes a set of diverse instances and leads into a general conclusion. deductive reasoning observes a general rule or principle and leads out from there to establish a set of instances based on the rule.

Divisibility: An Illustration In order to solidify our grasp of these types of reasoning, we will now take a look at some basic principles involving whole numbers and their divisors. As we all know, every even number is evenly divisible by two (with no remainder); you might call this a general rule. On that basis, then, you could go on to determine whether any specific number were even. For example, the number 39 when divided by 2 produces a quotient of 19 and a remainder of 1; consequently, we conclude that 39 is not even since it does not comply with the above rule. What type of reasoning is this: inductive or deductive? (Answer: deductive, since we are using a general rule to establish a specific instance.) Now consider the following multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, … 

If you will observe carefully, you will notice a pattern that appears in the ending digits of these numbers. What is this pattern?

Based on your observation, what can you infer about divisibility by 5? What type of reasoning is this: inductive or deductive?

Very good! You have noticed (I’m sure) the fact that all listed multiples of five have ending digits of either 0 or 5. Based on this observation, you concluded (I hope) that any number that ends in either 0 or 5 is a multiple of 5 and consequently is evenly divisible by 5; this is inductive reasoning—using specific examples to form a general rule or principle.

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

The Slippery Slope of Inductive Reasoning Consider numbers of the form n 2  1 where n is an even positive integer: n n2 1 2 5 4 17 6 37

Since the first three values for n 2  1 turned out to be prime numbers, it would appear as if n 2  1 will always turn out to be a prime number. ******************** It is often the case that three instances do establish a pattern that holds true in all further instances; however, as noted earlier, there is an element of risk involved with inductive reasoning.

******************** Even though we observe a number of instances that seem to follow a general rule, there may be some as yet unobserved instances that, if observed, would nullify our induced rule. In this case, the next value for n produces a value for n 2  1 that is not prime (n=8 yields n 2  1 =65); an instance such as this is referred to as a counterexample. And the house that was built upon the sand fell with a great crash… “Oh, dear!” you may be thinking, “Whatever will we do if we can’t depend on our inductive reasoning process? Will all be lost and will chaos ensue?” Not to worry. While our inductive reasoning may at times lead us astray, mathematicians demand that every conjecture and every line of reasoning be formally proved before being accepted as fact. We won’t pursue mathematical proofs in this text, but rest assured that they guarantee that mathematics rests on a solid foundation.

Analytical Thinking in Action As you are already discovering, this course is not "math as usual" and you'll embark on many mini-adventures as we proceed through the semester. In addition, some of the Exercises and Projects for Fun and Profit are not usual, either, in the sense that there may be no specific "cookie-cutter" examples to follow. In particular, some of the exercises include questions that require you to explore and investigate using your own mathematical reasoning skills without the benefit of exact patterns to follow.

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

Making a concerted effort to answer the questions is most important whether or not your answers are correct. So just go ahead and make some good guesses or conjectures—it's the effort at analytical and creative thinking that is of value here—not so much the final results. ******************** By the way, it's perfectly acceptable to make incorrect (but creative) guesses in mathematics—in fact, that is what most mathematicians do when faced with new and unusual problems. Eventually, they refine their incorrect (actually creative) guesses until in some cases they arrive at a "right" answer.

******************** If you think more about it, you may realize that this is exactly what we do in our personal lifedecision-making processes all the time. We make our best guess as to what the "correct" decision is regarding a particular situation and then often discover that the "correct" decision wasn't the best decision; so we modify that and make a revised decision that hopefully serves the situation more effectively.

******************** This process of trial and error, or guess and revise, is a basic pattern of mathematical reasoning. ********************

Rarely does anyone guess the "perfect solution" on their first attempt and many attempts are sometimes required before a satisfactory result is obtained. (Related trivia question: How many times did Thomas Edison try to make a light bulb before he got one to work successfully? ) In any event, please don't feel intimidated by problems that don't seem to yield immediate solutions—this is part of the nature of mathematics (and life); but do be encouraged to keep trying and making guesses until something satisfactory emerges or, in some cases, until you simply give up for awhile—and that's okay, too. I welcome with open arms your guesses, educated or wild, and your conjectures, partial or complete, knowing that in the process of forming them you are using your creative thinking skills and are developing an awareness of the nature of mathematical reasoning.

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

******************** Occasionally you will encounter a "Eureka!" moment that will make you realize that your struggles and efforts have been worthwhile. These are moments to be treasured—in fact, you may remember these moments years later, long after much of what else you learned during that period is forgotten.

******************** These moments, which are only encountered through persistent effort and a determined resolve not to easily give up on trying to find solutions or understand a concept, are memorable because they are intensely personal. In them, you overcome a seemingly insurmountable difficulty or dilemma and build confidence in your analytical reasoning and problem-solving abilities and even, at times, in your own self-worth. In conclusion, then, don't be dismayed at problems or exercises that at first don't yield solutions or that don't have "cookie-cutter" examples to guide you. Instead, be diligent and persistent in making guesses and conjectures, be willing to give up for awhile and come back later (your subconscious mind is a great ally), and be assured that it's okay to be "wrong" as long as you have made a good effort. One of my goals is to help you develop your mathematical reasoning skills; by the end of your experiences with this textbook, I hope that you will look back on this course of study as a very positive experience in that regard!

Insights and Conclusion In this section, we examined top-down and bottoms-up logic: deductive reasoning leads out from a general principle to establish specific instances of the principle whereas inductive reasoning begins with a set of facts or examples and leads into the development of a general principle. Through an example involving prime numbers we came to the realization that inductive reasoning, the bottoms-up variety, is inherently risky and prone to error. This, in turn, led to a realization of the need for an assurance of certainty; fortunately, mathematical proofs provide this assurance. The world would be a very shaky place if we couldn’t rely on anything! Underlying these forms of reasoning is the realm of creative thinking—the very heart of mathematics. We began with some riddles and a mystery case to engage your creative skills and to illustrate how enjoyable and invigorating it is to exercise your reasoning powers. The realm of creative thinking underlies all other mathematical activity and is the seedbed for many rich and fruitful mathematical ideas.

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

******************** I truly hope that throughout the rest of this course you come to fully enjoy the creative process and to build confidence in your mathematical reasoning skills—it’s a wonderful and delightful experience!

********************

**********

Hints for Questions and Activities in This Section p.33 Corny, lame or inspired? You be the judge. (“Correct” answers provided on p.47 following the Exercises and Projects for Fun and Profit.) 1. Think about various types of dog breeds. 2. Steve Jobs would be pleased. 3. Lots of acceptable answers for this one. 4. Words, words and more words… 5. Required legal term: tort

p.34 Bilbo’s Challenge 1. Canoeing, anyone? 2. Driftwood knows all about this. 3. Think of the beach. 4. A primitive element. 5. Be literal.

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

Exercises and Projects for Fun and Profit Exercises in Creativity Hey-diddle-diddle, the cat told a riddle, The cow fell down in a swoon. The little dog stared while the hamster glared And the fish flew away on a balloon. Get in a relaxed and creative mood, and try to solve the following riddles; there is more than one “correct” answer for some of these (see Helps and Hints for Exercises on p.432 in Appendix A):

1. What has a head like a cat, whiskers like a cat, feet like a cat, tail like a cat, but is not a cat? 2. It has schools but no children, Roads but no cars; Lakes but no water And neighborhoods but no yards. 3. What is black and white and red all over? 4. Walk I cannot But run I can; The earth gives me shape Though formless I am. 5. What is that which dwells twice in heaven, once in hell, and once in the life that knows it well? In the next set of exercises, you will have an opportunity to apply your analytical thinking and reasoning skills.

******************** I wish you all the very best of success as you work your way through the exercises and projects! Those "Eureka!" moments are priceless...  ******************** Trial and error combined with some analytical reasoning will lead you to success in the exercises that follow. In fact, all of these can be solved completely by trial and error and the use of a calculator; however, you can save a good deal of time and effort by using your analytical reasoning skills to shorten the testing process.

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

6. Four by five: Using each of the digits 5, 3, 6 and 7 exactly once (i.e., no duplicates allowed) to construct each four-digit number, construct as many four-digit numbers as possible that are evenly divisible by 5 (i.e., divisible by 5 with no remainder). As examples, 3675 and 3765 are two such numbers; please find all other solutions. 7. Four by six: Using each of the digits 5, 3, 6 and 7 exactly once (i.e., no duplicates allowed) to construct each four-digit number, construct as many four-digit numbers as possible that are evenly divisible by 6 (i.e., divisible by 6 with no remainder). 8. Four by seven: Using each of the digits 5, 3, 6 and 7 exactly once (i.e., no duplicates allowed) to construct each four-digit number, construct as many four-digit numbers as possible that are evenly divisible by 7 (i.e., divisible by 7 with no remainder).

The next exercise requires you to use your powers of observation, inductive reasoning and analytical thinking; if that sounds too intimidating, pretend that you didn’t read it and go on to the exercise anyway… 9. By four? Can you determine how to tell if a number is evenly divisible by 4 without actually dividing the number by 4? It will help to observe some examples of numbers that are divisible by 4 in various categories such as numbers between 1 and 100, numbers between 100 and 200, and numbers between 200 and 300 to get a better feel for what is involved. Look for patterns, even very general ones—anything that repeats. This repetition is a clue that will lead you to the secret involved in the rule. Press on Mr. Holmes!

This next exercise may be done by brute force (lots of trial and error) or you may reason your way to a shorter route to a solution. (It’s always quite satisfying when you see a way to shorten your work…) 10. Four by four: Using each of the digits 5, 3, 6 and 7 exactly once (i.e., no duplicates allowed) to construct each four-digit number, construct as many four-digit numbers as possible that are evenly divisible by 4. (If you haven’t yet answered #9 above, you may do so now or you may read the Rule for Divisibility by 4 provided on p.48 at the end of this section.)

********************

Creative Project: A Riddling Good Time

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

A RIDDLING GOOD TIME As noted in this section, solving riddles requires mathematical thinking. This project is designed to provide an enjoyable way for you to enter into this process and exercise your creativity as well. Assignment:

Create your own original riddles (minimum of three riddles but by all means do more if you get on a roll…). You might choose a topical theme as a focus or you might take an open-ended approach. Riddles need to be clean and wholesome—nothing suggestive or off-color, please. Please include your answers as well. Note: Riddles have an official and authoritative right to be “corny” and/or “lame,” so please don’t feel intimidated by this exercise in creativity. Your riddles will be just as valid as any others and, since there are no rules for what riddles should or shouldn’t be, you needn’t worry about measuring up to some predetermined standard. It is the effort in creative thinking that lies at the heart of this exercise—not the quality of the results. Please be persistent—you may pleasantly surprise yourself with your own hidden creativity!

Since this is a very open-structured project with no particular place to start, you may feel at somewhat of a loss at first. If so, you might begin by selecting an everyday object with which you are familiar, listing all of the object’s features, and then creating a riddle that relates to a feature of interest. Example:

  

Everyday object: chair Features: A chair has two arms (sometimes), four legs, a seat, a back, some rungs, a cushion (sometimes), etc. Possible riddle: What has legs but cannot walk, and a back but cannot bend?

Simplistic? Yes, but that’s okay. Creative? Definitely! Original? Yes, I did this all by myself.  Personally satisfying? Yes! There are intangible benefits connected to using your creative thinking powers to invent something new and unique. These benefits are both wide-ranging and deep, and are more easily experienced than described. You’ll see what I mean as you engage yourself in this project…

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

Answers to Riddles and Questions in this Section p.33 Corny, lame or inspired? 1. Because they were all Labs (i.e., Labradors, or Labrador retrievers) 2. an iPod 3. “We’d better be careful or we might fall into a whole.” 4. a dictionary 5. tort-a-leany (“tort” is a legal term; tortellini is a type of Italian pasta) p.34 Bilbo’s challenge 1. a river 2. ocean currents (or the wind) 3. the moon 4. fire 5. the letter “t” Sherlock Holmes case

p.35 ***** An Opportunity to Build your Mathematical Confidence ***** brought to you by Sir Arthur Conan Doyle Most of the dates are nine days after their respective predecessors; the last two dates are ten days after their predecessors. Consequently, the next date would logically be either September 14 (nine days later) or September 15 (ten days later). You could argue in favor of either choice... p.36 ***** Another Opportunity to Build your Mathematical Confidence ***** brought to you by Granada Television The successive differences between dates follow a distinctive pattern: 11 days, 9 days, 7 days, 5 days, 3 days. The next logical difference in this sequence is 1 day and the next date, therefore, would be October 21 (hence Mr. Holmes’ sense of urgency). p.37 ***** One More Opportunity to Build your Mathematical Confidence ***** brought to you by your enthusiastic textbook author The first date should have been 2, not 21, so that the sequence would have read: 2, 15, 26, 5, 12, 17, 20. The successive differences between the dates then follow a nice, orderly progression: 13 days, 11 days, 9 days, 7 days, 5 days, 3 days Perhaps the “21” was a typographical error in the film editing and production process… p.38 Do you see what makes this reasoning risky? The fact that we observe a number of examples that seemingly follow a pattern does not, in and of itself, guarantee that the pattern will always continue; there may be an instance that we have not observed that nullifies the apparent pattern.

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Math Is Everywhere! Explore and Discover It! 2.1 Deductive and Inductive Reasoning

p.45 Rule for divisibility by 4 If the ending two digits of a number are evenly divisible by 4, then the given number is evenly divisible by 4. Examples:

 

5736 has 36 as its final two digits. Since 36 is evenly divisible by 4, the original number, 5736, is also evenly divisible by 4. 9364 has 64 as its final two digits. Since 64 is evenly divisible by 4, the original number, 9364, is also evenly divisible by 4.

This rule can be induced by examining the multiples of four from 1–300 and noticing that the ending two digits repeat, beginning at 104. Hence, this pattern will continue and we need only to focus on the ending two digits in a given number. Pretty simple, isn’t it? Multiples of 4:

4, 104, 204,

8, 108, 208,

12, 112, 212,

16, 116, 216,

20, 120, 220,

24, 124, 224,

… 100, … 200, … 300,

********** Another way of thinking about this is shown in the example below: Example 1: 524 = 500 + 24 Every multiple of 100 is evenly divisible by 4, so 500 is evenly divisible by 4. The question of essence, then, is only whether the number formed by the ending two digits is evenly divisible by 4. In this case, 24 is evenly divisible by 4, so the number 524 must also be evenly divisible by 4.

As additional verification, since both numbers on the right are evenly divisible by 4, we could factor out a 4: 524 = 4 · (125 + 6) Then 524 is clearly a multiple of 4 and therefore must be evenly divisible by 4. Example 2: 386 = 300 + 86 Every multiple of 100 is evenly divisible by 4, so 300 is evenly divisible by 4. The question of essence, then, is only whether the number formed by the ending two digits is evenly divisible by 4. In this case, 86 is NOT evenly divisible by 4, so the number 386 must NOT be divisible by 4.

********** This same line of reasoning applies to all numbers, no matter how large.

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

2.2 Pattern Recognition: Searching for Connections “The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.” —Godfrey Hardy

Introduction Now that you’ve begun to put your logical and creative thinking skills into action, this section directs you to use your powers of observation to examine and explore numerical patterns of various sorts. In the process of doing so, you will come to appreciate the usefulness of some basic mathematical tools including pattern recognition, asking questions and making conjectures.

Pattern Recognition Let’s begin with some very simple patterns that we all can easily recognize. For example, 2, 4, 6, 8, … is a most familiar pattern, one that you probably learned in kindergarten or in early childhood. We all know that the next number in this sequence of numbers is 10. A mathematical sequence, by the way, is simply an ordered list of elements; these elements are often referred to as “terms.”

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections ***** Opportunities to Exercise your Analytical Reasoning Skills *****  Some other examples are: 3, 7, 11, 15, 19, …

and

14, 23, 32, 41, 50, …

With a bit of observation, I’m sure that you can determine the next terms for each of the above sequences. Please take a few moments to do this and put your mental gears in motion. (Answers to these and succeeding examples are provided on p.64 at the end of this section.)

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections common ratio (or common multiplier) between terms. Technically, the first three are known as arithmetic sequences and the latter two are known as geometric sequences. These two types of sequences are quite prevalent in life; we will encounter them again later in the textbook. Note: When used as an adjective, arithmetic is pronounced ă’-rĭth-mĕ’-tĭc with the heavier accent on the syllable me. While these two types of sequences are rather common, they are by no means the only types; there is an infinite variety of sequences, limited only by one’s imagination. Here are a few examples; the next terms are rather obvious. 7, 7, 7, 7, 7, … 1, –1, 1, –1, 1, … 1, 1, 2, 2, 3, 3, … 1, a, 2, b, 3, c, …

constant sequence alternating sequence sequence of doublets alphanumeric sequence

Successive Differences: An Analytical Tool As you may have already discovered, examining the differences in value between successive terms in a sequence can be quite informative. In the arithmetic sequences above, the difference between successive terms is the key to constructing the next terms in the sequence. Since this difference is the same for each pair of terms, we call this difference a common difference and simply ADD this common difference to a given term in order to construct the next term. In the sequence, 5, 17, 29, 41, … we can easily determine the differences between successive terms by subtracting a given term from the next term in the sequence: 17–5=12,

29–17=12,

41–29=12, and so on.

Since each difference is 12, we say that this sequence has a common difference (i.e., the difference is common to each pair of successive terms). The next term is found by simply adding this common difference to the previous term: 41+12=53. sequence terms: 5, 17, 29, 41, 53, … common differences: 12 12 12 12 As another example, in the sequence, 9, 16, 23, 30, …

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections there is a common difference of 7 and the next term is simply 30+7=37. Arithmetic sequences, due to their very simple structure, are easy to work with and we’ll examine them in greater depth in the next section on mathematical generalization.

A Different Type of Sequence In the following sequence, things are a little less simple: 1,

4,

9,

16,

25, …

***** Opportunities to Exercise your Analytical Reasoning Skills *****  Can you determine what the next term should be? Did you discover a common difference between the terms? Please take a moment to exercise your reasoning skills and answer these two questions; again, your efforts here will build your confidence in your mathematical thinking processes. (Answers may be found on p.64 at the end of this section.) ************ This is a case where there was not a common difference between the terms, although the successive differences themselves exhibited a pattern. It is often instructive to apply our analysis tool again to see what it might reveal. In the case of the successive differences (found by subtracting a given term from the one that follows it in the sequence), sequence terms: successive differences:

1,

4, 9, 16, 25, … 3, 5, 7, 9, …

there is an obvious pattern: the differences are increasing at a steady rate. If we examine the successive differences (again found by subtraction) in the secondary sequence,

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

3, 5, 7, 9, … we find that these terms are related by a common difference of 2: 5–3=2,

7–5=2,

9–7=2, and so on…

Ultimately, then, the original sequence depends on the value 2 for its structure; 2 is somehow inherent to the nature of this sequence of positive integer squares, i.e., 12 , 2 2 , 3 2... Interesting, isn’t it? sequence terms: successive differences: common differences:

1,

4, 9, 16, 25… 3, 5, 7, 9… 2 2 2

Digging Deeper Let’s take a look at another sequence: 1,

8,

27,

64,

125, …

***** Opportunities to Exercise your Analytical Reasoning Skills *****  Can you determine what the next term should be? Did you discover a common difference between the terms? (Answers may be found on p.64 at the end of this section.) ************

This is another case where there is not a common difference between the terms of the sequence although the successive differences themselves may contain a pattern. So we’ll apply our analysis tool again and look at the differences between the successive differences. sequence terms: 1, 8, 27, 64, 125, … successive differences: 7, 19, 37, 61, … second successive differences: 12, 18, 24, … This deeper sequence, 12, 18, 24, …

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

reveals a definite pattern and we can apply our analysis tool of successive differences once more. In other words, we’ll examine the differences of the differences of the differences. Whew! That was a mouthful, but it was fun to say; please forgive my fit of madness.  second successive differences: 12, 18, 24… common differences: 6 6 In this case, we arrive at a constant common difference of 6, much as we arrived at a constant common difference of 2 in the previous example involving whole number squares. Just as with the squares, we can see that there is a number, 6, that is somehow inherent to the nature of the sequence of positive integer cubes, i.e., 13 , 2 3 , 33... Very interesting, indeed! While all students are familiar with integer squares and cubes, very few students have investigated the underlying structures of these numbers.

********************** In the preceding investigation, as brief as it was, we were able to discover some previously hidden relationships and come away with a sense that there are some deeper connections here that may be interesting to explore; in fact, this may be the beginning of a real mathematical adventure! ********************** You’ll have opportunities to do some of this exploring in the Exercises and Projects for Fun and Profit at the end of this section and to make some of your own personal discoveries. ********************

Reverse Gear Let’s return to the sequence of cubes and the layers of successive differences as noted above. By beginning with the constant sequence of 6’s and working in reverse, we can construct the next terms in each of the layers of successive differences and eventually arrive at the original sequence and construct its next term. Pretty clever, eh? To construct the next term in the layer above the sequence of constant 6’s, we add 24 plus the common difference of 6 to get 30: 1,

8,

27, 64, 125 7, 19, 37, 61 12, 18, 24, 30 6 6 6

30 = 24 + 6

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

To go up another level, we add 61 plus the successive difference of 30 to get 91: 1,

8,

27, 64, 125 19, 37, 61, 91 12, 18, 24, 30 6 6 6

7,

91 = 61 + 30

To reach the top level of the original sequence we add 125 plus the successive difference of 91 to get 216: 1,

8,

27, 64, 125, 216 19, 37, 61, 91 12, 18, 24, 30 6 6 6

216 = 125 + 91

7,

This reversing process may be repeated as often as desired to construct more and more terms of the original sequence (although in this case it is much easier to directly calculate the integer cubes). In conclusion, the underlying structure of the original sequence has been analyzed to reveal its basis and, in turn, this basis has been used to extend the original sequence. ********************** This is a very effective and powerful form of reasoning and analysis that may be applied in various and sundry situations both in mathematics and in many other fields. **********************

Conjectures: Creative Guesswork Now that we have discovered underlying common differences for the sequences of squares and of cubes, we have before us a wonderful opportunity to engage in a very rich and fruitful area of mathematics: formulating questions and making conjectures based on a given set of facts or observations. These questions can be of infinite variety, but often take the form of:    

It appears as though there is a pattern. I wonder whether the pattern continues? What if the facts or situation were slightly different? How would that affect the outcomes? Why is that so? Why not?

Mathematicians (and students when they are encouraged ) often engage in this form of activity and it has been the basis for many exciting discoveries and rewarding experiences. I’ve taken

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections the liberty to recoin an old saying: “You learn something new every day, and every so often you make a discovery.” ********************** While learning is delightful, the personal discoveries that you make add richness, depth and joy to your experience of mathematics and bring a sense of newness to your relationship with math. **********************

One of my aims in writing this textbook was to lead you to make personal discoveries of this sort and you will have numerous opportunities to do so, including the one before you now.

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

Insights and Conclusion In this section, you began to develop your skills in formulating questions and making conjectures based on a given set of facts or observations; these skills are quite valuable and are applicable in many fields. We began with some simple number sequences to exercise your skills in

pattern recognition, a basic and fundamental mathematical endeavor. After encountering several types of sequences, we presented an analytical tool called successive differences to enable you to see underlying patterns more easily. ******************** Simple mathematical tools such as this often give us access to otherwise vague and murky areas; we’ll develop a number of these tools throughout the remainder of this textbook. ******************** Using this analysis tool, we were able to probe the depths of the nature of positive integer squares and cubes and discover some quite interesting and unexpected features. This is the type of investigation that can lead to new and unusual discoveries, even breakthroughs of sorts. It’s quite exciting to explore a mathematical topic at this depth—it’s almost like being on a new frontier! ********************

Exercises and Projects for Fun and Profit Exercises in Analysis Note: Please see Helps and Hints for Exercises on pp.432–433 in Appendix A. What Comes Next? Use your powers of observation to determine the next terms in each of the following sequences: 1. 4, 19, 34, ___, … 2. 17, 28, 39, ___, … 3. 42, 70, 98, 126, ___, … 4. 33, 52, 71, 90, ___, …

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

5. 2, 6, 18, ___, … 6. 16, 36, 81, ___, … 7. 81, 27, 9, ___, … 8. 125, 50, 20, ___, … Missing Terms and Analytical Reasoning Use your analytical reasoning skills to find the missing term in each of the following sequences: 9.

12, 19, ___, 33, …

10.

2, ___, 20, 29, …

11.

___, 2, 8, 32, 128, …

12.

6, 15, ___, 93.75, …

Interwoven Sequences: Two for the Price of One Exhibiting a technique used on occasion by cryptographers, the following sequences consist of two sequences woven together. The effect of the interweaving is to make pattern recognition more difficult in that the weaving method is not always obvious. In the case of alphanumeric sequences such as #13–14 below, the weaving technique is clear; however, in some of the other exercises you’ll have to search more diligently to find the correct weave. Use your analytical reasoning skills to find the next two terms in each of the following sequences: 13.

d, 17, g, 23, j, 29, ___, ___, …

14.

a, 12, f, 21, k, 30, p, ___, ___, …

15.

2, 9, 4, 13, 6, 17, ___, ___, …

16.

4, 2, 8, 4, 12, 6, ___, ___, …

Interwoven Sequences: Three at Any Price Use your analytical reasoning skills to find the next three terms in each of the following sequences in which three sequences have been woven together. You will need to make an initial conjecture concerning how the three sequences are woven together; the simplest conjecture will be the most effective…

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

17.

5, 8, 10, 11, 14, 7, 17, 20, 4, ___, ___, ___, …

18.

20, 17, 9, 13, 14, 11, 6, 11, 13, ___, ___, ___, …

19.

7, a, z, 15, b, y, 23, c, x, ___, ___, ___, …

20. 2, 6, f, 8, 15, k, 14, 24, p, ___, ___, ___, … A Mixed Bag: The Challenge of the Unexpected In this miscellaneous collection of sequences, use your powers of observation and critical thinking skills to find the missing terms. In each of these cases, you’ll need to make initial conjectures as to how many sequences are involved and in what way they are woven together. 21. 5, 3, 9, 7, 13, 11, ___, ___, … 22.

17, 13, 20, 16, 23, 19, 26, ___, ___, …

23.

9, –6, 14, –1, 19, ___, ___, …

24.

5, 8, 13, 11, 14, 19, 17, ___, ___, ___, … **********

Further Investigations 25. A fourth dimension: Following the examples given in the text for sequences of squares and cubes, use your computation skills (viz., subtraction) and investigative reasoning to determine the layers of successive differences and, ultimately, the underlying common difference for the sequence of numbers representing positive integer fourth powers. The sequence of positive integers raised to their fourth powers is: 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000… Warning: You’ll need to be very careful with your arithmetic here; any subtraction errors may cause the whole process to turn out to be hopelessly indecipherable.  It would be most advantageous to use a calculator…. Hint: You’ll need to calculate several layers of successive differences before you finally discover the row in which the differences are constant. 26. A fifth dimension: Following the examples given in the text for sequences of squares and cubes, use your computation skills (viz., subtraction) and investigative reasoning to determine the layers of successive differences and, ultimately, the underlying common difference for the

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

sequence of numbers representing positive integer fifth powers. The sequence of positive integers raised to their fifth powers is: 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000… Warning: You’ll need to be very careful with your arithmetic here; any subtraction errors may cause the whole process to turn out to be hopelessly indecipherable.  It would be most advantageous to use a calculator…. Hint: You’ll need to calculate several layers of successive differences before you finally discover the row in which the differences are constant. 27. Dimensional patterns: Record the underlying common differences from the preceding investigations with squares, cubes, and fourth and fifth powers. Construct a table of these values as shown below in order to more clearly see possible relationships—a table (or chart) is often a great help in recognizing patterns because it organizes the data in a very simple and basic structure and eliminates unnecessary verbiage: type of number square cube fourth power fifth power sixth power

power 2 3 4 5 6

underlying common difference 2 6

Do you observe a pattern in the common differences of these various powers (second power=squares, third power=cubes, fourth power, fifth power)? If so, what do you see? Based on your observation, make a conjecture concerning the underlying common difference for sixth power numbers. ************************ Searching for patterns is a fundamental mathematical activity, one that finds application in many areas and sometimes leads to new and exciting discoveries. While the above investigation may not seem exciting to you (after all, who cares much about the sixth powers of positive integers), if you have successfully discovered the pattern involved and made a correct conjecture about the underlying common difference for sixth powers, then you may give yourself a hearty congratulations and a big pat on the back! You are already beginning to expand and improve your analytical reasoning skills! 

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

Creative Projects: The Curious Date: 7-14-98 Another Curious Date: 10-1-01

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

THE CURIOUS DATE: 7-14-98 I was attending a workshop one summer and for some reason needed to record the date, July 14, 1998. As I wrote the standard date notation, 7-14-98, on my paper, I was struck by the curious nature of the numerical values contained in the date. Do you see what I mean? Since I had never noticed this before, I wondered whether this was because this sort of occurrence was relatively rare. Or perhaps I had not been very observant in the past…. Assignments: This observation immediately led to a number of related questions that now comprise this project. If you need help in determining the curious connection above, please see Note below. Afterwards, for each assignment please consider each of the ten years in the indicated decade (ten-year period) and answer the seven questions below: assignment Assignment #1 Assignment #2 Assignment #3 Assignment #4

years 1990–1999, inclusive 1980–1989, inclusive 1970–1979, inclusive 1960–1969, inclusive

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Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

years 1920–1929, inclusive 1930–1939, inclusive 1940–1949, inclusive 1950–1959, inclusive

1. Determine how many curious dates of this nature occurred and record them in a list. 2. Report the total number of dates that you found. 3. Determine the percentage of dates in the given period that have this property (i.e., divide the number of curious dates by the number of days in the given ten-year period). 63 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 2.2 Pattern Recognition: Searching for Connections

Answers to Questions in this Section p.50 Pattern Recognition The next terms are included below: 3, 7, 11, 15, 19, 23 add 4 to each term 14, 23, 32, 41, 50, 59 add 9 to each term 2, 6, 18, 54, 162

multiply each term by 3

12, 6, 3, 1.5, 0.75

multiply each term by 0.5, i.e., multiply by

1 2

p.52 A Different Type of Sequence 1, 4, 9, 16, 25, 36 the differences are changing at a steady rate; in case you haven’t noticed, these numbers are positive integer squares: 12 , 2 2 , 3 2 , 4 2 , ... p.53 Digging Deeper 1, 8, 27, 64, 125, 216

in case you haven’t noticed, these numbers are positive integer cubes: 13 , 2 3 , 33 , 4 3 , ...

p.56 Conjectures: Creative Guesswork Yes, that’s it! The most likely topic is:

in question form  Does the sequence of numbers representing positive integer fourth powers have an underlying common difference? in conjecture form  I would surmise at this point that the sequence of numbers representing positive integer fourth powers has an underlying common difference. Other conjectures (these may also be presented in question form):  There will be underlying common differences for the sequences of positive integer fourth powers, fifth powers, sixth powers, et al.  The underlying common difference for the sequence of positive integer fourth powers will be greater than 6.  Since the underlying common difference for cubes was 6, which is three times the common difference for squares, the underlying common difference for fourth powers might be 8, i.e., four times the common difference for squares.

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

2.3 The nth Term: The Power of Generalization “Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.” —Paul Dirac

Introduction We’ll now return to arithmetic sequences and take a closer look at their underlying structure. Through our investigation, you will come to appreciate the power of a fundamental mathematical skill: the ability to generalize. In addition, we’ll discuss an analytical reasoning tool involving relationships known as functions and you’ll learn how this tool was applied to planetary distances to reveal a hidden feature. Lastly, you’ll be faced with some non-numeric puzzles to which you can apply your developing skills in analysis.

Creating a Relationship In a mathematical sequence presented at the beginning of the previous section, you discovered a common successive difference of 4: sequence terms: common successive differences:

3,

7, 4

11, 4

15, 4

19, … 4

By simply adding four to each term, we can generate as many terms in the sequence as desired. If someone were to ask us for the value of the 50th term, we could compute the first fifty terms by successively adding 4’s and listing all fifty terms. However, it would be nice if there were some easier way of determining the value of the 50th term without calculating all forty-nine preceding terms. After all, who doesn’t like to save time when performing calculations? The next step in our investigative process is to see whether we can determine exactly how the number 4 is involved in the relationships between the terms and whether we can perhaps construct a general expression or formula that will make it easy to compute an arbitrarily selected term. So let’s construct the first several terms of the sequence by using the first term of the sequence, 3, as a starting point and adding the appropriate number of 4’s. 1st term: 2nd term: 3rd term: 4th term: 5th term:

3 3+4=7 3 + 4 + 4 = 11 3 + 4 + 4 + 4 = 15 3 + 4 + 4 + 4 + 4 = 19

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

It’s clear that each successive term involves a larger number of 4’s. A more concise way to write this would be: 1st term: 2nd term: 3rd term: 4th term: 5th term:

3 3+4=7 3 + 2 · 4 = 11 3 + 3 · 4 = 15 3 + 4 · 4 = 19

To achieve uniformity in style (many times this is of great advantage in recognizing relationships), we can rewrite the first two terms to conform to the pattern of the others: 1st term: 2nd term: 3rd term: 4th term: 5th term:

3+0·4=3 3+1·4=7 3 + 2 · 4 = 11 3 + 3 · 4 = 15 3 + 4 · 4 = 19

Now we’ll use our creative and analytical thinking skills and attempt to generalize these five individual relationships into one overall and comprehensive relationship. To simplify matters a bit and to avoid confusion, we’ll rewrite the above relationships like this: 1st term: 2nd term: 3rd term: 4th term: 5th term:

3+0·4 3+1·4 3+2·4 3+3·4 3+4·4

***** An Opportunity to Exercise your Powers of Observation *****  What do you notice about the above list of the first five terms of the sequence? Observe carefully the similarities and the differences between the terms and make a short list of your observations.

**********

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

******************** Observing similarities and differences is a most important mathematical skill and one that may be applied in all areas of academics and life itself. Not only that, but it’s simple and easy to do. Have fun! ********************

As usual, answers are provided at the very end of this section on p.84 after the Exercises and Projects for Fun and Profit.

Generalizing: A Powerful and Elegant Form of Reasoning We will now aim to find a general, comprehensive relationship (equation or formula) that is capable of describing any (and every) term in the sequence. In other words, we would like to construct an equation for a generalized term for the sequence that is representative of the inner nature of the sequence. This may sound a bit mysterious and abstract but, by taking one step at a time, we’ll be able to accomplish our goal. We’ll call this generalized term the nth term since the variable n is conventionally used to represent whole numbers in mathematics. In order to help ourselves discover this relationship, we’ll construct a table for the terms of the sequence as follows:

ordinal number of the term 1st 2nd 3rd 4th 5th

sequential number of the term 1 2 3 4 5

nth

n

description of the term a(1) = 3 + 0 · 4 a(2) = 3 + 1 · 4 a(3) = 3 + 2 · 4 a(4) = 3 + 3 · 4 a(5) = 3 + 4 · 4 a(n) =

Notation: A Matter of Convenience The individual terms in a sequence are often denoted by using the letter a as a variable and numerical subscripts to indicate which term is which. A typical sequence would look like this: a1 , a 2 , a3 , a 4 , a5 , a6 , a 7 , ... 67 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

The nth term of a sequence is often designated as a n . As indicated in the above table, the terms and their subscripts are sometimes designated as: a(1), a(2), a(3), a(4), a(5),… , a(n)… where the nth term is listed as a(n). This is very reminiscent of the function notation f(x) used in algebra classes and is quite appropriate in this situation.

***** An Opportunity to Exercise your Analytical Reasoning *****  Based on your list of similarities and differences from the previous exercise opportunity, make an attempt to complete the last entry in the above table. In this table, n is a variable (i.e., it can vary in value) and can represent any whole number; the description for a(n) will follow the patterns found in the first five term descriptions. Some parts of this pattern are simple and direct; the only one that is a bit indirect involves the multipliers of 4. If you can successfully recognize the connection between the sequential number of the term and the multiplier of 4 used in the description of the term, you will then have the necessary insight to construct a description of the nth term. You may do this in sentence form or, preferably, in equation form as shown in the table. Making connections such as this is a very fundamental and important skill in mathematics—in fact, this process lies at the heart of all mathematical creativity. Please enter into the process of discovery and reap the associated rewards—they are invaluable to your growth and development!

************

If you haven’t made the connection yet, please read the hint at the end of the section on p.76 immediately preceding the Exercises and Projects for Fun and Profit. Recognizing this connection is at the heart of constructing the generalization.

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Once you construct the description of the nth term, you have then established a relationship between the sequential number of a term (e.g., n = 2) and the description of the term (e.g., a(2) = 3 + 1· 4). This, in turn, establishes a relationship between the description of the term and the actual value of the term (e.g., a(2) = 7). We can display this information in terms of a table:

sequential number of the term: n 1 2 3

description of the term: a(n) = 3 + (n–1) · 4

value of the term: a(n)

a(1) = 3 + (1–1) · 4 a(2) = 3 + (2–1) · 4 a(3) = 3 + (3–1) · 4

3 7 11

As you can see, this looks very similar to the format for describing a mathematical function in which a table of x and y values is given. In fact, this is exactly the point!

******************** The information displayed in the table above actually represents a relationship known as a mathematical function, a powerful generalization tool that is used extensively in algebra settings but which may be used, as we shall learn, in almost any environment. ******************** In terms of our familiar charts for functions and their values, we could write: n 1 2 3

a(n ) 3 7 11

n

3 + (n–1) · 4

where n represents the independent variable (just as x does in algebra settings) and a(n) represents the dependent variable (just as y=f(x) does in algebra settings).

Mission Accomplished So you have now exercised your mathematical reasoning skills once again, this time to establish a general and very powerful relationship involving the terms of the original sequence, 3, 7, 11, 15, 19,…. The relationship,

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a(n) = 3 + (n–1) · 4 that you discovered concerning the general nth term unlocks the secrets of the sequence and opens the door to performing simple calculations to determine the value of any term in the sequence, including the 50th term. In the above relationship for a(n), we may choose to eliminate the parentheses by multiplying through by 4 and collecting like terms; this produces a simpler expression as an end result (but hides the underlying pattern and connections established earlier): a(n) = 3 + (n–1) · 4 = 3 + 4n – 4 = 4n –1 This simplified version of the nth term is valuable in terms of its efficiency when used to compute arbitrarily selected terms of the sequence. The 50th term, you say? No problem! a(50) = 4 · 50 – 1 = 200 – 1 = 199. The 105th term? Simple! a(105) = 4 · 105 – 1 = 420 – 1 = 419. Or if, for some strange reason, someone needed to know the 1232nd term of the sequence, we could just as easily compute that, too: a(1232) = 4 · 1232 – 1 = 4928 – 1 = 4927.

******************** Effective, isn’t it? In fact, it borders on being awesome! The skill of generalizing takes a bit of practice to master but is one of the most powerful tools in mathematics and is used by mathematicians everywhere (and scientists, businesspeople and others) in their search to establish new relationships and connections. The skill of simplifying mathematical expressions is also a very useful tool and allows us to perform computations more easily and efficiently. Together, these two skills embody the power and elegance that lie at the heart of mathematics. Use and develop them to good effect!

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

********************

Another Look at the Mysterious a(n) The generalized description of the nth term in a sequence tends to baffle students at first due to its unfamiliarity. However, this is no more mysterious than its algebraic cousin, f(x). When we write f(x) = 2x + 3, we are describing a general relationship between f(x) and x, or, since y=f(x), a general relationship between the variables y and x: y = 2x + 3. In this relationship, the y-value depends on the choice of an x-value. Once you choose a value for x, then you can calculate the corresponding value for y. By choosing successive values for x, you can construct a table of values: x 1 2 3

y=2x+3 y=2(1)+3=5 y=2(2)+3=7 y=2(3)+3=9

f(x)=2x+3 f(1)=2(1)+3=5 f(2)=2(2)+3=7 f(3)=2(3)+3=9

y 5 7 9

If we restrict our choices to the whole numbers beginning with 1, 2, 3, and so on, we then have the same situation that we have with sequences such as a(n)=2n+3; the only differences are the names of the variables used. If the input values of a sequence are represented by the variable n and the output values are represented by the variable p, then the table of values would look like this: n 1 2 3

p=2n+3 p=2(1)+3=5 p=2(2)+3=7 p=2(3)+3=9

a(n)=2n+3 a(1)=2(1)+3=5 a(2)=2(2)+3=7 a(3)=2(3)+3=9

p 5 7 9

Hence, a(n), also known as the nth term, is simply a way to describe the general rule of the sequence just as f(x) describes the general rule of the algebraic relationship between x and y. The expression for a(n) will always consist of variables and coefficients and will represent all possible ordered pairs of related values. Some examples of nth terms and some of their associated sequence values are:

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

n a(n)= 2n + 3 a(n)= 6n – 1 a(n)= –5n + 10

1, 2, 3, … 5, 7, 9, … 5, 11, 17, … 5, 0, –5, …

Given a specific description of an nth term, a(n), it is a simple matter to calculate the corresponding sequence values by substituting the values 1, 2, 3, and so on, for n in the rule described by a(n); the table above shows some examples.

Working in Reverse: Constructing the nth term, a(n), from its Sequence Values Constructing an nth term, a(n), from a set of sequence values is a bit more complicated but not very much so, especially when you recognize the basic elements involved in the structure of a(n). With regard to the original sequence at the beginning of this section, 3, 7, 11, 15, 19…. , and its nth term, a(n) = 3 + (n–1) · 4, described in the section entitled Mission Accomplished, we observed that 3 was the first term of the sequence and that 4 was the common difference between successive terms. We may use these observations in constructing nth terms for other sequences. The following example illustrates the simplicity of constructing the nth term, a(n), from the first four sequence terms of an arithmetic sequence: Example:

5, 11, 17, 23, ….

In this sequence, the first term is 5 and the common difference between terms is 6; therefore, following the model for a(n) described above, we may write: a(n) = 5 + (n–1) · 6

or a(n) = 5 + 6n – 6 = 6n – 1

******************** Hopefully, this construction process is now clearer and less mysterious!  ********************

The Missing Planet? One of the most interesting examples of the generalizing process and the search for patterns involves the planets in our solar system. Some centuries ago, as astronomers and astronomy students were studying planetary data, someone noticed an unusual aberration in the pattern of the distances of the planets from the sun. (They first had to observe the pattern, of course.) The facts were roughly as follows: 72 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

planet

Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

actual distance from the sun (in 10 6 km) 57.9 108.2 149.6 227.9 778.3 1427 2871 4497 5913

Due to the slow initial change in the distances followed by increasingly larger changes, a conjecture was made that this sequence of distances might be exponential in nature. (We will learn more about exponential growth in a later chapter.) Upon further examination, it was discovered that, indeed, the distances followed an exponential pattern with an equation that was approximately given by: distance from the sun = 32.53  1.7097217 n , where n represented the value of a planet’s numerical order from the sun. Beginning with 32.53 and multiplying successively by the growth factor 1.7097217 yields the following set of values for the exponential model:

planet

Mercury Venus Earth Mars

actual distance from the sun (in 10 6 km) 57.9 108.2 149.6 227.9

Jupiter Saturn Uranus Neptune Pluto

778.3 1427 2871 4497 5913

exponential model distance from the sun (in 10 6 km) 55.6 95.1 162.6 277.9 475.2 812.5 1389.2 2375.1 4060.8 6942.8

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

While the distances in the exponential model do not exactly match the actual distances, there is a high enough correlation (we’ll discuss this concept further in a later chapter) that we deem the model quite acceptable. In essence, then, the planetary orbits follow a simple mathematical pattern.

Pattern Recognition: Another Eureka! As can be seen in the table above, there is an exponential model distance in the sequence that does not correspond to a planet. After realizing this anomaly, astronomers ultimately conjectured that the asteroid belt that exists between the orbits of Mars and Jupiter and contains many thousands of asteroids may have at one time been a planet. Quite astounding, isn’t it? Incidentally, this discovery later became known as the Titius-Bode Law; an Internet search on this topic will produce numerous resources for further information if you are interested.

******************** This is just one example of how mathematical pattern recognition has facilitated and enabled discoveries in other fields. In fact, you might say that mathematics is a universal tool which crosses all boundaries; some call mathematics the “Queen of the Sciences,” an admirable title, indeed! ********************

Time to Get Creative Again We’ll finish this section with a look at a non-numeric pattern to stimulate your creative abilities. Mathematical reasoning can be applied to all sorts of situations including the following. Have fun!

***** An Opportunity to Exercise your Analytical Reasoning ***** 

See whether you can determine the next term in each of the following sequences:

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

These puzzling sequences require the use of your non-numeric reasoning skills and creative analysis. After the previous exercises that you’ve done in this section, I’m confident that you will be able to solve these, too! (There is a hint provided at the end of this section on p.76 immediately preceding the Exercises and Projects for Fun and Profit, if needed; answers are supplied on p.84 after the Exercises and Projects for Fun and Profit.) ************

Insights and Conclusion At the beginning of this section, we examined the structure of an arithmetic sequence, observing both its constant and variable features. Based on these observations, you were able to establish connections that led to a description of the nth term, a powerful generalization that described an individual term of the sequence in relation to the sequential number of the term. This was a major accomplishment! ******************** Generalization is a most powerful and effective tool and is used extensively in all areas of mathematics; it has application in all subjects, particularly in the sciences.

******************** The story of the missing planet and its mathematical underpinnings was used as an example of how the skills of observation and generalization can at times lead to some earth-shaking results. Lastly, we introduced some non-numeric patterns to reinforce the idea that mathematical reasoning is not restricted to numbers—it can be applied just about anywhere. We’ll see many more applications of this idea throughout the text and perhaps you, too, will then conclude that mathematics is the “Queen of the Sciences.”

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

Hints for Questions in this Section p.68 Generalizing: A Powerful and Elegant Form of Reasoning In seeking the generalization, we need to carefully observe the patterns involved. If you’ll notice, in the first five terms there are some features that remain constant and some features that change. The constant elements in each term are the initial number 3 and the final number 4; the elements that change are the term sequence numbers (1, 2, 3, 4, and 5) and the multipliers of 4 (0, 1, 2, 3, and 4). Your job as an observer is to establish a connection or relationship between these two changing features:

term number 1 2 3 4 5

multiplier of 4 (i.e., the multiplier that is applied to the constant value, 4) 0 1 2 3 4

Eventually, we would like to construct the generalized term, the nth term, and we would like to know what value to use as its multiplier. term number n

multiplier of 4 ?

Do you see a pattern in the above columns? If so, apply it to the nth term and you will have succeeded in generalizing the relationship!

p.74 Time to Get Creative Again First sequence: Round and round it goes; where it stops, does anyone know? 

Second sequence: Note the progressive orientations of the facial features…

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

Exercises and Projects for Fun and Profit Exercises in Analysis Note: Please see the Example Exercises provided at the end of this section on pp.83–84 and Helps and Hints for Exercises on p.433 in Appendix A. . Pattern Recognition in Arithmetic Sequences Use your powers of observation and analysis to find the next term in each of the following sequences:

1. 4, 8, 12, 16, ___, … 2. 10, 13, 16, 19, ___, … 3. 5, 11, 17, 23, ___, … 4. 15, 26, 37, 48, ___, … 5. 29, 38, 47, 56, ___, … 6. 32, 49, 66, 83, ___, …

Thinking in Reverse: Slightly More Complicated (But Not Much)

Using your analytical reasoning skills and thinking in reverse, find the missing first term in each of the following sequences: 7. ___, 15, 22, 29, 36, … 8. ___, 14, 29, 44, 59, … 9. ___, 26, 31, 36, 41, … 10. ___, 19, 36, 53, 70, … 11. ___, 26, 35, 44, 53, … 12. ___, 37, 50, 63, 76, …

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

Thinking In Between: Easy Version Use your reasoning skills to find the missing middle term in each of the following sequences.

13.

12, 23, ___, 45, 56, …

14.

3, 24, ___, 66, 87…

15.

8, 24, ___, 56, 72, …

16.

22, 29, ___, 43, 50, …

Thinking In Between: Less Is More (Complicated, That Is) Version Use your reasoning skills to find the missing middle term in each of the following arithmetic sequences. This is more complicated due to the fact that there is less information given; however, since the sequences are arithmetic in structure, you should be able to use your analytical reasoning skills to find solutions. You’ll need to use a bit of algebra here to find the common difference, e.g., in #17, write 4+d+d=52 and solve for d to find the common difference between terms. (See Example Exercises provided at the end of this section on pp.83–84.)

17.

4, ___, 52, …

18.

11, ___, 47, …

19.

9, ___, 49, …

20.

16, ___, 44, …

Double Trouble? Now you may extend your growing skills to find two consecutive missing terms in the arithmetic sequences below. Is this more difficult than the preceding set of exercises? You be the judge… (See Example Exercises provided at the end of this section on pp.83–84.)

21.

6, 22, ___, ___, 70, …

22.

23, 27, ___, ___, 39, …

23.

2, 25, ___, ___, 94, …

24.

11, 23, ___, ___, 59, …

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

Double Trouble? Less Is More Version These exercises require some extended reasoning. Are you up to the challenge? You’ll need to use a bit of algebra here to find the common difference, e.g., in #25, write 24+d+d+d=81 and solve for d to find the common difference between terms. (See Example Exercises provided at the end of this section on pp.83–84.)

25.

24, ___, ___, 81, …

26.

29, ___, ___, 53, …

27.

14, ___, ___, 95, …

28.

19, ___, ___, 67, …

Triple Indemnity If you can find two consecutive missing terms, then it seems logical that you should be able to find three… Give it a try! (See Example Exercises provided at the end of this section, p.83-84, and Helps and Hints for Exercises on p.433 in Appendix A.)

29.

9, ___, ___, ___, 69, …

30.

3, ___, ___, ___, 95, …

31.

6, ___, ___, ___, 82, …

32.

11, ___, ___, ___, 59, …

33.

5, ___, ___, ___, 89, …

34.

8, ___, ___, ___, 76, …

The nth Term Use your analytical reasoning skills to describe the nth term in each of the following sequences, i.e., write a mathematical formula for a(n). It may help to label each term with its sequence number (1, 2, 3, … ) and focus on the common difference as we did earlier in this section in order to more easily see the relationships involved. Constructing a table for the first few term numbers and the descriptions of the terms may also be quite helpful; this is the method that we used in the text to arrive at the description for the nth term and, ultimately, a formula for the nth term. It will be helpful to review the sections entitled Another Look at the Mysterious a(n) and Working in Reverse. (Also, see Example Exercises provided at the end of this section, pp.83– 84, and Helps and Hints for Exercises on p.433 in Appendix A.)

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35. 6, 11, 16, 21, … 36. 1, 9, 17, 25, … 37. 27, 23, 19, 15, … 38. 4, 16, 28, 40, … 39. 18, 35, 52, 69, … 40. 23, 41, 59, 77, …

Limits?? 41. In an arithmetic sequence, do you think that you could find any given number of consecutive missing terms? Is there an upper limit to the number that you could find? Please explain briefly.

How Many Are Enough? 42. In an arithmetic sequence, how many successive terms do you need to be given in order to establish a pattern? You might look back over the previous exercises for examples… ********** Non-numeric fun 43. Determine the next term in the following sequence. This is quite a challenge in visual pattern recognition and the solution may be rather obscure. Do persist, though, in your efforts to find the next term; this is one of those opportunities for a “Eureka!” moment.

********************

Creative Projects: Sequence Creations Generalization—A Powerful Tool ********************

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

SEQUENCE CREATIONS In this project, you are given opportunities to create some sequences of your own making and design. The first of these involves an arithmetic sequence—a very special type of sequence in which each term differs from both its predecessor and its successor by the same number of units; the second involves non-numeric sequences and open-ended creativity. Assignment:

A. Create with numbers Create three arithmetic sequences of at least four terms each and describe the nth term for each, i.e., state a mathematical formula for a(n). (Hint: You may want to do this in reverse order.) Example: arithmetic sequence: 7, 20, 33, 46, … nth term: a(n) = 7 + (n – 1) · 13 or a(n) = 13n – 6

******************** B. Create without numbers Create three non-numeric sequences consisting of a minimum of three terms each. Describe in sentence form the rule that governs each sequence and include a description of the next term (as yet unlisted) in the sequence. You may use letters, shapes, non-numeric keyboard characters, etc. The objective is to be creative!  Example: non-numeric sequence: A ! B @ C # … sequence rule: The successive letters of the alphabet, beginning with A, are alternated with the successive keyboard characters above the numbers 1, 2, 3, … on the keyboard. the next term: the letter D

There are additional examples of non-numeric sequences on pp.74–75 in the textbook.

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

GENERALIZATION—A POWERFUL TOOL Now that you have worked with arithmetic sequences and have learned how to construct the nth term for a given sequence, you are ready to take the next step in the process of mathematical generalization: develop a formula for the nth term that applies to any arithmetic sequence, not just a specific instance. ******************** This process of generalization is one of the most powerful tools in mathematics! It generally takes a bit of effort and concentration to achieve, but the results are well worth the effort. ********************

Let’s take a look at some arithmetic sequences and their nth term descriptions to help in getting started on this adventure into the realm of generalization. Again, a table is a useful assistant: arithmetic sequence 4, 7, 10, 13, … 9, 21, 33, 45, … 6, 13, 20, 27, …

common difference 3 12 7

a1 , a 2 , a3 , a 4 , ...

d

description of nth term a(n) = 4 + (n – 1) · 3 a(n) = 9 + (n – 1) · 12 a(n) = 6 + (n – 1) · 7 a(n) = ?

As you can see from the table, there are some characteristic features of these nth terms that are the same for all of the given sequences and their descriptions. Assignment:

Your goal in this project is to use your powers of observation and analytical reasoning skills to construct a description of a(n), the nth term of the generalized arithmetic sequence, a1 , a 2 , a3 , a 4 , ... In doing so, please complete the final entry in the table above. (See Helps and Hints for Generalization—A Powerful Tool on p.434 in Appendix A.) In order to provide a handle on terminology, let’s agree to label the common difference between terms in an arithmetic sequence with the letter d (d for difference). Then d will be able to vary from sequence to sequence, i.e., d will serve as a variable but in each sequence will represent the common difference between terms in that given sequence. And now you’re all set to use your observation and reasoning skills to construct a formula for the nth term of an arithmetic sequence, no matter what values the individual terms may have. With the work that we have done already, you’re right on the brink of taking a major step in the process of mathematical generalization. Have fun!

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

EXAMPLE EXERCISES Thinking In Between #18. 11, ___, 47, ...

In this exercise, the missing term is between two given terms. To proceed sequentially from 11 to the missing term, we would need to add some number (the common difference between terms) to 11; to proceed from the missing term to 47, we would need to add the same number to the missing term. Let's label this number with the variable "d". To get from 11 to 47, then, we know that 11 + d represents the missing term and 11 + d + d = 47. Solving this algebraic statement will allow us to determine the missing term: 11 + 2d = 47, so 2d = 36 and d = 18 (this is the common difference between terms). We can then determine the missing term by adding d to the first term. 11 + 18 = 29; 29 is the missing term. Double Trouble? #22. 23, 27, ___, ___, 39, ...

In this exercise, there are two missing terms; however, you can determine the common difference between terms by finding the difference between the first two terms: 27 – 23 = 4. Then add this common difference to 27 to get the next term, and so on. 27 + 4 = 31; 31 + 4 = 35; 35 + 4 = 39 Double Trouble? Less Is More Version #26. 29, ___, ___, 53, ...

In this exercise, there are two missing terms but no direct way to determine the common difference between terms. Using the technique described above in #18, we can write the equation: 29 + d + d + d = 53. Solving this equation for d will determine the common difference which may then be added to each term to construct the succeeding term. 29 + d = first missing term; then, first missing term + d = second missing term. Triple Indemnity #30. Use the same technique described above in #26. The nth Term #36. 1, 9, 17, 25, ...

Finding the nth term often presents the most difficulty to students as it the most abstract of the exercises. Please review the sections entitled Another Look at the Mysterious a(n) and Working in Reverse. We'll now construct a table for Exercise #36; the bottom row of the table shows the desired value for a(n). In #36, the first term is "1" and the common difference between terms is "8".

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Math Is Everywhere! Explore and Discover It! 2.3 The nth Term: The Power of Generalization

EXAMPLE EXERCISES—continued number of the term

description of the term

value of the term

1

a(1) = 1 + (1 – 1) · 8

1

2

a(2) = 1 + (2 – 1) · 8

9

3

a(3) = 1 + (3 – 1) · 8

17

4

a(4) = 1 + (4 – 1) · 8

25

n

a(n) = 1 + (n – 1) · 8

8n – 7

Limits?? #41. In #25–27, you found two consecutive missing terms; in #29–33, you found three consecutive missing terms. This question is asking whether you could find any given number of consecutive missing terms or whether there might be some limit to the number that you could find.

********************

Answers to Questions in this Section p.67 Creating a Relationship The similarities are:  each term begins with the number 3  each term includes a multiple of 4

The differences are:  The multipliers of 4 are different in each term

p.74 Time to Get Creative Again The next term in the first sequence is either an arrow pointing up or an arrow pointing down; the underlying pattern is that of a 90-degree rotation as evidenced by the rectangular bar. Whether you rotate to the left or to the right is open to personal preference since there is not sufficient data to either require or prohibit either option.

The next term in the second sequence is formed by inverting the mouth (it becomes a frown…). The pattern here is: 1) invert the eyes; 2) invert the nose; 3) invert the mouth.

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

2.4 Logical Thinking and Reasoning Backwards: Puzzles for Fun and Profit “I wonder if I’ve been changed in the night? Let me think: was I the same when I got up this morning? I almost think I can remember feeling a little different. But if I’m not the same, the next question is “Who in the world am I?” Ah, that’s the great puzzle!” —Lewis Carroll (Alice’s Adventures in Wonderland)

Introduction In this section, we will engage ourselves with some puzzles designed to further strengthen your burgeoning skills in making conjectures and using analytical reasoning. These puzzles are simple in structure and have a gradually increasing level of difficulty, beginning with the very easiest. I have found that when students are instructed in how to solve simple versions of a more complicated puzzle, they encounter much more success when attempting the complicated version.

Alphanumeric Puzzles: Alphabet Addition The following puzzles are variously known as cryptarithms, alphanumeric puzzles, and wordsum puzzles. The basic idea is that each letter in one of these puzzles represents a single digit between 0 and 9, inclusive; repeated instances of a particular letter in a puzzle represent the same digit in all instances in that puzzle. The strategies involved in solving one of these puzzles include trial and error, analytical reasoning, and the processes of determining the possible and eliminating the impossible. Some of the puzzles have a unique solution (only one answer) while others have multiple solutions and still others may have no solution at all. Often there are several possibilities to consider and you’ll need to assign temporary numerical values to some of the letters and follow through with them until you find that they either will or won’t work as a solution. ******************** It’s a bit of a mental challenge, but I’m sure that you will meet with success! Have fun!

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

The Very Simplest Case: It Really Doesn’t Get any Easier than This… In the simplest of alphanumeric puzzles, there is only one solution: A +A A

***** An Opportunity to Exercise Your Analytical Reasoning Skills *****  Use logical reasoning to determine the only possible value for A in this case. Remember, each instance of A represents the same digit. **********

Congratulations—I knew that you could do it! As you have determined, the only possible value for A is 0; any other value results in a sum that is greater than the original value for A and would need to be represented by a letter different from A.

A Small Step Up in the Level of Difficulty (Still Very Easy…) Let’s try the next simplest puzzle: A +A B

In this case, the two addends are the same but the sum is different.

***** Another Opportunity to Exercise Your Analytical Reasoning Skills *****  What are the possible values for A and B in this case? (Multiple solutions are possible.)

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

Remember, each instance of A represents the same digit; the sum, B, represents a single digit that is different from A (A and B represent different values). Try reasoning backwards to determine the values; recognizing the single-digit nature of B will lead you to some conclusions about A. While I am confident that you can solve this puzzle on your own, answers are provided at the end of this section on p.101 after the Exercises and Projects for Fun and Profit so that you can check your solution if desired. **********

You should have discovered four solutions for the above puzzle and, if so, give yourself a pat on the back—you’re on the way to becoming an accomplished puzzle solver. The next simplest puzzle has a total of nine solutions: A +B B

In this case, the two addends represent different digits while the sum represents the same digit as one of the addends. Rather than have you solve this one at the moment, I’ll include it as the first exercise in the Exercises and Projects for Fun and Profit at the end of this section; I know that you’ll be up to the challenge.

The Most Complex of the Single-digit Addition Puzzles The last of the single-digit addition puzzles involves the following arrangement: A +B C

Of the puzzles that we have considered thus far, this one is the most complex in that there are many possible solutions (32 solutions to be exact). One strategy to find these solutions is to simply make a list of all possible value combinations for A and B in a methodical manner; i.e., begin by letting A=0, trying all possible values for B, checking the result C, and eliminating those that don’t meet the criteria of A, B and C being different values and the sum C being a single digit. Then let A=1, try all possible values for B, check the result C, and eliminate those that don’t meet the criteria. The beginning steps in applying this strategy are shown in the three groups below:

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

A +B C

0 0 0 0 … 0 1 2 3 … 0 1 2 3 …

A +B C

1 1 1 1 … 0 1 2 3 … 1 2 3 4 …

A +B C

2 2 2 2 … 0 1 2 3 … 2 3 4 5 …

Some of the above combinations of values do not meet the criteria and may be eliminated as solutions. In particular, all possibilities in the first of the three groups are disqualified since they have two duplicate digits (in 0+0=0 there are actually three duplicate digits). In the second group, the first two possibilities, 1+0=1 and 1+1=2, are likewise disqualified since A, B and C must represent different digits; if we were to extend this set of possibilities to its full extent, the combination 1+9=10 would also be eliminated due to the double-digit nature of the sum. Again, I’ll let you put your analytical thinking skills to work on this puzzle as one of the Exercises and Projects for Fun and Profit at the end of this section; I’m confident that you can find all 32 solutions with relative ease. ******************** Having a methodical approach in mathematics is a key that often unlocks solutions and reveals patterns that can then shorten the investigative process.

********************

One Final Issue Concerning Single-digit Addition Puzzles There is only one category of puzzles involving single-digit addends that we have not yet considered: those with two-digit sums. The level of complexity of this category is no greater than that of the previous puzzles, but these puzzles do involve a “carry-over” from the ones’ column to the tens’ column. Here is an example: A +A BC

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

***** An Opportunity to Exercise Your Analytical Reasoning Skills *****  Using the experience that you have gained in this section, determine the solutions for this puzzle. Remember, A, B and C represent different digits. While this is a simple exercise, please take the few minutes required to think it through; the process of doing so will continue to build confidence in your ability to use your mathematical reasoning skills and allow you to have fun while doing so! As usual, answers are provided at the end of this section on p.102 after the Exercises and Projects for Fun and Profit so that you can check your solution if desired. **********

You’ll have an opportunity to investigate other puzzles in this category in the Exercises and Projects for Fun and Profit at the end of this section.

The Next Level of Complexity: Two-digit Addends Now we’ll take another step to increase the complexity of these puzzles by allowing two-digit addends and two-digit or three-digit sums. As an example, let’s take a look at this puzzle together: TE +E TH

Putting our analytical thinking powers to work, we can observe that the T in the tens’ column is the same both in the addend and in the sum; by reasoning backwards we can conclude that the addition of E + E must not have resulted in a “carry-over” from the ones’ column to the tens’ column. In other words, the sum of E + E must be less than or equal to 9. This realization leads to several possibilities for the value of E.

***** An Opportunity to Exercise Your Analytical Reasoning Skills *****  All right, then, what do you think—what are the possible values for E? I trust that you’ll have little trouble figuring them out. **********

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

Very good! As you ascertained, the only possible values are: 0, 1, 2, 3 and 4. These choices for E lead to the following five cases: TE +E TH

T0 +0 T0

T1 +1 T2

T2 +2 T4

T3 +3 T6

T4 +4 T8

In the first of these cases, where E=0, the occurrence of 0 in the position represented by the letter H violates the criterion requiring each different letter to represent a different digit and hence may be eliminated as a source of possible solutions. For the remaining four cases, we may now continue with our analytical reasoning processes.

***** Another Opportunity to Exercise Your Analytical Reasoning Skills *****  In each of the four remaining cases, how many possible values can T represent? (This requires a bit more thought.) Please take time now to analyze the case E=1 by writing down the specific instances in an orderly fashion—recording each possibility helps immensely in observing general patterns and structure. Answers are provided on p.102 after the Exercises and Projects for Fun and Profit at the end of this section.

********** Since T can assume any of seven values in each of the four cases, it is sensible to conclude that there are a total of 7 · 4 = 28 possible solutions to this puzzle. Now we can give ourselves a pat on the back and a word of congratulations! While we haven’t written all 28 solutions out in detail, we have logically reasoned our way to a definitive assessment of the nature of the solutions and, if necessary, could easily write out all of the solutions. ******************** Many times in mathematics our goal is to determine the nature of a relationship or structure and our concern doesn’t lie in specifying all possible outworkings of the relationship; such is the case here. ********************

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

An Extension: Two-digit Addends Involving “Carry-overs” As we continue to gradually increase the level of puzzle difficulty, we’ll consider a puzzle involving two-digit addends and the process of “carrying over.” In this instance, B B addend + C C addend A D E sum Once again, we’ll put on our logical thinking caps and make some observations about the structure of the puzzle:  

Since there is a digit, A, in the third column (the hundreds’ column) of the sum, there must have been a “carry-over” from the sum in the second column (the tens’ column). Since the addend entries in the first and second columns are identical and yet have different entries for their sums, there must have been a “carry-over” from the first column to the second column; otherwise, the sums in both of the columns would have been the same.

*****An Opportunity to Exercise Your Analytical Reasoning Skills*****  You may now apply some further reasoning to determine the value of A; in doing so, you will continue to develop and sharpen your analytical thinking skills. This is a most valuable process—one that you will encounter many times throughout this text. Please don’t skip over it! And the value for A is: ________ 

To confirm your results, you may consult the answer provided on p.102 after the Exercises and Projects for Fun and Profit at the end of this section. **********

Now that you have determined the value for A, you may continue with an investigation of the possible values for B and C. Here again a methodical, trial-and-error approach will quickly yield the sought-after results. Since the sums of B and C involve “carry-overs,” let’s start by giving B a specific value and then testing all values of C that result in sums that are greater or equal to 10.

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

11 22 22 33 33 33 44 +99 +88 +99 +77 +88 +99 +66 110 110 121 110 121 132 110

44 44 44 … +77 +88 +99 … 121 132 143 …

In the above partial listing, each of the entries except the next-to-the-last one is invalid due to duplicate digits (i.e., two or more different letters represent the same digit). As you can see, there may not be too many solutions to this puzzle…

The Top of the Ladder of Complexity While it appears that we are still in the very beginnings of the learning process in terms of solving this type of puzzle, we are actually very near the top of the ladder of complexity. ******************** As is the case in many areas of mathematics, these puzzles require only a limited set of basic skills for their solutions. ******************** We have already learned how to:    

recognize addends that represent 0 use duplicate column entries to advantage determine the existence of “carry-overs” or the absence thereof methodically investigate all possible combinations of addend values and their sums

The only other puzzles that offer more complexity are those involving multi-digit addends or those involving sums of more than two addends. In both of these cases, however, the same basic skills are required: recognizing addends of 0, using duplicate column entries to advantage, determining “carry-overs,” and constructing methodical investigations. Some of these more extensive puzzles might offer fewer clues in terms of duplicate entries or “carry-overs” and might require more trial and error (and hence more time and persistence) to solve. Nonetheless, you have now learned the skills necessary to meet with success. I trust that the Exercises and Projects for Fun and Profit at the end of this section will provide you with opportunities to further build self-confidence in your puzzle-solving abilities. By the time you have completed those exercises you may be well on your way to new heights in applying your analytical thinking skills!

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

The Ultimate Challenge Up until now, we have been analyzing simple alphanumeric puzzles in order to establish the basic skills and strategies required for finding solutions; now it’s time to investigate a more complex puzzle that truly classifies as a “word-sum” puzzle. This is where the fun begins because not only can you find solutions to these sometimes humorous puzzles, but you can also wax creative and construct and solve some of your very own word-sum puzzles using words or phrases of your choosing. Afterwards, you may offer them to friends, family or co-workers and impress them with your mathematical and analytical skills. ******************** It’s always most satisfying to create something that is original and unique—it brings out the artist in you… ******************** Now, let’s take on the ultimate challenge of using our logical thinking skills and reasoning backwards when necessary to solve the following word-sum puzzle: PLAY +BALL GAME In the following analysis, we’ll try to focus on the critical features of the puzzle as outlined in earlier sections in order to expedite a solution.

***** An Opportunity to Exercise Your Analytical Reasoning Skills *****  What feature of the puzzle stands out as an immediate clue? ********** You may have come up with more than one correct answer here since by this point your reasoning skills are in high gear; if so, you’ve done very well! Rather than list all possible answers, I’ll lead you through one possible sequence of logical reasoning: 1) One distinctive feature is that there are two columns involving L+A, each with different sums. Since the sums are different, there must be a “carry-over” involved from either or both of the first and second columns.

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

2) In the third column from the right (the hundreds’ column), L+A=A would normally indicate the fact that L=0; however, since the first column shows that Y+L=E, and E is a different digit than Y, then L cannot have the value of 0. 3) The only other possible way that L+A=A in the third column is connected to the idea of a “carry-over” from the second column. As discussed earlier in the answers provided for the section entitled An Extension: Two-digit Addends Involving “Carryovers”, the only possible “carry-over” value is “1”. 4) We have now reached a critical point in our analysis: How can 1+L+A=A in the third column? The answer is that L must assume the value of 9! Then, 1+L+A=1+9+A=10+A and the value for A would be entered as the sum in the third column while the extra ten would be carried over to the fourth column as a “1.”

This is the realization that leads directly to the puzzle solution. While this is a bit of a challenge, I hope that by now this type of reasoning has become familiar to you and that you feel comfortable in following along with this reasoning sequence.

5) Let’s focus now on the second and third columns using the value of 9 for L: 9A +A 9 AM

Choosing a value for A at random, let’s say A=3, we can examine the results: 1 93 +3 9 (1) 3 2

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

This fits perfectly with the requirement that the three values in the positions for A are the same and it looks like we may have found a possible solution for the puzzle. Hurrah!

6) In order to confirm this conjecture, we need to consider the entire puzzle and make sure that these values for A and L fit together with the other values in the puzzle to form a solution: PLAY + BALL GAME

P93Y + B399 G32E

*****A Final Opportunity to Exercise Your Analytical Reasoning Skills*****  Do you see any problem with this proposed solution so far? ********** 7) Unfortunately, the best laid plans of mice and men oft go astray. As you can see, the first column, Y+9=E, requires that Y does not equal 0 and, as a result, there must necessarily be a “carry-over of 1” from the first column to the second column.

8) In the second column, A+L=M, we’ll have the following carry-over plus addends: 1 +3 + 9 = M This is the same as: 10 + 3 = M (or, in general, 10 + A = M) and will result in the value of 3 for the sum digit, M, in the second column (and a carry-over of “1” to the third column). 9) “But wait!” you’re probably thinking, “Then M=3 will have the same value as A—a violation of the puzzle criteria.” And slowly the realization dawns that, no matter what value for A is chosen, the “carry-over” from the first column will always result in the following developments in the second-column addends: 1 + (A + L) 1 + (L + A) 1+9+A 10 + A

carry-over of 1 from the first column commutative property of addition substitution property (L=9) substitution property (1+9=10)

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

Moreover, 10+A=M will always result in M having the same digit value as A. Alas and alack, we conclude that there is no possible solution to this word-sum puzzle and we’ll have to go our way with a bit of an empty feeling about the result.  On the other hand, we have very successfully reasoned our way through a challenging puzzle and arrived at a valid conclusion!  This is quite an accomplishment and I must compliment you on your skill and persistence.

******************** The analytical reasoning process, which involves logical thinking in both forward and backward directions, is of fundamental importance in mathematics as well as in life in general. ******************** Exercising these skills in contrived situations such as puzzles provides opportunities to sharpen and develop your problem-solving abilities; these problem-solving abilities will be of benefit throughout your entire life. I hope that you’ve enjoyed our excursion in this section!

Insights and Conclusion Puzzle solving has long been a favorite pastime of mathematicians and others; in fact, puzzles themselves are categorized as Recreational Mathematics in the overall scheme of things. In this section, you were engaged with some alphanumeric and word-sum puzzles in order to further strengthen and develop your skills in making conjectures and using analytical reasoning—two fundamental mathematical processes. The puzzles were carefully chosen to elucidate basic strategies involved in finding solutions including trial and error, determining the possible and eliminating the impossible, and reasoning backwards. Developing a methodical approach to examining possible solutions was also demonstrated and espoused for its efficacy—orderliness can be of much assistance in the search for solutions. The puzzles were presented beginning with the very simplest level of difficulty and moving progressively up the ladder of complexity. While the simplest puzzles may have seemed overly simple, I have found that starting from the very beginning and gradually increasing the level of difficulty has achieved excellent results in terms of student understanding and conceptual development.

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

Rather than being overwhelmed by complicated puzzles, students who learn puzzle solving in this manner come to realize that there are only a few basic skills involved and find it easier to apply these basic skills to complicated-looking puzzles. On the whole, they meet with much more success and are consequently happier and more satisfied with the whole process.

I trust that in the exercises and projects below you also will meet with much success and will enjoy the challenge of solving puzzles. Press on! ********************

Exercises and Projects for Fun and Profit Exercises in Analysis Note: Please see the Example Exercises provided at the end of this section on pp.100– 101 and Helps and Hints for Exercises on pp.434–435 in Appendix A. Single-digit addends Use your analytical thinking skills to determine ALL of the solutions to each of the following alphanumeric puzzles and write out all solutions in a list or table. (Note: Some of the puzzles may have no solutions; if so, please indicate as such.) Enjoy the mental challenge and have fun!  1.

A +B B

2.

A +B C

3.

A +B CC

4.

A +A AB

5.

A +A BB

6.

7.

A +B CD

8.

A +B CA B +C DD

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

Double-digit addends—A Bit More Complicated Here you may apply the same basic skills to solve the following puzzles. Please find ALL solutions for each exercise and record them in a list or table. 9.

GA + A FY

10. +

TA S TY

Double-digit addends—A Bit More Complicated?? You be the judge. Whether or not these puzzles are any more difficult than the preceding set remains to be seen. Again, your basic skills will carry you through—then you may pass judgment on the level of complexity. Please find ALL solutions for each exercise and record them in a list or table. 11.

DA + A VEE

12.

YU + U BET

13.

BL + A STS

14.

DA + U BLE

15. This is the puzzle that we began to investigate earlier in the section entitled, An Extension: Two-digit Addends Involving “Carry-overs,” and will now revisit so that you can construct the full solution: BB +C C ADE 16. One more double-digit puzzle for good measure: TA + R RY Some Three-digit Addend Challenges While these puzzles begin to look more complicated, once again your basic skills are all that is required. 17.

ONE ONE +ONE TWO

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

18.

ABC +CBA DEFG

19. An Old Classic and a favorite of college students away from home (this one requires a good deal of persistence!): SEND +MORE MONEY ********************

Creative Project: Puzzle Constructions ********************

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

EXAMPLE EXERCISES Solving the Exercises in Analysis requires making a determined effort to find all possible solutions to each exercise. This requires a combination of analytical thinking and trial-anderror. To give you an idea of what this involves, some exercises and detailed solutions are provided below: 2. A+B=C In this example, the only restriction is that each digit be different, i.e., no duplicates allowed, since each of the indicated letters, A, B and C, is different. Starting with A=1, we can begin to list some solutions: 1+2=3, 1+3=4, 1+4=5, 1+5=6, 1+6=7, 1+7=8, 1+8=9 Note that we can't use 1+9=10 since C must be a single-digit value. Letting A=2, we can also list the solutions: 2+1=3, 2+3=5, 2+4=6, 2+5=7, 2+6=8, 2+7=9 Note that we can't use 2+2=4 since A and B must be different digits; note also that 2+1=3 is a different (and valid) solution than 1+2=3. Letting A=3, we can list the solutions: 3+1=4, 3+2=5, 3+4=7, 3+5=8, 3+6=9 And so on. As you have noticed, there are fewer and fewer solutions as we use larger values for A; before too long, you will have listed all possible solutions. It helps to be orderly and methodical in this process of determining solutions. Random trial-anderror is far less efficient and quite tedious...  6.

A +B CA

In this case, the A's represent the same digit. Logically, then, B must be 0. But if B=0, then there will be no carry-over from the first column to the second; consequently, there can be NO solution for this exercise. We're assuming, as usual, that numbers don't begin with 0, like “0 4”. 8.

B +C DD

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

In this example, D and D represent the same digit. So the sum D D must be 11 or 22 or 33, etc. Since B and C are both single-digit values, the only possible choice for D D is 11. All that is needed, then, is to find the combinations of values whose sum is 11: 2+9=11, 3+8=11, 4+7=11, 5+6=11, 6+5=11, 7+4=11, 8+3=11, 9+2=11 These are the only possible solutions for this exercise. 12.

YU +U BET

In this example, U and U represent the same digit. In addition, the Y is different than the E in the sum, so there must have been a carry-over from the first column to the second. This logically implies that U+U is greater than or equal to 10; consequently, there are only five choices for U: 5, 6, 7, 8 and 9. Since B indicates that there is a carry-over from the second column to the third, Y must equal 9 so that B=1 and E=0. The possibilities, therefore, are: 95+5=100 96+6=102, 97+7=104, 98+8=106, 99+9=108 Of these, 95+5=100 is disqualified due to the duplication of 0's; 99+9=108 is disqualified due to the fact that Y must be a different digit than U. As a result, there are only three solutions: 96+6=102, 97+7=104 and 98+8=106

Hopefully, the above examples are enough to help you get started on finding solutions for the odd-numbered exercises. Have fun!  *****************

Answers to Questions in this Section p.87 A Small Step Up in the Level of Difficulty (Still Very Easy…) Since the sum, B, is a single digit, it has a value less than or equal to 9. Since both addends have the same value, the largest value for A is 4. The value of 0 for A is invalid since B would then not have a different value (B would have the same value as A). Therefore, the only possible values for A are 1, 2, 3 and 4 and there are consequently four possible values for B (and hence four solutions to this puzzle). The values for B are: 2, 4, 6 and 8; the complete solution set is: A +A B

1 +1 2

2 +2 4

3 +3 6

4 +4 8

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Math Is Everywhere! Explore and Discover It! 2.4 Logical Thinking and Reasoning Backwards

p.89 One Final Issue Concerning Single-digit Addition Puzzles Since there is a “carry-over” from the first column to the second, and since A is repeated as an addend, there are only five choices for A: 5, 6, 7, 8 and 9. Each of these results in values for the digits in the sum that are different from the digit for A; hence, each of these five generates a solution to the puzzle. 5 +5 10

6 +6 12

7 +7 14

8 +8 16

9 +9 18

p.90 The Next Level of Complexity: Two-digit Addends We don’t allow the letter T to assume the value of 0 in order to maintain simplicity in our numerical naming structure. The numbers 5, 05, 005 and 0005 all have the same numerical value of 5 but as a convention we don’t include leading 0’s when naming numbers. Technically, T could be 0 but we will not allow that usage here. As a result, the letter T can potentially assume any of the values from 1 to 9, inclusive. However, T must have a value different from those of the letters E and H. In each case, then, T can assume 9 – 2 = 7 different values as follows: In case 1 where E=1 and H=2, T can assume the values 3,4,5,6,7,8 and 9. In case 2 where E=2 and H=4, T can assume the values 1,3,5,6,7,8 and 9. In case 3 where E=3 and H=6, T can assume the values 1,2,4,5,7,8 and 9. In case 4 where E=4 and H=8, T can assume the values 1,2,3,5,6,7 and 9. This results in 28 possible solutions. p.91 An Extension: Two-digit Addends Involving “Carry-overs” As I hope you have already determined, the letter A must represent a value of 1 since there are only two columns involved in “carry-overs” and the maximum amount of either of the “carryovers” is 1. To see this more clearly, imagine the worst-case scenario in which the addend entries, B and C, in the first two columns had the maximum values of 9 and 8, respectively. (They both can’t be 9 since B and C represent different values.) Then the sum in the first column (ones’ column) would be 17, of which ten (i.e., a “1”) would get carried over to the second column; the sum in the second column would then become 18, of which ten (i.e., a “1”) would likewise get carried over to the third column. This is the maximum value of the carry-over and hence the value of A must be 1.

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Math Is Everywhere! Explore and Discover It! 2.5 Magic Squares: Sum Fun!

2.5 Magic Squares: Sum Fun! “The shallow consider liberty a release from all law, from every constraint; the wise man sees in it, on the contrary, the potent Law of Laws…” —Walt Whitman

Introduction Now that you have begun to develop your analytical thinking skills, you will have opportunities in this section to apply them to new situations, culminating in an investigation of what are known as magic squares, a fascinating area of mathematics. In the process, you will broaden your skill base to include analysis of relationships that involve specific constraints. Constraints are limitations or restrictions placed on the values of the variables in a relationship and generally serve to focus solution efforts in a particular direction and reduce the number of possible solutions.

Relationships Involving Constraints: One Variable In the following examples, we'll use the operation of addition along with some variables to illustrate the analytical process. ******************** Once again, we'll begin with the simplest case and gradually increase the level of complexity until we reach our goal, magic squares. ******************** In this simplest case, a single variable is involved in a relationship along with some instructions: Solve for a: a +2 7 As can be readily seen, the only value for the variable a that "solves" this relationship is the value 5. In this case, there was no explicit external constraint, i.e., limitation or restriction, on the variable a; the instructions simply asked you to "solve for a." In some instances, instructions may contain restrictions, or constraints, on the value of the variable(s) in a relationship. For example, a modification of the above example might contain the following instructions:

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Solve for a, where a is a whole number between 6 and 10: a +2 7 In this instance, the constraint that the variable a must be a whole number between 6 and 10 changes the outcome dramatically. Alas, what seemed like such a simple and easy solution has been precluded from consideration and we are left with no solution at all.

The Second Simplest Case: Two Variables with Constraints We will now add a second variable to the mix and examine another simple relationship. In particular, let's consider the following: Solve for a and b: a + b 10 Without any externally imposed constraints, there are an infinite number of solutions for this relationship, e.g., a 1 2 –17 100

b 9 8 27 –90

Imposing constraints generally limits the number of solutions and provides a bit more of a challenge in finding them; if, for example, the constraint is that both a and b are whole numbers between 1 and 6, inclusive, then there are only a limited number of solutions.

***** An Opportunity to Exercise Your Analytical Reasoning Skills *****  Use your analytical reasoning skills to determine all possible solutions for the relationship based on the above-mentioned constraint, i.e., both a and b are whole numbers between 1 and 6, inclusive. I trust that you will have little difficulty in finding these solutions (there are only three of them)…

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As usual, answers are provided on p.118 following the Exercises and Projects for Fun and Profit at the end of this section. **********

Next Step: Three Variables with Constraints Continuing to develop the level of complexity, we'll next consider a three-variable addition: Solve for a, b and c: a +b c Without constraints on the values of the variables, this relationship allows for an endless number of solutions; to make things a bit more interesting, let's apply a very specific set of constraints to the variables and their values: the values for the variables must be positive consecutive integers (e.g., 5, 6 and 7) and may be assigned to the variables in any order (e.g., a=6, b=5, c=7) The constraints provide a bit more of a challenge in finding solutions for this relationship.

***** Another Opportunity to Exercise Your Analytical Reasoning Skills *****  Use your analytical reasoning skills to determine all possible solutions for the relationship based on the above-mentioned constraints, i.e., the values for the variables must be positive consecutive integers (e.g., a=6, b=5, c=7). Again, I trust that you will have little difficulty in finding the solution… (Answers may be found on p.118 at the end of this section.) **********

Going Up! Next Level, Please! Now that you've completed the first few levels of difficulty with flying colors, we'll continue to broaden our endeavors. Since there is no way to form an addition relationship involving two addends and a sum when using four variables along with the constraint that the values for the variables must be positive consecutive integers less than 10 (Can you see why? If not, see p.118), we'll go on to use five variables. Consider the following relationship and its constraints:

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Solve for a, b, c, d, and e, where the values for the variables must be positive consecutive integers less than 10 (e.g., 2, 3, 4, 5 and 6) and may be assigned to the variables in any order (e.g., a=4, b=2, c=3, d=5, e=6):

ab +c de

***** A Grand Opportunity to Exercise Your Analytical Reasoning Skills *****  Use your analytical reasoning skills to determine all possible solutions for the relationship based on the above-mentioned constraints, i.e., the values for the variables must be positive consecutive integers less than 10 (e.g., a=4, b=2, c=3, d=5, e=6). This requires some analytical thinking combined with basic trial and error. I'm confident that once again you'll find the solutions (there are only six of them). In case you'd like to compare your reasoning process and answers with the author’s solutions, these solutions are provided on p.118 at the end of this section immediately following the Exercises and Projects for Fun and Profit. ********** You'll encounter the next levels of complexity in the six-, seven- and eight-variable cases (which are easier than you might think, for the most part) in the Exercises and Projects for Fun and Profit at the end of this section. Meanwhile, we're ready to move on and apply the skills that you're developing to a new structure: the mathematical magic square.

Introduction to Magic Squares Everyone is familiar with checkerboards (or chessboards) and their basic structure: a square that has been subdivided into a number of smaller squares of equal area. In the case of chessboards, there are eight small squares along each side and a total of sixty-four small squares on the board. In this section, we are going to investigate squares like these in which numerical values are placed in each of the smaller squares according to a few simple rules; when completed, these squares are known as magic squares. These squares come in different sizes and are often denoted by indicating their length and their width (even though these values are the same). For examples, a square having three smaller squares along its edges is called a 3x3 (three-by-three) square, and a square such as the chessboard is called an 8x8 (eight-by-eight) square. To construct a general description of this type of square, we can use a variable to hold the place of the length and width and indicate a

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square’s size as nxn (pronounced n-by-n), where n is the variable of choice. (Mathematicians often use the variable n when referring to integer quantities.) Note: Other naming conventions include referring to the above squares as a “square of order 3,” a “square of order 8,” and a “square of order n,” respectively.

Basic Rules for Magic Squares The rules for constructing an nxn magic square are: 1) A total of n·n, or n 2 , different numerical values (entries) must be used; no duplicate values (entries) are allowed. 2) Each of the small squares must contain exactly one value (entry). 3) The sum of the entries in a particular row, column or main diagonal must be the same for each row, column and main diagonal (left and right). Note: By convention we use only positive whole numbers as entries; however, you could use negative whole numbers and/or zero if you wished. The third rule, of course, is the crux of the matter. While it might seem like a simple task to fill in the squares with suitable values, it is a bit more difficult than it seems and once again requires some analytical reasoning and some trial and error. In magic squares, we are dealing with both geometry in regard to the smaller squares and their positions and with arithmetic in regard to the sums of the rows, columns and diagonals. ******************** Once again, we’ll begin with the simplest of cases and gradually add to the level of complexity so that you can see and appreciate the developmental logic and build confidence in your analytical reasoning. ********************

The Very Simplest Magic Square The simplest square is, of course, the smallest. The smallest possible magic square will have one unit along each of its edges, i.e., it is simply one square with no smaller squares contained within. To illustrate, we’ll consider a 1x1 magic square with one entry consisting of the value 1.

1

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This case is SO simple that it’s almost confusing; for example, how can you have the sum of the entries in a row when there is only one entry? The same question applies to the column, and the diagonals appear to be nonexistent; we are now faced with a bit of a dilemma and some choices to make. We can declare that this square is “too small” and doesn’t meet the minimum requirements for classification as a magic square and move on to the next larger square, or we can consider this as a sort of degenerate case in which we allow the single entry to represent the “sum” of the row, column and diagonals. (We will also have to allow that the diagonal sums can contain only one entry and that both diagonals are represented by the given square.) If we choose the latter alternative, then the sum of the entry in the only row is 1, the sum of the entry in the only column is 1, the sum of the left diagonal is 1, and the sum of the right diagonal is 1. According to the rules for magic squares, this degenerate case qualifies as a 1x1 magic square. On the other hand, if we choose the former alternative, this case may be ignored and we can proceed to the next level of difficulty. I’ll let you be the judge and jury on this matter; my personal preference is to include this case in the interest of being thorough.

The Second Simplest Square: 2x2 We’ll now take a look at the next level of difficulty: a 2x2 square.

1

2

3

4

I have chosen the first four whole numbers as entry values and, as you can see, these entries do not comprise a magic square. The sums of the diagonals are the same, 1+4=5 and 2+3=5, but the sums of the rows and columns do not have this same value of 5: 1+2=3, 3+4=7, 1+3=4, and 2+4=6.

***** An Opportunity to Exercise Your Analytical Reasoning Skills *****  Use your analytical reasoning skills to determine whether there are any numerical values that could be used to construct a 2x2 magic square. We’ll limit ourselves to values that are whole numbers greater than zero and, as a reminder, the four values must be unique (no duplicates). Give it a try!

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You may also use trial and error if you wish, but afterwards apply your logical thinking skills to convince yourself of your conjectured conclusion. (I’ve provided the answer on p.119 after the Exercises and Projects for Fun and Profit at the end of this section in case you want to check your reasoning processes.) **********

Meanwhile, we’ll move on to the next level of difficulty—the 3x3 square. This is where the real fun begins!

The Third Simplest Square: 3x3

a

b

c

d

e

f

g

h

i

To conform to a long-standing tradition, we’ll constrain ourselves in this case to a very simple set of values for the variable entries: the numbers from 1 to 9, inclusive. The goal is to arrange these nine values in the square, one in each smaller square with no duplicate values allowed, so that the sums of each of the rows, each of the columns and each of the main diagonals are equal. In the lettered square above, this means that the following eight sums must have equal value: (rows) (columns) (diagonals)

a+b+c a+d+g a+e+i

d+e+f b+e+h c+e+g

g+h+i c+f +i

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***** A Challenging Opportunity to Exercise Your Analytical Reasoning Skills *****  You are now facing your greatest challenge yet in this chapter: the construction of a 3x3 magic square as outlined above. Above all, this endeavor requires persistence and determination; however, using some analytical reasoning will be of great advantage in your attempts to find a solution. I’ll let you begin the process of trial and error before giving you any advice; there is great value in attempting this on your own without any outside help or intervention! While you may experience a measure of frustration in the process, this is a normal facet of active, exploratory mathematics—solutions do not always readily appear. Even failed attempts, though, have value in that you begin to see what doesn’t work and can then modify your strategies and approaches until you eventually arrive at some insights that lead to a solution.

******************** As noted above, this exercise is a definite challenge and when you succeed you will be quite satisfied, perhaps even exhilarated, with your accomplishment. So do enter into the endeavor with a measure of courage and a resolve to persist until you meet with success!

As they say in Paris, “Bonne chance, mes amis!”

********************

Taking It to the Next Levels: To Infinity and Beyond! It is of some interest to continue this investigation and examine methods of construction for 4x4 magic squares, 5x5 magic squares, and beyond. As you might imagine, patterns of construction become apparent in these investigations, and mathematicians over the years have established some construction methods that are specific to squares of even order (i.e., squares containing an even number of smaller squares) and methods that are specific to squares of odd order (i.e., squares containing an odd number of smaller squares).

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Rather than describing these methods here, I’m going to include these as Internet research investigations in the Creative Projects at the end of this section; it’s quite fascinating to see how these squares can be constructed. In addition, there are a number of variations on this theme including panmagic squares, multiplication magic squares, bimagic squares, semimagic squares, and many more. There are even some famous historical personalities who investigated magic squares including the statesman Benjamin Franklin and the artist Albrecht Dürer. I trust that you’ll enjoy your research endeavors and that you’ll develop an appreciation for this fascinating mathematical topic.

Insights and Conclusion In this section, we continued to expand and exercise the use of your analytical reasoning skills through the investigation of number puzzles. The concept of constraints on variable values was introduced in order to facilitate understanding of the structure of magic squares and to provide some familiarity with the nature and effect of constraints. Constraints appear in many real-world applications as well as in recreational mathematics. ******************** We also employed the analytical approach of increasing levels of complexity in our investigation of magic squares. Once again, we began our investigation at the very simplest of levels and gradually increased the complexity; while this approach may seem simplistic at times, it is in many instances a valuable tool for analysis and can reveal some of the underlying structural elements in a mathematical relationship. ******************** At the end of the section, you were presented with a challenge: the construction of a 3x3 magic square. After some attempts and failures, many students tend to view this as a seemingly impossible task; however, with perseverance and determination many go on to meet with success. This is the most difficult assignment thus far in this text and if you haven’t yet succeeded in the construction, please don’t be dismayed; the probing question hints provided on p.119 in lieu of a solution after the Exercises and Projects for Fun and Profit at the end of this section are designed to help you focus and sharpen your analytical reasoning skills and point you in the direction of the solution. I purposely refrained from providing the solution so that you have the full opportunity of working this challenge out on your own and reaping the personal rewards that come with successfully constructing the square.

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Congratulations to those of you who have completed the construction—you are to be complimented on your skill and perseverance; and to those of you who are still working on it, be encouraged—the hints will hopefully lead you to success! **********

Exercises and Projects for Fun and Profit Exercises in Analysis Note: See Helps and Hints for Exercises on pp.435–436 in Appendix A. Two-variable Relationships with Constraints Use your analytical thinking skills to determine all of the solutions to the following relationships: (Note: Some of these may have no solutions; if so, please indicate as such.) Enjoy the mental challenge and have fun!  1. Solve for a and b, where both a and b are positive whole numbers less than 8:

a +b 11 2. Solve for a and b, where both a and b are positive whole numbers less than 9:

a +b 13

Three-variable Relationships with Constraints Here you may apply the same basic skills to solve the following relationships: 3. Solve for a, b and c, where the values of the variables are positive whole numbers less than 6 and each variable represents a different value (no duplicates allowed): a +b c 4. Solve for a, b and c, where the values of the variables are positive whole numbers less than 7 and each variable represents a different value (no duplicates allowed): a +b c

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5. Solve for a, b and c, where the values of the variables (in any order) are positive consecutive integers less than 10 (e.g., a=5, b=4 and c=6 are possible assignments for a, b and c): a +b c 6. Solve for a, b and c, where the values of the variables (in any order) are positive consecutive integers less than 8 (e.g., a=5, b=4 and c=6 are possible values for the variables):

a +b c

Six-variable Relationships with Constraints Whether or not these relationships are any more difficult than the preceding set remains to be seen. Again, your basic skills will carry you through—then you may pass judgment on the level of complexity. 7. Solve for a, b, c, d, e and f, where the values of the variables (in any order) are positive consecutive integers less than 10 (e.g., a=4, b=2, c=3, d=5, e=7 and f=6 are possible values for the variables): ab +cd ef

Seven-variable Relationships with Constraints Whether or not these relationships are any more difficult than the preceding set remains to be seen. Again, your basic skills will carry you through—then you may pass judgment on the level of complexity. 8. Solve for a, b, c, d, e, f and g, where the values of the variables (in any order) are positive consecutive integers less than 10 (e.g., a=4, b=2, c=5, d=3, e=6, f=8 and g=7 are possible values for the variables):

abc + d ef g 9. Solve for a, b, c, d, e, f and g, where the values of the variables (in any order) are positive consecutive integers less than 10 (e.g., a=4, b=2, c=5, d=3, e=6, f=8 and g=7 are possible values for the variables):

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ab +cd efg

Eight-variable Relationships with Constraints Whether or not these relationships are any more difficult than the preceding set remains to be seen. Again, your basic skills will carry you through—then you may pass judgment on the level of complexity. 10. Solve for a, b, c, d, e, f, g and h, where the values of the variables (in any order) are positive consecutive integers less than 8 (e.g., a=4, b=5, c=3, d=2, e=1, f=6, g=8 and h=7 are possible values for the variables): abc + de fgh 11. Solve for a, b, c, d, e, f, g and h, where the values of the variables (in any order) are positive consecutive integers less than 9 (e.g., a=4, b=5, c=3, d=2, e=1, f=6, g=8, h=7 are possible values for the variables):

abc + de fgh Multiplication Puzzles Now that you have developed an understanding of the structure of these relationship puzzles, you may try your hand at solving some that involve the operation of multiplication. 12. Three-variable multiplication: Solve for a, b, and c, where the values of the variables are positive whole numbers less than 10 and each variable represents a different value (no duplicates allowed):

a xb c 13. Four-variable multiplication: Solve for a, b, c and d, where the values of the variables are positive whole numbers less than 10 and each variable represents a different value (no duplicates allowed): a x b cd

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14. Five-variable multiplication: Solve for a, b, c, d and e, where the values of the variables (in any order) are positive consecutive integers beginning with 1 (i.e., use each of the values from 1 to 5, inclusive; no duplicates allowed):

ab x c de 15. Seven-variable multiplication: Solve for a, b, c, d, e, f and g, where the values of the variables (in any order) are positive consecutive integers beginning with 1 (i.e., use each of the values from 1 to 7, inclusive; no duplicates allowed):

abc x d efg 16. Puzzling: Now that you have developed an understanding of the structure of these number relationship puzzles, create a minimum of two of your own using the operation of division and constraints of your choice for others to solve. You can present them to your friends, family, co-workers, etc., and impress them with your level of puzzle-solving and puzzlecreating ability!  c The simplest example of this structure is: a | b where the dividend (b), divisor (a), and quotient (c) each consist of a single digit, but you are free to use more digits in your constructions.

********************

Creative Projects: Magic Squares Magic Square Extensions

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MAGIC SQUARES A magic square is a square that has been sectioned off into smaller squares of equal area where the smaller squares have been equally distributed in rows and columns. For example, a 3x3 magic square contains nine smaller squares in three rows and three columns. In addition, if numbers are placed in each of the nine squares, the sums of the entries in each column, each row and each main diagonal are equal. a b c d e f g h i In the magic square above, this means that each of the following eight sums must have the same numerical value: rows:

a+b+c d+e+f g+h+i

columns:

a+d+g b+e+h c+f+i

diagonals:

a+e+i c+e+g

Note: Constructing a 3x3 magic square can be quite a challenge, especially if you simply use the trial-and-error approach. This project affords an excellent opportunity to use your analytical reasoning skills to effectively narrow the possibilities. A very good analytical line of reasoning would begin by recognizing that a 3x3 magic square has the same structure as the childhood game of tic-tac-toe. A good analytical question is: What is the most important square in the 3x3 magic square? (To answer this, you might recall the most important square in the game of tic-tac-toe.) A second and related question is: What is the most important number in the list of entry numbers for the 3x3 magic square? I'll let you put the answers to these two questions together as you carry out your investigation and I'll trust that these will lead you in a fruitful direction. (See Helps and Hints for Magic Squares on p.436 in Appendix A.) Assignments: For each assignment below, use each of the given numbers exactly once to construct a 3x3 magic square such that all sums as indicated above are equal. Trial and error, a good bit of persistence and your analytical reasoning are all that is needed to meet with success. Have fun! assignment Assignment #1 Assignment #2 Assignment #3 Assignment #4

numbers to be used 4, 7, 10, 13, 16, 19, 22, 25 and 28 5, 9, 13, 17, 21, 25, 29, 33 and 37 8, 11, 14, 17, 20, 23, 26, 29 and 32 9, 13, 17, 21, 25, 29, 33, 37 and 41

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MAGIC SQUARE EXTENSIONS For each assignment, write a short report (minimum of one page) on your findings:

Assignments: Assignment #1: Magic Squares on the Web. Do some Internet research on magic squares. Explain briefly the details of interest and include the Web addresses (URL’s) of the sites that you found to be of interest Assignment #2: Famous Magic Squares on the Web. Do some Internet research on the magic squares of Benjamin Franklin and Albrecht Dürer. Explain briefly the details of interest in each square and include the Web addresses (URL’s) of the sites that you found to be of interest. Assignment #3: Magic Squares of Even Order. Do some Internet research on magic squares of even order and construct a 4x4 magic square based on one of the construction methods that you discover. Explain briefly the details of interest and include the Web addresses (URL’s) of the sites that you found to be of interest. Assignment #4: Magic Squares of Odd Order. Do some Internet research on magic squares of odd order and construct a 5x5 magic square based on one of the construction methods that you discover. Explain briefly the details of interest and include the Web addresses (URL’s) of the sites that you found to be of interest.

********************

Just for Fun: Sudoku—A Japanese Brainteaser: Sudoku is a variation on magic squares that was developed some years ago and has now become a popular puzzle around the world. Visit the Sudoku website at www.sudoku.com to learn more about this puzzle and try your hand at it if you wish. It may be more engaging than you suspect! A number of newspapers now include a Sudoku puzzle as a daily feature. Have fun! 

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Answers to Questions in this Section p.104 The Second Simplest Case: Two Variables with Constraints The only solutions are: a +b 10

4 6 10

5 5 10

6 4 10

p.105 Next Step: Three Variables with Constraints The only solutions are: a=1, b=2, c=3 and a=2, b=1, c=3. Since the values for a, b and c are consecutive integers and c (the sum) is the largest of the three values, c will always be one unit greater in value than the second largest consecutive integer; consequently, the smallest of the three consecutive integers must be 1. p.105 Going Up! Next Level, Please! The only possible arrangement of four variables is: a + b = c d, where “c d” represents a twodigit sum. Since the maximum carry-over from the first column to the second is 1, c must equal 1; this requires that the other three positive consecutive digits are 2, 3 and 4. However, no two of these are large enough to create a carry-over when used as addends in the first column. Therefore, there is no solution to this puzzle involving four variables. p.106 Going Up! Next Level, Please! There are only five possible choices for the values of the variables as indicated in the sets below: set 1 2 3 4 5

values 1, 2, 3, 4, 5 2, 3, 4, 5, 6 3, 4, 5, 6, 7 4, 5, 6, 7, 8 5, 6, 7, 8, 9

Given that the variables a and d in the tens’ column represent different values, there must have been a “carry-over” from the ones’ column to the tens’ column. This implies that b+c is greater than 10 (b+c can’t equal 10 because 0 is not an acceptable value for the variable e due to the constraints).

ab +c de

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Consequently, set 1 may be eliminated since it contains no values that can satisfy this condition. Set 2 has only two values, 5 and 6, that will produce a sum greater than 10, but 5+6=11 and this requires that e=1. Since 1 is not a member of the set of values for set 2, set 2 may be eliminated. Skipping over to set 4, there are only four possible arrangements for b+c: 6+5 6+7 6+8 7+8  6+5 will require that e=1; since 1 is not one of the values in set 4, 6+5 may be eliminated.  6+7 will require that e=3; since 3 is not one of the values in set 4, 6+7 may be eliminated.  6+8 will require that e=4; this is an acceptable possibility but the only remaining values for a and d are 5 and 7, respectively, and these do not produce a valid solution.  7+8 will require that e=5; this is an acceptable possibility but the only remaining values for a and d are 4 and 6, respectively, and these do not produce a valid solution. In the final analysis, then, set 4 does not allow for any solutions to the addition puzzle. Using this same type of analytical reasoning and a bit of trial and error, you hopefully discovered that the only solutions to the addition puzzle are found in sets 3 and 5 as follows:

+

47 6 53

46 + 7 53

59 + 8 67

+

58 9 67

+

79 6 85

76 + 9 85

p.108 The Second Simplest Square: 2x2 As you have realized, there simply are no 2x2 magic squares under the given constraints on the values for the variables. According to the rules for constructing a magic square, the following six sums in a 2x2 magic square must be equal:

a+b = a+c = a+d = b+c = b+d = c+d If you consider the first two sums on this list, you will quickly see that they require that b=c, a violation of one of the rules for a magic square; consequently, there is no possible solution.

pp.109–110 The Third Simplest Square: 3x3 The construction of this square is a challenge indeed and often requires much persistence and perseverance. A large part of the value of this exercise of your analytical reasoning skills lies in spending the time required, working through your initial frustrations at not finding a quick and easy solution, and continuing the pursuit until you meet with success. This “Eureka!” moment is well worth the effort required and is an accomplishment that you can look upon in the future as something of significance. In order to allow you to experience this “Eureka!” moment, I’m not going to provide the answer but instead ask you some probing questions to help you along in your quest for a solution:

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Math Is Everywhere! Explore and Discover It! 2.5 Magic Squares: Sum Fun! 

A very good analytical line of reasoning would begin by recognizing that a 3x3 magic square has the same structure as the childhood game of tic-tac-toe. In view of this fact, a good analytical question is: Which square is the most important square in the 3x3 magic square? (To answer this, you might recall the most important square in the game of tic-tac-toe.)

A second and related question is: Which number is the most important number in the list of entry numbers for the 3x3 magic square?

I hope that these questions will lead you to further analysis and, ultimately, to the successful construction of the 3x3 magic square. Be of good courage and cheer and press on!

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Math Is Everywhere! Explore and Discover It! 3.0 Introduction and Objectives

Chapter Three Math from Other Times and Places: Historical Investigations 3.0 Introduction and Objectives The thumbnail sketches below represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

3.1 The Greeks: Students and Investigators

3.2 Pascal’s Amazing Triangle: Simple yet Profound 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

3.3 Number Systems through the Ages: Stylistic and Diverse

MCMXLVIII Babylonian

Egyptian

Roman

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Math Is Everywhere! Explore and Discover It! 3.0 Introduction and Objectives

Learning Objectives for Chapter 3 1. To enable students to develop their analytical reasoning skills through pattern recognition and problem solving. 2. To enable students to use their deductive and inductive reasoning skills in recognizing patterns of various types. 3. To enable students to appreciate the power of generalization and to develop the ability to use inductive reasoning to form generalizations from sets of specific facts or data. 4. To allow students to exercise their creativity by conducting Internet research and writing reports on various mathematical topics. 5. To enable students to gain familiarity with Web-based mathematical resources and to develop their critical thinking skills in assessing and evaluating these resources. 6. To allow students to investigate math history and numeration in other cultures. 7. To engage students in active learning through the use of projects and activities. 8. To enable students to develop confidence in their mathematical abilities. 9. To empower students to become lifelong learners.

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Math Is Everywhere! Explore and Discover It! 3.1 The Greeks: Students and Investigators

3.1 The Greeks: Students and Investigators “All human knowledge thus begins with intuitions, proceeds thence to concepts, and ends with ideas.” —Immanuel Kant

Introduction Over 2500 years ago, Greek civilization was near its zenith and mathematics was being explored for its abstract value rather than exclusively for its practical value. The Greeks had developed the skills of analysis and conceptualization and went on to apply them in various fields; with regard to mathematics in particular, they made significant contributions in Number Theory and Geometry. In this section, we’ll take a brief look at some notable Greek mathematicians and their accomplishments. In the Exercises and Projects for Fun and Profit at the end of this section you’ll have further opportunities to explore and investigate the Greeks and their mathematical legacy.

Pythagoras and his Famous Triangle Theorem In the sixth century B.C., there was a Greek philosopher named Pythagoras (~569 B.C. – ~465 B.C.) who founded a school of sorts. His students studied four subjects—arithmetic, music, geometry and astronomy—and were subject to a rather rigid set of rules and requirements. The students focused primarily on mathematics and viewed other principles in light of their mathematical understanding. One of the well-known theorems of mathematical number theory attributed to Pythagoras (although probably known by others before his time) is the famous theorem involving the lengths of the sides of a right triangle: The Pythagorean Theorem: In a right triangle in which the length of the hypotenuse is designated by the variable c and the lengths of the two legs are designated by the variables a and b, a2  b2  c2 This theorem is taught in all high school geometry classes and has many applications in various settings.

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Math Is Everywhere! Explore and Discover It! 3.1 The Greeks: Students and Investigators

Euclid’s Elements The Greek mathematicians began to organize their knowledge on the basis of fundamental axioms. Euclid (~325 B.C. – ~265 B.C.), perhaps the most famous Greek mathematician, compiled many of these axioms into a series of books entitled The Elements. In this broad compendium of the mathematics of the day, Euclid provided a logical ordering of then current mathematical knowledge and established a work that has come down to us today as one of the outstanding accomplishments of the period. While he wasn’t a great mathematician in terms of making new discoveries and expanding the mathematical knowledge base of his time, he certainly was accomplished as an organizer of existing knowledge and as a teacher. His five postulates for Geometry are the basis for what has become known as Euclidean geometry, the plane geometry that many high school and college students study.

Archimedes: Man of Math and Science Archimedes (287 B.C. – 212 B.C.) is generally considered to be one of the most creative men of his day. His interests included both mathematics and mechanical inventions and he is famous for his role in the defense of Syracuse during the Second Punic War. Some of his inventions, such as the Archimedean screw and the compound pulley, are still in use today; some of his discoveries became general principles, such as the principle concerning floating bodies and their weights. Altogether, Archimedes was a remarkably accomplished individual in his time and someone who has had an effect on all succeeding generations.

********************

Exercises and Projects for Fun and Profit Further Investigations 1. Pythagoras: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on this Greek mathematician and write a short report (minimum of one page) on your findings. Please include the highlights of his mathematical accomplishments. 2. Euclid: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on this Greek mathematician and write a short report (minimum of one page) on your findings. Please include the highlights of his mathematical accomplishments. 124 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 3.1 The Greeks: Students and Investigators

3. Archimedes: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on this Greek mathematician and write a short report (minimum of one page) on your findings. Please include the highlights of his mathematical accomplishments. 4. Math Instruction in Ancient Greece: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on how mathematics was taught in ancient Greece and write a short report (minimum of one page) on your findings. 5. The Pythagorean Theorem—Proofs by the Dozens: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on the various proofs of the Pythagorean theorem and write a short report (minimum of one page) on your findings. ********************

Creative Project: Polygonal Numbers—A Geometric View

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Math Is Everywhere! Explore and Discover It! 3.1 The Greeks: Students and Investigators

POLYGONAL NUMBERS—A GEOMETRIC VIEW Since the early Greeks had not yet developed symbols for numbers, it is likely that the Pythagoreans and other Greek mathematicians viewed numbers in terms of the number of objects or dots required to represent a number, e.g., 1 = 1 dot, 2 = 2 dots, 3 = 3 dots, 4 = 4 dots, etc. This eventually led to a classification of numbers in terms of the regular geometrical shapes that the dots could assume. For examples, three dots can form an equilateral triangle and four dots can form a square. This idea can be extended to larger numbers by constructing similar figures with appropriately spaced dots. In the case of the square numbers, we have the following:

The single dot (actually a small circle) represents a degenerate square in which each side has a length of zero units, the array containing four dots represents a square in which each side has a length of one unit, and the array containing nine dots represents a square in which each side has a length of two units. Hence, the numbers 1, 4 and 9 are considered to be square numbers. Each successive square may be formed from the previous one by extending two adjacent sides by a length of one unit and then adding dots at equally spaced intervals of one unit to complete the larger square. Without any loss of generality, we can always choose to extend the left edge of the square in an upwards direction and to extend the bottom of the square to the right. In a similar fashion, we can construct triangular numbers as shown below:

The single dot represents a degenerate triangle in which each side has a length of zero units, the array containing three dots represents a triangle in which each side has a length of one unit, and the array containing six dots represents a triangle in which each side has a length of two units. Hence, the numbers 1, 3 and 6 are considered to be triangular numbers. As with the square numbers, each successive triangle may be formed from the previous one by extending two adjacent sides by a length of one unit and then adding dots at equally spaced intervals of one unit to complete the larger triangle. Without any loss of generality, we can always choose to extend both the left edge and the right edge of the triangle in a downwards direction and then fill in the appropriate number of dots along the new bottom edge.

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Math Is Everywhere! Explore and Discover It! 3.1 The Greeks: Students and Investigators

Math Is Everywhere! Explore and Discover It! 3.2 Pascal’s Amazing Triangle: Simple yet Profound

3.2 Pascal’s Amazing Triangle: Simple yet Profound “In the pure mathematics, we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven.” —Edward Everett, American statesman

Introduction The array of numbers below shows the first six rows of Pascal’s Triangle. It is named in honor of the French mathematician, Blaise Pascal (1623–1662).

1 1 1 1 1 1

3 4

5

1 2

1 3

6 10

1 4

10

1 5

1

The numerical entries in the triangle have a mathematical structure of much distinction (as you shall soon discover) and have been studied and explored by various mathematicians down through the centuries. While Pascal did not invent the triangle that bears his name, he did study it and write about it during his lifetime; his writings were considered to be of much importance with regard to illuminating the triangle’s properties. The earliest written records of this amazing arrangement of numbers date back in history at least to ancient China; however, the triangle may actually have its origins further back in time. As you may have suspected, the triangle does not end with the six rows shown above—it continues indefinitely onward with rows of ever-increasing length and numerical values that become progressively larger in successive rows. There are some features of the triangle that are immediately obvious such as the symmetric nature of the numerical entries in each row; there are other features that are less obvious but nonetheless of great interest. It is these features that you will explore in greater depth in the Creative Project at the end of this section.

Connections to Other Branches of Mathematics As you will soon discover, Pascal’s Triangle has connections to several other branches of mathematics including algebra, probability and number theory. It contains the triangular numbers that were presented and discussed in the previous section and it also contains, in an unusual way, the Fibonacci sequence of numbers that we will explore in a subsequent section. Altogether, this simple and unassuming triangular structure has many interesting mathematical 128 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 3.2 Pascal’s Amazing Triangle: Simple yet Profound

features both on its surface and at its deeper levels; in fact, I’m not sure that all of its features and connections have yet been discovered… ********************

Exercises and Projects for Fun and Profit Creative Project: Pascal’s Triangle ********************

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Math Is Everywhere! Explore and Discover It! 3.3 Number Systems through the Ages: Stylistic and Diverse

3.3 Number Systems through the Ages: Stylistic and Diverse “On two occasions I have been asked, 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?' I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.” —Charles Babbage

Introduction In this section, we will take a look at number systems of various civilizations including our own. As cultures and civilizations came into being, the need to count and keep records arose quite naturally. Since numbers and counting are so fundamental, one might think that a standard system for counting and recording would have developed; however, that was not the case. As it turned out, each civilization created its own unique methods and styles of numeration. We’ll focus on just a few of these to give you a sense of the broad diversity of ancient number systems and then take a closer look at our own present-day system and its elegance and efficiency. What future civilizations may devise is an open question…

Sumerians and Babylonians: Inventive and Accomplished In ancient Sumeria (~3500 B.C.), base 60 was used (rather than our familiar base 10) in counting, but the Sumerians did not have a system of place value like ours. Theirs was an additive system in which each number was represented by collections of symbols whose values were simply added together to produce the total value of the number. The Sumerian numbers were formed by combining the symbols for 1 and 10 as shown below (figures are very rough approximations): a vertical wedge, representing the value 1: a horizontal wedge, representing the value 10: Some examples of Sumerian numbers are: 2:

61:

Notice that larger numbers need increasingly larger numbers of symbols to represent them.

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Math Is Everywhere! Explore and Discover It! 3.3 Number Systems through the Ages: Stylistic and Diverse

When the Babylonians took control of the Mesopotamian region (~2000 B.C.), they adopted the Sumerian base 60 and used it in a positional place value system, thus making the recording process much more efficient. The Babylonian place values began with 60 0  1 in the rightmost column; each successive column to the left had a value that was 60 times larger than the column on its right. To illustrate, the first four columns and their place values are shown below: column place value in base 60

4 60  216,000 3

3 60  3600 2

2 60  60 1

1 60  1 0

This place value system made the representation of large numbers much more efficient.

Example 1: To represent the number 2, they would make two marks like these: Note: Both of these wedges are considered to be in the ones’ column.

Example 2: To represent the number 61, they would make two marks like these:

Note: In order to be sure that these two characters weren’t mistaken for the value 2, they would leave extra space between them; then the leftmost wedge was seen as being in the 60’s column (i.e., the second column from the right that has a place value of 60), and the whole number was viewed as (1 · 60) + (1 · 1) = 61.

Example 3: To represent the number 11, they would mark on a clay tablet the characters:

In this case, the horizontal wedge representing the value 10 is still considered to be in the ones’ column along with the vertical wedge representing the value 1 since there is no distinguishing space between the characters. Example 4: To represent the number 31 they would mark on a clay tablet the characters:

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Math Is Everywhere! Explore and Discover It! 3.3 Number Systems through the Ages: Stylistic and Diverse

In this case, the horizontal wedges representing the value 10 are still considered to be in the ones’ column along with the vertical wedge representing the value 1 since there is no distinguishing space between the characters. Example 5: To represent the number 601, they would mark on a clay tablet the characters:

In this case, the leftmost wedge is seen as being in the 60’s column and represents a value of 10 · 60 = 600; the second wedge is in the ones’ column. Hence, the whole number has a value of (10 · 60) + (1 · 1) = 601. ******************** As you can see, this was a definite improvement over the Sumerian system that had no place value. There are some interesting theories as to why both of these cultures chose 60 as a base value rather than the base 10 that we use today and you’ll have an opportunity to explore this topic further in the Exercises and Projects for Fun and Profit at the end of this section.

********************

The Artisans of Egypt Ancient Egyptians had a well-developed number system by 3000 B.C. They developed a more artistic set of numerals and were quite ingenious at working with fractions but did not have a place value system like that of the Babylonians. In order to write large numbers, they simply used more of their symbols in an additive fashion. The symbols themselves involved the powers of 10 (i.e., the Egyptians used base 10 in their number system rather than powers of 60 in base 60) and in this way they had a mechanism for constructing large numbers although it was somewhat limited in efficiency. The Egyptian numerals included the following characters and continued with other symbols for larger powers of 10: 1: 10:

100:

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Math Is Everywhere! Explore and Discover It! 3.3 Number Systems through the Ages: Stylistic and Diverse

Example 6: To write the number 325, the Egyptians would have to use ten symbols:

Can you imagine trying to write 999 with Egyptian numerals?

This lack of place value made for a rather cumbersome system of numeration compared to our present-day system. On the other hand, the Egyptians were quite successful at devising ways to work with fractions and at multiplying by repeatedly adding. The pyramids attest to their mathematics and engineering skills despite a rather awkward number system.

The Romans and their Famous Numerals When ancient Rome grew to become a world power, its influence was felt far and wide. Even today, we use Roman numerals on clock faces, in copyright notices in books and movies, in book chapter headings and subheadings, in page numbers for prefaces, etc. While their numeration system left a lot to be desired, it clearly has retained quite a bit of appeal. After all, 2000 years is a long time… The Roman system was similar to that of the Egyptians in that it employed a set of characters for certain values and repeated these characters additively to construct larger numbers; there was no place value. The characters are the familiar set consisting of: 1 I

5 V

10 X

50 L

100 C

500 D

1000 M

Example 7: To represent the number 4,878, the Romans would write (in descending order):

MMMMDCCCLXXVIII. As with the Egyptian method, writing Roman numerals was quite tedious and cumbersome. To make matters worse, there were no simple ways to multiply or divide these numerals and calculations were difficult, at best. The additive system employed here is far less effective than our place value system of today.

The Mayans and the Number Twenty The Mayans of Central America developed a rather unique system using place value and base 20. Their system was unusual in that it employed a symbol for the number 0 (zero). For more

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Math Is Everywhere! Explore and Discover It! 3.3 Number Systems through the Ages: Stylistic and Diverse

information on this topic (and others in this section), please visit the MacTutor History of Mathematics Archive at http://www-history.mcs.st-and.ac.uk

Our Present-Day System of Place Value: Hindu-Arabic Elegance and Efficiency Today, we are accustomed to the ease and simplicity of a very efficient number system based on place value. This system was popularized by the Hindus of India and the Arabs of Mesopotamia and Northern Africa and is perhaps the simplest and most powerful number system. There are ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, and, in a given numerical expression, each column has a predetermined place value associated with base 10. To represent four thousand, eight hundred and seventy-eight, we simply write 4,878. Having practiced writing numbers in this way since elementary school, we hardly recognize the elegance and efficiency of this numeration system. What we really mean when we write those four digits is the number having the value: (4 · 1000) + (8 · 100) + (7 · 10) + (8 · 1). In other words, each digit is automatically assigned a place value determined by the column in which it is found. Each of these place values is based on a power of 10, beginning with the zero power in the rightmost column. We could have written: (4 · 10 3 ) + (8 · 10 2 ) + (7 · 10 1 ) + (8 · 10 0 ) Note: Since 10 0 = 1, the rightmost column has a place value, or positional value, of 1. Example 8: 392 = (3 · 10 2 ) + (9 · 10 1 ) + (2 · 10 0 )

Expressing numbers as the sum of multiples of powers of 10 is called expanded notation and more fully describes what the given digits represent in terms of value. In the next chapter, we will see the usefulness of this system with regard to bases other than 10.

Addition and Multiplication Made Easy The Hindu-Arabic system greatly simplifies the operations of addition and multiplication that were so cumbersome for other cultures. When we add two numbers, we have learned that we can “carry” values over to the column to the left of the one in which we are working. For example, if we were to add 37 and 26, we would begin by adding the 7 and the 6 and obtaining the result of 13. Since 13 is larger than our underlying base, 10, we enter the 3 in the ones’ column and “carry” a 1 (which actually represents ten 1’s) over into the next column on the left, the tens’ column. Continuing on, we add the 3 and the 2 along with the carried over 1 to obtain 6 as the result in the tens’ column. The sum, of course, is 63.

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Math Is Everywhere! Explore and Discover It! 3.3 Number Systems through the Ages: Stylistic and Diverse

37 + 26

1 37 + 26 3

37 + 26 63

If you were alive during the Roman times and wanted to add these two numbers, you would have had to combine the Roman numerals together to come up with the result: XXXVII + XXVI = XXXXXVVIII= LXIII Actually, this isn’t much more involved than addition in our system, but it does require combining the various numeral types together to form larger numerals where appropriate, e.g., V+V=X. When it comes to multiplication, however, the power and efficiency of the Hindu-Arabic system becomes quite obvious. When we multiply 37 · 26, we find the partial products and add them together; this is a very simple procedure that we all learned in elementary school: 37 x 26 222 + 74 962

partial product partial product

******************** In the Roman system, however, attempting to multiply XXXVII · XXVI is a much more daunting task! Attempting to find partial products and then adding them together is quite laborious. Thank goodness that we aren’t still using the Roman number system today… ********************

Since the Roman system is an additive one, i.e., XXXVII = X+X+X+V+I+I and XXVI = X+X+V+I, multiplying these two numbers together involves the use of the distributive property just as in the FOIL method for multiplication of polynomials. We begin by multiplying and distributing: (X+X+X+V+I+I) · (X+X+V+I) = (X·X + X·X + X·X + X·V + X·I + X·I) + (X·X + X·X + X·X + X·V + X·I + X·I) + (V·X + V·X + V·X + V·V + V·I + V·I) + (I·X + I·X + I·X + I·V + I·I + I·I)

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Math Is Everywhere! Explore and Discover It! 3.3 Number Systems through the Ages: Stylistic and Diverse

We then convert each partial product (e.g., X·X=C) to get the following intermediate result: (X+X+X+V+I+I) · (X+X+V+I) = C + C + C + L + X + X +C+C+C+L+X+X + L + L + L + XXV + V + V +X+X+X+V+I+I We combine like terms and convert as necessary:

C + C + C + C + C + C = DC L + L + L + L + L = CCL X + X + X + X + X + X + X = LXX XXV + V + V + V = XXXX I + I = II

And then we combine and convert some more. Aaaarghhh! DC + CCL + LXX + XXXX + II = DCCCCLXII

This is the result: 962.

In case you haven’t noticed, our place value system allows us to perform this multiplication by simply adding two intermediate partial products. Perhaps we should all write thank-you letters to the developers of the Hindu-Arabic number system for their wonderful contributions to mathematics… Note: CM is a Roman shortcut for DCCCC; they used a limited subtractive principle to make things a bit more efficient. Consequently, DCCCCLXII = CMLXII. As you can easily see, though, their efficiency in numeration is sadly lacking…

The Abacus: A Calculator with a Memory To get an in-depth look at our elegant and efficient place value system, we are going to embark shortly on a project that will give us some hands-on experience. By way of introduction, once upon a time, in a place far away, some inventive person(s) came up with the idea of creating a device to assist with the processes of addition and multiplication (and subtraction and division, for that matter). Although its precise origins are a bit obscure, the abacus came into being many centuries ago and is still in use today in some locales. One of my former students reported that he recently observed the use of an abacus by a merchant in New York City’s Chinatown.

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Math Is Everywhere! Explore and Discover It! 3.3 Number Systems through the Ages: Stylistic and Diverse

The abacus is a collection of beads strung on rods in such a way that the operation of addition, for example, can be carried out by moving the appropriate beads forwards or backwards. The sum is easily seen in the arrangement of the beads, and an additional number can be added to this sum to produce a second sum that represents the grand total. Of course, this process can be repeated indefinitely, and after each operation the abacus “remembers” what the subtotal is. You will have an opportunity to explore and investigate this remarkable calculator in one of the Creative Projects in the Exercises and Projects for Fun and Profit at the end of this section. In fact, there are virtual abacuses (actually, the plural of abacus is abaci in Latin) now available on the Internet that allow you to operate one electronically. ********************

Exercises and Projects for Fun and Profit Creative Projects: The Abacus Numeration in Other Cultures ********************

THE ABACUS We will now engage ourselves in a project involving the use of the abacus and examine its structure in more detail. The abacus is a wonderful physical representation of the place value number system that we use today in mathematics and very nicely reveals the underlying nature of our system. Most students know what an abacus is but few have actually used one; this project will be a hands-on encounter with the world’s first calculator with a memory. Have fun! Assignment:

Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on the Chinese abacus. Visit at least two sites and write a short report (minimum of one page) on your findings. Note: It is best to use the search expression Chinese abacus although the results from abacus are interesting with regard to how many businesses use the word abacus as part of their company name.

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Math Is Everywhere! Explore and Discover It! 3.3 Number Systems through the Ages: Stylistic and Diverse

NUMERATION IN OTHER CULTURES Down through history, cultures and civilizations have used various types of numeration systems and mechanical computation devices. Assignment:

Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on the numeration systems and mechanical calculators listed below. Choose one numeration system OR choose one mechanical calculator that particularly interests you and write a report (minimum 2 pages, double-spaced, with a maximum of 12pt font size and 1 inch margins) on your findings. You may include pictures and/or diagrams where appropriate.

As you carry out your research, please construct a webliography of the three best sites that you visit. This should include:  the Web address of each site (i.e., the URL—Uniform Resource Locator)  a brief description of each site’s content and features  an evaluation of each site’s effectiveness (excellent, very good, good, …) Your complete report should include a minimum of one page of text, not including pictures and diagrams and not including your webliography. With pictures and diagrams and webliography included, your report should be a minimum of two pages in length.

******************** Numeration Systems

Mechanical Calculators

Chinese numeration Sumerian numeration Babylonian numeration Egyptian numeration Greek numeration Mayan numeration

Napier’s rods Leonardo daVinci’s adding machine Blaise Pascal’s adding machine Charles Babbage’s analytical engine

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Math Is Everywhere! Explore and Discover It! 4.0 Introduction and Objectives

Chapter Four Number Bases: Surprising Versatility 4.0 Introduction and Objectives The thumbnail sketches below represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

4.1 Understanding Other Number Bases: Places of Value 12,345 = (1  10 4 )  (2  10 3 )  (3  10 2 )  (4  101 )  (5  10 0 ) and 12345 5 = (1  5 4 )  (2  5 3 )  (3  5 2 )  (4  51 )  (5  5 0 )

4.2 The Binary System: Secret Language of Computers A coded message:

1001000 1000101 1001100 1001100 1001111

4.3 Number Base Applications: More Fun with Math! Hexadecimal color codes for webpages:

Red = #FF0000

White = #FFFFFF

Blue = #0000FF

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Math Is Everywhere! Explore and Discover It! 4.0 Introduction and Objectives

Learning Objectives for Chapter 4 1. To enable students to develop their analytical reasoning skills through pattern recognition and problem solving. 2. To enable students to develop their deductive reasoning skills through encoding and decoding messages and symbols. 3. To allow students to apply their knowledge of our base 10 number system to number systems in other bases. 4. To allow students to exercise their creativity by conducting Internet research and writing reports on various mathematical topics. 5. To enable students to gain familiarity with Web-based mathematical resources and to develop their critical thinking skills in assessing and evaluating these resources. 6. To demonstrate the usefulness of mathematics through practical applications involving mathematical number bases. 7. To encourage students to persevere in finding solutions to challenging problems in order to further develop confidence in their mathematical abilities. 8. To engage students in active learning through the use of projects and activities.

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

4.1 Understanding Other Number Bases: Places of Value “One is hard pressed to think of universal customs that man has successfully established on earth. There is one, however, of which he can boast: the universal adoption of the Hindu-Arabic numerals to record numbers. In this we perhaps have man's unique worldwide victory of an idea.” —Howard Eves

Introduction In the last chapter, we presented the concept of expanded notation of a number; in this section, we’ll begin by reviewing that concept and then extending it to a more general form. This more general form will then allow us to consider other number bases and come to a deeper understanding of their structure.

The Underlying Structure Revealed: Expanded Notation To review, when we represent a number like 12,345, we know from experience what we mean; however, if you had not yet learned about numbers, this representation would be quite meaningless. In order to explain the meaning of 12,345 to a kindergartner, let’s say, or early elementary school student, you would of necessity have to explain about place value. You might begin by explaining that each of the columns (places) in the number has a different value: the ones’ column is on the right, the tens’ column is to its left, the hundreds’ column is to the left of the tens’ column, and so on. This is a very fundamental and important aspect of our base ten number system. The value of the number represented by the numerals 12,345 isn’t found by simply adding the digits together like this: 1+2+3+4+5=15; nor is the value found by multiplying the digits together like this: 1·2·3·4·5=120. As we all know, the value is found by using its place values together with multiplication and addition: 12,345 = (1  10 4 )  (2  10 3 )  (3  10 2 )  (4  101 )  (5  10 0 ) or 12,345 = (1 · 10,000) + (2 · 1000) + (3 · 100) + (4 · 10) + (5 · 1) Note: In this expanded notation, each product involves the base 10 and an appropriate exponent. In particular, 10 raised to the zero power is equal to 1: 10 0  1 .

In other words: 12,345 = 10,000 + 2,000 + 300 + 40 + 5. 141 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

This fundamental structure is so often taken for granted that it is unfortunately overlooked by elementary students when it comes to performing the basic operations of arithmetic: addition, multiplication, subtraction and division. As an example, elementary school teachers often find this common error in students’ work: 25.732 + 3.79 26.111 The error here, of course, lies in not realizing that the columns have specific place values and need to be aligned properly: 25.732 + 3.79 29.522 Another common mistake is seen in the following example: 14.57 – 8.039 6.549

minuend subtrahend difference

Here the error stems from not realizing that the minuend contains an unseen yet understood digit, zero, in the third decimal place that represents (0 · 0.001) or (0 · one one-thousandth). Gaining a solid understanding of the concept of place value, then, is important for further work with numbers and their operations. We’ll now investigate an interesting connection between expanded notation and the representation of polynomials.

Numbers as Polynomials In algebra classes you learned that polynomials are the workhorses of algebra—they are used in almost every area of algebra and are a wonderful tool in the realm of mathematics. When we write polynomials, we generally use a standard format such as descending order or degree. For example, a typical polynomial involving the variable x might look like this:

x 4  2 x 3  3x 2  4 x  5 or, to be a little more thorough and complete: 1x 4  2 x 3  3x 2  4 x 1  5 x 0 or, even more expansively: (1  x 4 )  (2  x 3 )  (3  x 2 )  (4  x 1 )  (5  x 0 )

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

This should be very reminiscent of what we saw above concerning expanded notation: 12,345 = (1  10 4 )  (2  10 3 )  (3  10 2 )  (4  101 )  (5  10 0 ) The connection here is that our everyday numbers can actually be viewed as polynomials involving the base 10, where 10 represents a specific value of the variable, x. Not only that, but since x is able to vary in value (i.e., it is a variable), we can now begin to use other number bases in this same polynomial structure. For instance, we could choose to use the number 5 as our base and then represent a number in base 5. Example 1: 234 in base 5 could be viewed in the same polynomial fashion as above:

(2  5 2 )  (3  51 )  (4  5 0 ) In this case, the base number 5 is being used in place of the variable and the result is an expanded notation that allows us to easily compute the value of this number with regard to our usual base 10 system: 234 in base 5 = (2  5 2 )  (3  51 )  (4  5 0 ) or (2 · 25) + (3 · 5) + (4 · 1) = 50 + 15 + 4 = 69 in terms of our usual base 10. Note: The subscript 5 in the number 234 5 is a standard mathematical notation that indicates the number’s base. Note: As soon as we write 25 in place of 5 2 , we have switched back to our standard decimal notation in base 10, i.e., the “2” is in the tens column and the “5” is in the ones column. Each of the numbers 50, 15, 4 and 69 are hence in base 10.

We can say, then, that 234 in base 5 is equal in value to 69 in base 10: 234 5  6910  69 Note: The subscript 10 is unnecessary here but is included for the sake of being thorough. When a number has no attached subscript, the number is understood to be in base 10.

You might be wondering why in the world anyone would ever want to write numbers using base 5 or any base other than 10, for that matter. It does seem like a rather arcane practice— something found only in dry and perhaps dusty mathematics textbooks; however, as we shall see

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

in later sections of the book, there are a number of valuable, even astounding, applications for other number bases. You’ll be amazed at how widely they are used in our daily lives!

Conversion Practice To gain some familiarity with the process of converting a number that is given in some other base to a value in our usual base 10 system, we’ll consider a few examples: Example 2: Convert 3122 in base 4 to a value in base 10 In expanded notation, or polynomial form, 3122 4 = (3  4 3 )  (1  4 2 )  (2  41 )  (2  4 0 ) = (3 · 64) + (1 · 16) + (2 · 4) + (2 · 1) = 192 + 16 + 8 + 2 = 218 10 = 218 in our usual base 10 number system Example 3: Convert 524 in base 7 to a value in base 10 524 7 = (5  7 2 )  (2  71 )  (4  7 0 ) = (5 · 49) + (2 · 7) + (4 · 1) = 245 + 14 + 4 = 263 10 = 263 in our usual base 10 number system Example 4: Convert 10101 in base 2 to a value in base 10 10101 2 = (1  2 4 )  (0  2 3 )  (1  2 2 )  (0  21 )  (1  2 0 ) = (1 · 16) + (0 · 8) + (1 · 4) + (0 · 2) + (1 · 1) = 16 + 0 + 4 + 0 + 1 = 21 10 = 21 in our usual base 10 number system

In each of the above examples, the pattern is the same: the original number has place values, or positional values, that are determined by the base in which the number is given. Exponents are applied to the base beginning with 0 as the exponent in the rightmost column of the original number and increasing by one as you move to each succeeding column to the left. Each digit of the original number is then multiplied by the base and exponent combination that belongs to its column. Afterwards, these products are added together to produce the value of the number with respect to our everyday base 10 system.

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

Some Healthful Exercise Try your hand at converting the following numbers into base 10 values (answers provided on p.155 at the end of this section): a) b) c) d)

231 in base 4 572 in base 9 2345 in base 6 10111 in base 2 ********************

Reversing the Process Now that we have learned how to convert numbers from other bases into base 10 values, we would also like to learn how to convert base 10 numbers into values in other bases. Again, you may be wondering why someone would have any interest in doing so; trust me—it will lead us to some very powerful applications in later sections of our textbook. The reverse process is a bit more complicated to carry out, but with a bit of diligence and persistence you will soon be experts at it! There are two methods for the reverse conversion process: the long method and the short method. We’ll examine both, and then you may use whichever you prefer.

Reversing the Process: The Clear but Somewhat Tedious Long Method Example 5: Convert the base 10 number 359 into a base 5 number

As an overview, we would like to parcel out the 359 units we have been given into appropriate multiples of powers of 5. Step 1: In the long method, we begin by determining the highest power of 5 that will divide into 359. Since the powers of 5 are 5 0  1, 51  5, 5 2  25, 5 3  125, 5 4  625, and so on, we can see by inspection that the highest power of 5 that divides into 359 is 5 3 =125.

Then we divide 359 by 125:

359 125

Note: This is often written as: 359/125.

359/125 = 2 with a remainder of 109 since 2 · 125 = 250 and 359 – 250 = 109.

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

Both the quotient and remainder here are important. We make note of the fact that 359 contains two 125’s and then use the remainder, 109, in the next step as we continue to separate this remainder into appropriate powers of 5. Note: If you perform the division 359/125 on your calculator, you will obtain a result of 2.872. The remainder needed in this case is NOT 0.872! 0.872 is the decimal part of the quotient and represents a fractional value based on the number of decimal places shown. In this case, the fractional value is 872/1000, and the complete quotient in the division of 359 by 125 is: 2 and 872/1000. The decimal part of the quotient is a very different concept than that of a remainder as we normally consider it. It helps to think back to those long-ago days when you first learned to divide (without the use of a calculator, and before you learned about decimals) and you would think like this: 13/5 = 2 with a remainder of 3. Step 2: We now work with the remainder, 109, and determine the highest power of 5 that divides into the remainder. By inspection, we see that 5 2 =25 divides into 109 and we subsequently divide 109 by 25:

109/25 = 4 with a remainder of 9, since 4 · 25 = 100 and 109 – 100 = 9. We make note of the fact that 109 contains four 25’s and then use the current remainder, 9, in the next step. Step 3: We continue the process with this latest remainder, 9, and we see that the highest power of 5 that divides into 9 is 51 =5.

Dividing 9 by 5 yields 9/5 = 1 with a remainder of 4. We make note of the fact that 9 contains one 5 and then use the current remainder, 4, in the next step. Step 4: At this point, the only power of 5 that divides into the remainder 4 is 5 0 =1 and it divides evenly 4 times with no remainder. The zero power in any base will always equal 1, so this step is rather simplistic. We make note of the fact that 4 contains four 1’s, and since there is no remainder, we have completed the process of parceling the number 359 into various parts, each of which involves a power of 5. Step 5: Lastly, we collect our notes about the number and type of multiples that we used and put all of this information together to construct a number that represents 359 in base 5.

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

We used the following multiples of powers of 5: 2 · 53 4 · 52 1 · 51 4 · 50

= 2 · 125 = 4 · 25 =1·5 =4·1

= 250 = 100 = 5 = 4 359

The number that we have just constructed through our repetitive division process is (in expanded notation): (2  5 3 )  (4  5 2 )  (1  51 )  (4  5 0 ) More simply, we can write this number as: 2414 5 (2414 in base 5). At last, we have accomplished our goal!

While it is a bit tedious, this conversion process can be carried out in a similar fashion for any number base; it’s basically (pardon the play on words…. ) a matter of finding how many multiples of each power of the base are contained in the original base 10 number. We’ll now consider a further example to illustrate the case where a particular power of the base happens to have NO multiples.

Example 6: Convert the number 39 in base 10 to a number in base 6 Step 1: Determine the highest power of 6 that divides into 39:

Since 6 0  1, 61  6, 6 2  36, 6 3  216 , and so on, it’s clear that 6 2 =36 is the highest power that is contained in 39. Hence, 39/36 = 1 with a remainder of 3. Make note of the 1 and use the 3 in the next step. Step 2: Determine the highest power of 6 that divides into 3:

6 0 =1 is the highest power and it divides into 3 exactly 3 times with no remainder. We have completely converted the original number 39 into multiples of the powers of the base 6. However, we skipped over 61 =6 since there was no opportunity to use this power. In this case, we need to make special note of the fact that we used no (zero) multiples of 61 ; as you will see, this is important in constructing our final base 6 number.

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

Step 3: Collect our notes and construct the final base 6 number. We used the following multiples:

1 · 62 0 · 61 3 · 60

= 1 · 36 =0·6 =3·1

= 36 = 0 = 3 39

Notice how we included the product 0 · 61 as part of our construction. This is important because each place (position) in the number has a specific value that is based on sequential powers of the base (without skipping any powers). In expanded notation, our new base 6 number is: (1  6 2 )  (0  61 )  (3  6 0 ) More simply we can write: 103 6 (103 in base 6). (Again, the zero is an important placeholder.) We have shown then, that 39 in base 10 has the same value as the number 103 in base 6: 3910  1036 .

Reversing the Process: The Elegant yet Mysterious Short Method We will now examine a nice shortcut to the conversion process that we carried out in the above examples. This elegant procedure does the same job much more quickly and efficiently but obscures the underlying processes so that it isn’t readily apparent why this method works. We’ll investigate this procedure and analyze its nature in the Exercises and Projects for Fun and Profit at the end of this section. For now, we will present a simple description of the method. Example 7: Convert 359, the same base 10 number as above, to a number in base 5

This example may help you to come to an understanding of why and how this shortcut works since you can compare it with what we did earlier in the long conversion method. We begin by dividing the given number 359 by the desired base number 5 and finding the quotient and the remainder; then we’ll repeatedly divide the ensuing quotients by 5 and obtain new quotients and remainders until we reach a quotient of zero. (When the quotient becomes less than the base number 5, 5 is too large to divide into the quotient but we can say that 5 divides into the quotient zero times with a remainder equal to the quotient; you can see this in the illustration below.) 359/5 = 71/5 = 14/5 = 2/5 =

quotient of 71 with a remainder of 4 quotient of 14 with a remainder of 1 quotient of 2 with a remainder of 4 quotient of 0 with a remainder of 2

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

Since we have done our divisions sequentially in a downward manner, one below the other, we now read our remainders in the opposite direction, i.e., upwards: 2414. These remainders, read in the proper order, are the correct digits for the base 5 number that is equal in value to the original base 10 number of 359! 2414 5  35910 Amazing, isn’t it? And somewhat mysterious as well! As mentioned earlier, you will have an opportunity to examine the inner workings of this mysterious method in the Exercises and Projects for Fun and Profit at the end of this section.

Example 8: Convert 39 in base 10 into a number in base 6

Using our new and elegant short conversion method, we simply begin dividing 39 by 6: 39/6 = quotient of 6 with a remainder of 3 6/6 = quotient of 1 with a remainder of 0 1/6 = quotient of 0 with a remainder of 1 When we reach the quotient of 0, the resulting remainders in reverse order (upwards) represent the digits of the converted number in base 6: 103 in base 6 is equal in value to 39 in base 10. “Outstanding!” you might be thinking to yourself, “But what relevance does this have to my daily life??” The answer to your musings will appear in the next section. Stay tuned! ********************

More Healthful Exercise Try your hand at converting the following numbers to other bases by means of the short method (solutions provided on p.155 at the end of the section): a) b) c) d)

Convert 3214 in base 10 to a base 6 number. Convert 157 in base 10 to a base 3 number. Convert 295 in base 10 to a base 4 number. Convert 90 in base 10 to a base 2 number.

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

Exercises and Projects for Fun and Profit Exercises in Analysis Note: See Helps and Hints for Exercises on pp.438–439 in Appendix A. Convert the following numbers in various bases to numbers in base 10:

1. Convert 456 in base 7 to an equivalent number in base 10. 2. Convert 2133 in base 4 to an equivalent number in base 10. 3. Convert 112211 in base 3 to an equivalent number in base 10. 4. Convert 2632 in base 8 to an equivalent number in base 10. 5. Convert 728 in base 9 to an equivalent number in base 10. Convert the following numbers in base 10 to numbers in other bases as indicated:

6. Convert 345 in base 10 to an equivalent number in base 4. 7. Convert 231 in base 10 to an equivalent number in base 6. 8. Convert 737 in base 10 to an equivalent number in base 5. 9. Convert 93 in base 10 to an equivalent number in base 2. 10. Convert 230 in base 10 to an equivalent number in base 3. Note concerning the typing of superscripts: The “^” symbol conventionally represents the concept of “raising to the power of” so that 5^2 represents 5 raised to the power of 2 which equals 5·5=25. The “^” symbol is generally located above one of the number keys on your computer keyboard; you need to press the Shift key to access it.

If you have Microsoft's Equation Editor installed on your computer, you may use that to type official superscripts. To type the expression 5 2 using an actual superscript, position your cursor where you want the expression to be located, then click on Insert at the top of your Word document. Click on Object and then scroll down the list and select Microsoft Equation 3.0. This will bring up a textbox and a Tool Palette. In the textbox (where the blinking cursor is located), type the digit 5. Then click on the third box on the bottom row in the Tool Palette. When you hold you mouse over this box, it will say “Subscript and superscript templates.” Click on this box and then click on the top left entry in the dropdown menu. This will allow you to enter the exponent 2 in the textbox. To finish, click outside of and to the right of the textbox to close the box and the Tool Palette. If you need to edit the base and/or exponent later, double click on the image to reopen the textbox and Tool Palette and then click inside the textbox to edit.

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

Further Investigation 11. The Mysterious Short Method: Examine carefully the workings of the short conversion method as described in this section. Using your powers of observation and your analytical reasoning skills, describe in detail why this method works and produces correct results. You might begin by explaining exactly what is happening in each step and what each quotient and remainder represents. Please type your answer in paragraph form and give an example to illustrate your remarks.

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Creative Project: The Banker’s Dilemma

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

THE BANKER’S DILEMMA Once upon a time, a wealthy man decided to deposit some of his money in a local bank. Being mathematically inclined and of the belief that bankers also should be adept at math, he decided to present a challenge to the banker and his staff. His challenge was as follows: He brought to the bank a sum of paper money consisting of bills of the same denomination along with ten empty money boxes equipped with locks but no keys. (Each box was presently open but once an amount of money was put into it and it was shut, no one could open it except the wealthy man.) His instructions to the banker were to distribute the money among the boxes by that evening at 9:00 p.m. (and then shut and lock them) in such a way that the banker could at some future time give the wealthy man whatever amount of money he requested without opening any of the boxes. In other words, the wealthy man could request a withdrawal of any amount of dollars between the minimum value of the given paper money denomination (i.e., \$1, \$5, \$10 or \$20) and his total amount of money, inclusive, and the banker would only be able to pick up some number of the locked boxes and hand them to him without opening them or changing the amount of money in them. The wealthy man will ask for an EXACT amount (such as \$280) and the banker will need to give him EXACTLY THAT AMOUNT—NO MORE and NO LESS. The perplexed banker stood for quite a while staring blankly at the boxes, having no idea as to how to proceed. No matter how he planned it, it didn’t seem to work out. Time wore on and the day grew late and still the banker had no solution. He was beginning to wonder whether a solution existed or whether this was some sort of cruel hoax initiated by the wealthy man. Assignments:

Before things get too much worse, can you please help the distraught banker resolve this dilemma? For each assignment, please determine how to distribute the indicated deposit into the ten money boxes so as to satisfy the wealthy man’s request (see Hints and Additional Hints provided on pp.153–154): assignment Assignment #1 Assignment #2 Assignment #3 Assignment #4

amount of deposit \$1000 in the form of 1000 \$1 bills \$5000 in the form of 1000 \$5 bills \$10,000 in the form of 1000 \$10 bills \$20,000 in the form of 1000 \$20 bills

Project Extension: There are actually a number (albeit limited) of different solutions to this dilemma. For each assignment, find as many of these solutions as possible.

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

I would encourage you to pursue this with persistence, since persistence is a necessary ingredient for obtaining a solution. However, if you are at your wit’s end, you may read the hint below… 

******************** Hint for the Banker’s Dilemma (Assignment #1):

Use the same strategy as described in the Hint for Assignment #1. If the banker asks for \$5 (the minimum amount using \$5 bills), then you must have a box with exactly \$5 in it. If the banker asks for \$10, then you must either have a second box with \$5 in it or have a second box with \$10 in it. Which is the better choice in terms of efficiency? And so on… Hint for the Banker’s Dilemma (Assignment #3):

Use the same strategy as described in the Hint for Assignment #1. If the banker asks for \$10 (the minimum amount using \$10 bills), then you must have a box with exactly \$10 in it. If the banker asks for \$20, then you must either have a second box with \$10 in it or have a second box with \$20 in it. Which is the better choice in terms of efficiency? And so on… Hint for the Banker’s Dilemma (Assignment #4):

Use the same strategy as described in the Hint for Assignment #1. If the banker asks for \$20 (the minimum amount using \$20 bills), then you must have a box with exactly \$20 in it. If the banker asks for \$40, then you must either have a second box with \$20 in it or have a second box with \$40 in it. Which is the better choice in terms of efficiency? And so on… 153 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

Additional Hint for the Banker’s Dilemma (Assignment #1):

The secret is to be ruthlessly efficient. In other words, do NOT put an amount into a box if you can already construct that amount from your previous boxes. To begin, you need a box containing \$1 in case the wealthy man asks for exactly \$1. Do you need a box with \$2 in it? Now you have a choice: either you create a box with exactly \$2 in it or you create another box with \$1 in it. The first choice is better since it is more efficient as you'll see below. So now you have a box containing \$1 and a box containing \$2. Do you need a box with \$3 in it? The answer is No, because you could give the wealthy man the first two boxes if he asked for exactly \$3. (This wouldn't be the case if you made two boxes with \$1 in each of them.) Do you need a box with \$4? Yes. Do you need a box with \$5? I'll let you decide, and then continue with that logical reasoning process for the rest of the boxes. Hope this helps!  Additional Hint for the Banker’s Dilemma (Assignment #2):

The secret is to be ruthlessly efficient. In other words, do NOT put an amount into a box if you can already construct that amount from your previous boxes. To begin, you need a box containing \$5 in case the wealthy man asks for exactly \$5. It will be more efficient to next have a box containing \$10. You don’t need a box with \$15 since you can give the man both of the first two boxes if he asks for \$15. Do you need a box with \$20? Yes. Do you need a box with \$25? I’ll let you decide and then continue on with your logical reasoning process. Additional Hint for the Banker’s Dilemma (Assignment #3):

The secret is to be ruthlessly efficient. In other words, do NOT put an amount into a box if you can already construct that amount from your previous boxes. To begin, you need a box containing \$10 in case the wealthy man asks for exactly \$10. It will be more efficient to next have a box containing \$20. You don’t need a box with \$30 since you can give the man both of the first two boxes if he asks for \$30. Do you need a box with \$40? Yes. Do you need a box with \$50? I’ll let you decide and then continue on with your logical reasoning process. Additional Hint for the Banker’s Dilemma (Assignment #4):

Please follow the Additional Hint for Assignment #3 and use \$20 as the minimal amount. ******************** The Banker’s Dilemma is a true challenge—the solution at first seems quite impossible but with enough persistence you, too, may arrive at the solution. At that point, you will have encountered a “Eureka!” moment and will have reaped the rewards of your diligence and persistence. Best wishes for your complete success!

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Math Is Everywhere! Explore and Discover It! 4.1 Understanding Other Number Bases: Places of Value

Answers to Healthful Exercises in This Section p.145 Some Healthful Exercise Solutions

a) 231 in base 4 = 45 as a base 10 value (2 · 4 2 )+(3 · 41 )+(1 · 4 0 ) = (2·16)+(3·4)+(1·1) = 32+12+1 = 45 b) 572 in base 9 = 470 as a base 10 value (5 · 9 2 )+(7 · 91 )+(2 · 9 0 ) = (5·81)+(7·9)+(2·1) = 405+63+2 = 470 c) 2345 in base 6 = 569 as a base 10 value (2 · 6 3 )+(3 · 6 2 )+(4 · 61 )+(5 · 6 0 ) = (2·216)+(3·36)+(4·6)+(5·1) = 432+108+24+5 = 569 d) 10111 in base 2 = 23 as a base 10 value (1 · 2 4 )+(0 · 2 3 )+(1 · 2 2 )+(1 · 21 )+(1 · 2 0 ) = (1·16)+(0·8)+(1·4)+(1·2)+(1·1) = 16+0+4+2+1 = 23 p.149 More Healthful Exercise Solutions (using the short method)

a) 3214 = 22514 in base 6 3214/6=535 r4; 535/6=89 r1; 89/6=14 r5; 14/6=2 r2; 2/6=0 r2 Reading the remainders in reverse order from right to left and recording them gives 22514 in base 6 b) 157 = 12211 in base 3 157/3=52 r1; 52/3=17 r1; 17/3=5 r2; 5/3=1 r2; 1/3=0 r1 Reading the remainders in reverse order from right to left and recording them gives 12211 in base 3 c) 295 = 10213 in base 4 295/4=73 r3; 73/4=18 r1; 18/4=4 r2; 4/4=1 r0; 1/4=0 r1 Reading the remainders in reverse order from right to left and recording them gives 10213 in base 4 d) 90 = 1011010 in base 2 90/2=45 r0; 45/2=22 r1; 22/2=11 r0; 11/2=5 r1; 5/2=2 r1; 2/2=1 r0; 1/2=0 r1 Reading the remainders in reverse order from right to left and recording them gives 1011010 in base 2

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Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers

4.2 The Binary System: Secret Language of Computers “Computers are composed of nothing more than logic gates stretched out to the horizon in a vast numerical irrigation system.” —Stan Augarten

Introduction As you may have noticed in the preceding section, the base 2 number system is quite simple and uses only the digits 0 and 1. This is ideal for applications in which only two states or entities are involved. As the age of technology dawned in the twentieth century and electronic devices with inherent states of “off” and “on” came into their own, it was only natural that the base 2 number system would be applied to these devices in an attempt to model their behaviors. The base 2 number system is more commonly referred to as the binary number system. Claude Shannon, an MIT undergraduate and graduate student in the 1930s, wrote a famous paper entitled A Mathematical Theory of Communication while working at Bell Labs in 1948. In this theory, he showed how information could be transmitted electronically and recovered through the use of mathematical coding theory and some error-correcting codes. The paper marked the beginning of the field of information theory and has been widely acknowledged as one of the shining accomplishments of the twentieth century. Shannon was the first to recognize that all information can be conveyed through the use of binary digits.

Computers and the Binary Number System A computer is basically made up of a large number of electronic switches, some of which are combined to form logic gates so that the computer can make decisions based on information that is given to it as input. In essence, the only things that an electronic switch (and hence a computer) understands are the two electrical states, off and on; these states are conveniently represented by the two binary digits, 0 and 1, respectively. In the early days of computers when magnetic tape was used as a storage device, data was stored as sequences of magnetically off and magnetically on configurations on the magnetically sensitive coating on cellulose tape. (This is the same kind of tape found in old-fashioned audiocassette cartridges.) When computer programmers needed to record data, they naturally chose the binary number system since the digits 0 and 1 readily lent themselves to this application as Claude Shannon had indicated. The digit 0 was chosen to represent the magnetically off state and the digit 1 represented the magnetically on state.

Amazing Speed The amazing speed at which computers operate is due in part to the simplicity of the binary number system and its use in transmitting information. Since all information can be represented by 0’s and 1’s that correspond to electrical off’s and on’s, information can thus travel at the speed of electricity through an electrical circuit on a computer microprocessor. As I’m sure you

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Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers

are aware, the speeds at which these microprocessors operate continue to increase with each new generation of computers. When you stop to think about it, the speed at which computers accomplish their tasks is most impressive. The fact that you can type an email message and click the Send button and have it arrive at its destination computer is pretty amazing, but even more so is the fact that it can arrive on the other side of the world within a few seconds or less! And not just text but full-color graphics as well! If you have searched for information on the Internet using Google.com or some other search engine, you will have seen this same speed in action. A typical search takes only a few seconds or less and in that short space of time can locate millions of webpages that are related to your search expression. This type of speed is truly remarkable!

Information Interchange With regard to representing information as 0’s and 1’s, early computer manufacturers each had their own sets of binary codes for representing information. As you might imagine, this made it impossible for computers with different codes to communicate with each other and it created an enforced and undesirable isolation between computers of different manufacturers. In 1961, Bob Bemer at IBM proposed the creation of a standard information interchange code to supersede the dozens of disparate codes in existence at the time. After much work and collaboration with others, Bemer succeeded in creating and promoting the American Standard Code for Information Interchange (ASCII) code that is now in use around the world. Consequently, a piece of information such as the letter “A” or the word “math” can be encoded and recorded as a recognized and accepted sequence of 0’s and 1’s. Today, we take electronic transmission of information for granted and think little about the underlying processes that make it possible. In the Exercises and Projects for Fun and Profit that follow, we will take a closer look at the means by which computers handle and process information and, through a project involving the ASCII code, how your word processor works.

********************

Exercises and Projects for Fun and Profit Creative Projects: Chinese Trigrams and Binary Numbers The ASCII Code Course Logo: A Hidden Message 157 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers

CHINESE TRIGRAMS AND BINARY NUMBERS In ancient China, a system of broken and solid lines was devised to convey the universal images of life; these lines were assembled in groups of three and were known as trigrams. This system of lines can easily be interpreted in terms of its binary structure, i.e., the fact that there are only two distinct symbols in the system—a solid line and a broken line. In this project, we’ll investigate the binary connection. Each trigram is made up of three symbols, where each symbol is either one solid line or one broken line consisting of two sections. If you interpret each solid line as the binary digit “1” and each broken line as the binary digit “0” (i.e., each PAIR of broken pieces taken together represents the single binary digit “0”), then each trigram contains three binary digits that represent a binary number. This binary number may then be converted into a decimal number.  

A solid line, __________ , represents the binary digit 1. A broken line, ____ ____ , represents the binary digit 0.

******************** Assignments: In each of the assignments below, read each of the trigrams and determine both its binary value and its decimal value. Read the lines as binary digits beginning at the top of each trigram and proceeding downwards. Example: In Assignment #1, the first trigram consists of a broken line followed by two solid lines (when read from top to bottom). This represents the three-digit binary number 011 (in base 2). Using expanded notation in base 2, this number has the decimal value of: (0  2 2 )  (1  21 )  (1  2 0 ) = (0 · 4) + (1 · 2) + (1 · 1) = 0 + 2 + 1 = 3 The objective here is to decode each trigram as if it were a three-digit binary number and then convert the binary number to a decimal value by using expanded notation; you should have eight three-digit binary numbers and their single-digit decimal equivalents when you are finished. ********************

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Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers

For each trigram, follow the example on the preceding page to accomplish the following:  read the three symbols from the top down and convert each solid line or pair of broken lines to a binary digit to obtain a three-digit binary number  use expanded notation to convert each three-digit binary number to an equivalent singledigit decimal number Note: The columns are arranged vertically and each assignment consists of eight trigrams in its respective column. For each assignment, please convert each of the eight trigrams (a through h) and record your answers in the proper order. trigram a.

Assignment #1 _____ _____ _____________ _____________

Assignment #2 _____________ _____ _____ _____________

Assignment #3 _____________ _____________ _____________

Assignment #4 _____ _____ _____ _____ _____________

b.

_____________ _____________ _____________

_____ _____ _____________ _____ _____

_____ _____ _____ _____ _____________

_____________ _____________ _____ _____

c.

_____ _____ _____________ _____ _____

_____________ _____________ _____ _____

_____________ _____ _____ _____ _____

_____ _____ _____________ _____________

d.

_____________ _____ _____ _____ _____

_____ _____ _____________ _____________

_____________ _____________ _____ _____

_____ _____ _____

e.

_____ _____ _____ _____ _____________

_____ _____ _____

_____ _____ _____

_____ _____ _____________ _____ _____

_____________ _____ _____ _____________

f.

_____________ _____________ _____ _____

_____ _____ _____ _____ _____________

_____________ _____ _____ _____________

_____ _____ _____________ _____ _____

g.

_____________ _____ _____ _____________

_____________ _____ _____ _____ _____

_____ _____ _____

_____ _____ _____

_____________ _____________ _____________

h.

_____ _____ _____

_____________ _____________ _____________

_____ _____ _____________ _____________

_____________ _____ _____ _____ _____

_____ _____ _____

_____ _____ _____

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Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers

THE ASCII CODE American Standard Code for Information Interchange Introduction When computer scientists first began using the digits 0 and 1 to encode information, there arose a need for a standardized coding system so that all computers would “speak the same language.” Eventually, the ASCII character code was agreed upon. The ASCII code uses decimal numbers to represent keyboard characters such as letters and numbers as well as special characters like the comma, period, etc. A representation of this code can be found on p.165; note that both uppercase and lowercase letters are included in the character set. ASCII code decimal numbers (in base 10) are then converted into binary numbers (in base 2). This binary (or base 2) coding system was chosen since it was such a natural way to represent the electromagnetic states of “on” and “off”—the only information that a computer really understands. (As smart as they appear to be, computers can’t read words or numbers—they can only sense electrical currents and their effects.) Due to the fact that the standard computer keyboard contains 94 characters, a set of at least 94 binary numbers is required in order to form a one-to-one coding system in which each binary number represents a unique keyboard character. The largest six-digit binary number is 111111; in expanded notation, this binary number represents the decimal number 63: (1 · 25) + (1 · 24) + (1 · 23) + (1 · 22) + (1 · 21) + (1 · 20) = (1 · 32) + (1 · 16) + (1 · 8) + (1 · 4) + (1 · 2) + (1 · 1) = 32 + 16 + 8 + 4 + 2 + 1 = 63 This means that six-digit binary numbers can represent 64 values: the numbers from 1 through 63 and the number 0 that is represented by the six-digit binary number, 000000. Since 64 binary numbers are not enough to accommodate all 94 keyboard characters, larger 7-digit binary numbers are needed to make up enough decimal number equivalents to cover the 94 standard characters. Since the largest 7-digit binary number is 1111111, which is equivalent to 127 as a decimal number, seven-digit binary numbers can represent 128 values: the numbers from 1 through 127 and the number 0. Computer designers decided to use the extra 34 (128 – 94 = 34) binary numbers for special control characters to control devices such as printers.

Encoding Keyboard Characters as Decimal Numbers and Again as Binary Numbers Each letter, number and character on a computer keyboard has its own coded decimal representation as well as its binary representation equivalent. As an example, the letter “C” has a decimal code value of 67 and a binary number representation of 1000011. 160 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers

The process of encoding a keyboard character as a binary number consists of two steps: Step 1: Encode a Keyboard Character as a Decimal Number To encode a keyboard character as a decimal number, you begin by locating the character in the ASCII table (see table on p.165 at the end of this project) and determining its row and column numbers. You then append the column number to the end of the row number to form a decimal number (i.e., an integer value in our everyday base 10 number system); this decimal number is the character’s decimal code. Examples:  The letter “C” is found in row 6 and column 7; its decimal code is 67.  The number “5” is found in row 5 and column 3; its decimal code is 53.  The letter “p” is found in row 11 and column 2; its decimal code is 112. Step 2: Convert a Character’s Decimal Code into a Binary Number To convert a decimal code number into a binary number (i.e., a number in base 2), we can use the elegant and mysterious short method described earlier in this section. As an example of this method, we’ll convert the decimal code for “C”, 67, into a binary number by successively dividing by 2 and keeping track of the remainders: Note: These remainders are NOT the remainders found by dividing with a calculator. The calculator converts integer remainders into decimal parts of a whole unit. To get the proper remainders, you need to perform pencil and paper division (as you used to do in elementary and middle school). 67/2 = quotient of 33 with a remainder of 1 33/2 = quotient of 16 with a remainder of 1 16/2 = quotient of 8 with a remainder of 0 8/2 = quotient of 4 with a remainder of 0 4/2 = quotient of 2 with a remainder of 0 2/2 = quotient of 1 with a remainder of 0 1/2 = quotient of 0 with a remainder of 1 Since we divided in a downward direction, we need to read our results in an upward direction in order to obtain the correct result: 67 = 1000011 in base 2. Note: The process is complete when you have obtained your first quotient of zero. 2 is too large to divide into 1 and obtain a positive whole number quotient but we can correctly say that the quotient is 0 and the remainder is 1. ********************

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Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers

Example 1: To demonstrate the entire process, we’ll encode the letter “p” as a decimal value and then convert that decimal value into a binary number. Step 1: From the ASCII table, “p” has a decimal value of 112 since it is in row 11 and column 2; 2 appended to the end of 11 results in the decimal value 112. Step 2: To convert 112 into a binary number, we’ll use our elegant and mysterious short method once again to successively divide by 2 and keep track of the remainders: 112/2 = quotient of 56 with a remainder of 0 56/2 = quotient of 28 with a remainder of 0 28/2 = quotient of 14 with a remainder of 0 14/2 = quotient of 7 with a remainder of 0 7/2 = quotient of 3 with a remainder of 1 3/2 = quotient of 1 with a remainder of 1 1/2 = quotient of 0 with a remainder of 1 We have now obtained a quotient of 0 and the process is finished. Since we divided in a downward direction, we’ll read our results in an upward direction: 112 = 1110000 in base 2. **********

Adding Check Digits for Accuracy and Error Detection As a way to check for electronic transmission errors, computer designers decided to add an eighth digit to each binary number to serve as a measure of the parity of the number of 1’s that the number contains (parity is the state of being either “odd” or “even”). This digit is known as a check digit and is placed at either the beginning or the end of the 7-digit binary number.  

If there is an even number of 1’s in the binary number, then the check digit is 0. If there is an odd number of 1’s in the binary number, then the check digit is 1.

Note: For purposes of this project, we will place a check digit at the beginning of a number (in the leftmost position). Examples:  The check digit for the letter “C” is 1 since its binary value, 1000011, contains an odd number of 1’s; the complete 8-digit representation, including the appended check digit, is 11000011.

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Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers 

The check digit for the letter “p” is 1 since its binary value, 1110000, contains an odd number of 1’s; the complete 8-digit representation, including the appended check digit, is 11110000.

Reversing the Process: Decoding Binary Numbers into Keyboard Characters To decode a binary number into its keyboard character, we reverse the above process. Example 2: Decode the binary number 11000011 To begin with, we ignore the leftmost digit, 1, since it is the check digit and is not part of the original binary number. This leaves us with the original 7-digit binary number, 1000011. Step 1: In expanded notation, this number has the value: (1 · 26) + (0 · 25) + (0 · 24) + (0 · 23) + (0 · 22) + (1 · 21) + (1 · 20) = (1 · 64) + (0 · 32) + (0 · 16) + (0 · 8) + (0 · 4) + (1 · 2) + ( (1 · 1) = 64 + 0 + 0 + 0 + 0 + 2 + 1 = 67 Step 2: This indicates that the keyboard character associated with 11000011 is located in row 6 and column 7 of the ASCII table, viz., the letter “C”. ******************** Example 3: Decode the binary value 11110000 We begin by ignoring the leftmost digit, 1, since it is the check digit and is not part of the original binary number. This leaves us with the original 7-digit binary number, 1110000. Step 1: In expanded notation, this number has the value: (1 · 26) + (1 · 25) + (1 · 24) + (0 · 23) + (0 · 22) + (0 · 21) + (0 · 20) = (1 · 64) + (1 · 32) + (1 · 16) + (0 · 8) + (0 · 4) + (0 · 2) + (0 · 1) = 64 + 32 + 16 + 0 + 0 + 0 + 0 = 112 Step 2: This indicates that the keyboard character associated with 11110000 is located in row 11 and column 2 of the ASCII table; viz., the letter “p”.

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Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers

164 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers

Assignment #1

Part I Part II

cryptology 11100010 11100001 11110011 01100101 10110010 01100011 01101111 11100100 01101001 11101110 11100111

Assignment #2

Part I Part II

encodings 01100011 11101000 01100101 01100011 11101011 11100100 01101001 11100111 01101001 01110100 11110011

Assignment #3

Part I Part II

conversions 01101001 01101100 01101001 11101011 01100101 01100011 01101111 11100100 01101001 11101110 11100111

Assignment #4

Part I Part II

algorithms 11100010 01101001 11101110 11100001 01110010 11111001 11110000 01101111 01110111 01100101 01110010

The ASCII Character Set: 0 1 2 3 4 5 6 7 8 9 10 11 12

0 nul lf dc4 rs ( 2 < F P Z d n x

1 soh vt nak us ) 3 = G Q [ e o y

2 stx ff syn sp * 4 > H R \ f p z

3 etx cr etb ! + 5 ? I S ] g q {

4 eot so can “ , 6 @ J T ^ h r |

5 enq si em # 7 A K U _ i s }

6 ack dle sub \$ . 8 B L V ‘ j t ~

7 bel dcl esc % / 9 C M W a k u del

8 bs dc2 fs & 0 : D N X b l v

9 ht dc3 gs ‘ 1 ; E O Y c m w

The numbers in the leftmost column of the table (boldface) represent the left digit(s) of an ASCII decimal code; the numbers in the top row of the table (boldface) represent the rightmost digit of an ASCII decimal code. Examples:  The decimal code for a lowercase “a” is 97 (row 9, column 7)  The decimal code for a lowercase “z” is 122 (row 12, column 2) Each of these decimal codes is found by appending the character’s column number to the end of the character’s row number.

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Math Is Everywhere! Explore and Discover It! 4.2 The Binary System: Secret Language of Computers

COURSE LOGO: A HIDDEN MESSAGE Using your newly acquired skills involving the ASCII code, decode the encrypted message contained within the logo for this textbook and course of study as shown below. (You will need to complete the ASCII Code project on pp.160–165 before beginning this project.) IMPORTANT: To read the binary numbers in their correct orientation, please imagine that you are standing in the center of the logo and rotating in a clockwise direction as you read each binary number; begin in the top left-hand corner with the number 1001101. Note: As you rotate in a clockwise direction at the center of the logo and read each binary number from left to right, you’ll actually be reading the numbers at the bottom of the logo backwards from the way that you would normally read them. Note: There are no check digits in the coded binary numbers below; each digit in each binary number needs to be used in the conversion calculations. When you are finished, you will have decoded the binary numbers into an English-language sentence that contains a hidden message. ******************** Enjoy the challenge! 

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Math Is Everywhere! Explore and Discover It! 4.3 Number Base Applications: More Fun with Math!

4.3 Number Base Applications: More Fun with Math! “In symbols, one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.” —Gottfried Wilhelm Leibniz

Introduction While number bases other than our own familiar base 10 may seem a bit arcane, they really do have applications in our daily lives although their presence is generally hidden behind the scenes. The hexadecimal number system, otherwise known as base 16, is used primarily in computer applications; in particular, hexadecimal values are used in the construction of color codes for webpages. In this section, you will learn about the “basics” of the base 16 number system (pardon the play on words! ); you may then use this knowledge in completing the Creative Project entitled Hexadecimals in the Exercises for Fun and Profit at the end of this section.

The Ordinary and Familiar Base 10 To develop an understanding of base 16, we’ll begin with something that we all know well: the decimal number system, otherwise known as the base 10 number system. In this system, there are ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9; in addition, there is a specific place value assigned to each digit in a decimal number. For example, the decimal number 329 (in base 10) represents the sum of the following set of products in expanded notation: (3·100) + (2·10) + (9·1) = 300 + 20 + 9 = 329 We could also have written this expanded notation by using powers of the base, i.e., powers of 10: (3  10 2 )  (2  101 )  (9  10 0 ) = 300 + 20 + 9 = 329 Note: Any positive number raised to the 0 power is equal to 1; hence, 10 0  1 .

Notice that the position of each digit in the number 329 represents a different place value; there is only one digit allowed in each place-value position. Notice also that each place value is a power of the base, beginning with the zero power in the rightmost position, and that the powers increase by one in each successive position to the left.

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Math Is Everywhere! Explore and Discover It! 4.3 Number Base Applications: More Fun with Math!

place value digit decimal value of position

3rd position 10 2 =100 3

2nd position 101 =10 2

1st position 10 0 =1 9

3·100=300

2·10=20

9·1=9

The Unfamiliar Base 16 with Familiar Structure This behavior is typical of modern place-value number systems, including base 16; in these systems, each place value depends on the given base. In base 16, the number 329 would have the following structure:

place value digit decimal value of position

3rd position 16 2 =256 3

2nd position 161 =16 2

1st position 16 0 =1 9

3·256=768

2·16=32

9·1=9

Notice that the place values are powers of 16 rather than 10; the actual value of 329 in base 16 is thus quite different than 329 in base 10. We can calculate the equivalent decimal value of 329 in base 16 by multiplying each base 16 digit by its respective place value and adding the products together: (3· 16 2 ) + (2· 161 ) + (9· 16 0 ) = (3·256) + (2·16) + (9·1) = 768 + 32 + 9 = 809 As you can see, 329 in base 16 is equivalent to the base 10 number 809 where 809 = (8· 10 2 ) + (0· 101 ) + (9· 10 0 ) = (8·100) + (0·10) + (9·1) As outlined above, the process of using expanded notation to evaluate a base 16 number produces a decimal value in our familiar base 10 system. This process is simple and straightforward and is illustrated further in the following examples: Example 1: 76 in base 16 76 in base 16 = (7· 161 ) + (6· 16 0 ) = (7·16) + (6·1) = 112 + 6 = 118 in base 10

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Math Is Everywhere! Explore and Discover It! 4.3 Number Base Applications: More Fun with Math!

Example 2: 584 in base 16 584 in base 16 = (5· 16 2 ) + (8· 161 ) + (4· 16 0 ) = (5·256) + (8·16) + (4·1) = 1280 + 128 + 4 = 1412 in base 10

Base 16 and its Unusual Alphabetic “Digits” In all of the above examples involving base 16 numbers, each digit occupied a unique placevalue position; this is a simple and efficient means of avoiding confusion and is a mathematical convention when writing numerical values. A problem presents itself, however, when using number bases greater than 10. In base 16, for example, the place values of the first few positions, often referred to as columns, are as follows:

place value

3rd column 16 2 =256

2nd column 161 =16

1st column 16 0 =1

When attempting to record the decimal number 12 as a number in base 16, the dilemma is that 12 is too small in value to be carried over to the second column and therefore must be recorded entirely in the first column:

place value digits

3rd column 16 2 =256

2nd column 161 =16

1st column 16 0 =1 12

Unfortunately, someone reading this base 16 number, 12, might erroneously assume that the “1” was in the second column and that only the “2” was in the first column, resulting in a different value than was intended, i.e., (1·16) + (2·1) = 16 + 2 = 18. To avoid this source of confusion, mathematicians invented a simple coding system for number bases greater than 10. This coding system replaces each double-digit value less than the value of the base with a single symbol. The alphanumeric code for base 16 is as follows: value 10 11 12 13 14 15

symbol code A B C D E F

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Math Is Everywhere! Explore and Discover It! 4.3 Number Base Applications: More Fun with Math!

 

Using this code, the base 16 number 12 may be written as: C (in base 16) Likewise, the base 16 number 15 may be written as: F (in base 16)

Note: In base 16, the number 16 is written as 10 (in base 16) and therefore doesn’t need an alphanumeric code. 10 in base 16 is equivalent to (1·16) + (0·1) = 16 + 0 = 16 in base 10.

In other number bases, the digits are as follows; note the encoded digits in bases greater than 10: number base system 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

number of digits 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

digits

0,1 0,1,2 0,1,2,3 0,1,2,3,4 0,1,2,3,4,5 0,1,2,3,4,5,6 0,1,2,3,4,5,6,7 0,1,2,3,4,5,6,7,8 0,1,2,3,4,5,6,7,8,9 0,1,2,3,4,5,6,7,8,9,A 0,1,2,3,4,5,6,7,8,9,A,B 0,1,2,3,4,5,6,7,8,9,A,B,C 0,1,2,3,4,5,6,7,8,9,A,B,C,D 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

Decoding and Converting in Base 16 Interpreting some base 16 numbers thus involves both alphanumeric decoding and the use of the base 16 place-value system. Some examples follow: Example 3:

3 B in base 16 3 B = (3 · 161 ) + (B · 16 0 ) = (3·16) + (11·1) = 48 + 11 = 59 in base 10

Example 4:

A 7 in base 16 A 7 = (A · 161 ) + (7 · 16 0 ) = (10·16) + (7·1) = 160 + 7 = 167 in base 10

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Math Is Everywhere! Explore and Discover It! 4.3 Number Base Applications: More Fun with Math!

Example 5:

F C in base 16 F C = (F · 161 ) + (C · 16 0 ) = (15·16) + (12·1) = 240 + 12 = 252 in base 10

Example 6:

E 9 D in base 16 E 9 D = (E · 16 2 ) + (9 · 161 ) + (D · 16 0 ) = (14·256) + (9·16) + (13·1) = 3584 + 144 + 13 = 3741 in base 10

Conclusion Now that you have learned the basic structure of base 16 numbers and the methods by which to decode and convert these numbers into decimal (base 10) values, you may proceed with the Hexadecimals project. In this project, you will investigate the use of base 16 values in color codes for webpages and will be able to determine how much of each color is being combined to form the resulting color display.

******************** It’s an interesting project and one that you may eventually put to personal use if and when you design your own webpages. Incidentally, some Internet service providers such as Tampa Bay Roadrunner offer free personal website hosting as part of their service agreement for providing high-speed Internet access via cable modem. In any event, have fun! 

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Exercises and Projects for Fun and Profit Creative Project: Hexadecimals

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Math Is Everywhere! Explore and Discover It! 4.3 Number Base Applications: More Fun with Math!

HEXADECIMALS A BASE 16 APPLICATION In this technological age that we live in, several number bases have gained prominence in terms of use in applications. Base 16 is one of these and has been employed by the computer industry to define webpage color codes. Each hexadecimal color code contains six digits and/or characters in base 16, e.g., #38FC9A. Each color on your computer screen is a mixture of the colors red, green and blue, sometimes referred to as the RGB color base. Computers can now represent many millions of colors by mixing various amounts of these three colors. Assignments:

Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on hexadecimal color codes and determine how the amounts of each color are encoded using base 16, the hexadecimal number base. Afterwards, please answer the following three questions:

1. Describe what each pair of digits in a color code represents. Be specific—there are three pairs of digits/characters involved and you should describe the role of each pair. 2. Determine which color codes are used for the following colors (i.e., which base 16 digits and/or characters are used to represent the following standard colors) and include these color codes in your report:     

Red (pure red, not a mixture) Green (pure green, not a mixture) Blue (pure blue, not a mixture) Black White

3. For each assignment in the table below, please calculate the amounts (number of units in base 10) of red, green and blue used in the following webpage colors (see Examples 3, 4 and 5 on pp.170–171 for converting a pair of hexadecimal characters into base 10 values):

a. b. c.

Assignment #1 # B47FA9 # AABB66 # DD33EE

Assignment #3 #C85D69 #2ECF53 #9966FF

Assignment #4 #FF0066 #25D6A8 #FF99CC

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Math Is Everywhere! Explore and Discover It! 5.0 Introduction and Objectives

Chapter Five Modulus Arithmetic and Its Many Uses 5.0 Introduction and Objectives The thumbnail sketches on this page represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

5.1 Clock Systems: The Cycles of Life

5.2 The Modulus in Action: Check Digits and Error Detection International Standard Book Numbers (ISBN): 978-1-85326-058-2

978-0-7382-0442-0

5.3 Cryptography: The Mathematics of Privacy Secret messages: TBIZLJB QL ZOVMQLDOXMEV

33 15 9 10 24

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Math Is Everywhere! Explore and Discover It! 5.0 Introduction and Objectives

Learning Objectives for Chapter 5 1. To enable students to develop their analytical reasoning skills through pattern recognition and problem solving. 2. To enable students to develop their deductive and inductive reasoning skills through encoding and decoding various codes and ciphers. 3. To allow students to apply their knowledge of cryptography to construct codes and ciphers. 4. To allow students to exercise their creativity by conducting Internet research and writing reports on various mathematical topics. 5. To enable students to gain familiarity with Web-based mathematical resources and to develop their critical thinking skills in assessing and evaluating these resources. 6. To demonstrate the usefulness of mathematics through several practical applications involving modulus arithmetic. 7. To engage students in active learning through the use of projects and activities. 8. To enable students to develop confidence in their mathematical abilities. 9. To empower students to become lifelong learners.

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Math Is Everywhere! Explore and Discover It! 5.1 Clock Systems: The Cycles of Life

5.1 Clock Systems: The Cycles of Life Introduction All of us are familiar with the concept of keeping time and have read clock faces and “told time” since our kindergarten days. However, few are aware of how mathematically valuable this “clock” concept is and in how many different applications it is used. In this section we will examine clock systems and their arithmetic properties and then later in this chapter we will investigate some of their many applications. Representing time on a clock involves dividing the day into equal segments and numbering them sequentially. Traditionally, we have used twelve equal divisions on a clock face rather than twenty-four, since the twelve divisions can easily be used twice: once to represent the morning hours and once again to represent the afternoon and evening hours. Hence we have clock faces containing the numbers (or sometimes Roman numerals or other markings) from one to twelve.

The Modulus: A Very Useful System When the time of day is after noon, we begin counting from one to twelve again so that the thirteenth hour of the day is equivalent to 1:00 p.m., the fourteenth hour of the day is equivalent to 2:00 p.m., the fifteenth hour of the day is equivalent to 3:00 p.m., and so on. In mathematics, this cyclic behavior represents a modulus and in this case we would say that our clock represents a modulus 12 system in which one complete cycle contains twelve equal divisions and each division represents one hour. We could have also represented time in a different modulus such as modulus 6. If this were the case, our clock faces would contain the numerals from one to six, and we would use them four times each in the course of a twenty-four-hour day. Of course, our a.m./p.m. (antemeridian/ postmeridian) dichotomy would have to be replaced with four distinctive labels in order to allow us to distinguish to which part of the day a clock time referred. We could use something like early a.m., late a.m., early p.m. and late p.m., and use abbreviations like e.a.m, l.a.m, e.p.m and l.p.m, but as you can easily see, this is more cumbersome and complicated than is necessary. A time like 9 a.m., for example, would become 3 l.a.m and 10 p.m. would become 4 l.p.m in this modulus 6 system. ******************** Aren’t you glad that we don’t use modulus 6?  ********************

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Math Is Everywhere! Explore and Discover It! 5.1 Clock Systems: The Cycles of Life

Military Time Military clocks are even simpler than this and are based on a modulus 24 system in which a complete cycle contains 24 divisions and the hours of the day are numbered from 0 to 23 rather than from 1 to 12. For military personnel, 1:00 p.m. is equivalent to 13 o’clock and 3:00 p.m. is 15 o’clock (or more precisely, 1300 hours and 1500 hours, respectively). Military time is indicated by a series of four digits where the first two represent the hour of the day and the last two represent the number of minutes past the hour. Note: Leading zeros are used for the early morning hours that contain only one digit. Some conventional times and military equivalents are shown below: conventional time (modulus 12) 12:30 a.m. 7:00 a.m. 10:30 a.m. 1:00 p.m. 7:20 p.m. 11:45 p.m.

military time (modulus 24) 0030 hours 0700 hours 1030 hours 1300 hours 1920 hours 2345 hours

The military clock begins again after a span of twenty-four hours. In order to convert military time into our conventional modulus 12 time, we need to subtract twelve hours (from midnight to noon) from a given military time that is greater than 1200 hours. Example 1:

1600 hours = 16 hours and 0 minutes past midnight – 12 hours 4 hours and 0 minutes past noon 1600 hours is equivalent to 4:00 p.m.

Example 2:

2040 hours = 20 hours and 40 minutes past midnight – 12 hours 8 hours and 40 minutes past noon 2040 hours is equivalent to 8:40 p.m.

For both conventional and military clocks, it is the cyclic nature of the system that gives it its usefulness. Assuredly, we wouldn’t want to continually count the hours of a week or month or year without reverting to some sort of cyclic system; I can’t imagine saying that it is 124 o’clock or 2,345 o’clock, can you?

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Math Is Everywhere! Explore and Discover It! 5.1 Clock Systems: The Cycles of Life

Modulus Conversions: Clock Systems of Various Sizes In a given modulus, there is only a limited set of valid numerical values; all numbers that are larger than the modulus may be converted (reduced) to equivalent values within the modulus. For example, if we choose to work with modulus 12, as in clock faces, then we have twelve valid numerical values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Note: In mathematical applications, we choose to begin our set of values with the number zero so that in modulus 12 the twelve valid numerical values would be: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11; the inclusion of zero makes the modulus easier to work with in applications. Now that we have the basic ideas, we can select a modulus and convert other numerical values into values that belong to the modulus. In modulus 7, for example, the values are 0, 1, 2, 3, 4, 5 and 6, and some conversions using modulus 7 are as follows: 0 is equivalent to 0 in modulus 7 1 is equivalent to 1 in modulus 7 2 is equivalent to 2 in modulus 7 3 is equivalent to 3 in modulus 7 4 is equivalent to 4 in modulus 7 5 is equivalent to 5 in modulus 7 6 is equivalent to 6 in modulus 7 7 is equivalent to 0 in modulus 7 since it is the value at which we begin counting again 8 is equivalent to 1 in modulus 7 since 8 – 7 = 1 (1 full cycle of 7 plus 1 unit left over) 9 is equivalent to 2 in modulus 7 since 9 – 7 = 2 (1 full cycle of 7 plus 2 units left over) and so on…. Larger values and values in other moduli (plural of modulus) may also be converted:  34 is equivalent to 6 in modulus 7 since 34 contains four full cycles of 7 units and has 6 units left over (34 – 28 = 6)  37 is equivalent to 2 in modulus 7 since 37 contains five full cycles of 7 units and has 2 units left over (37 – 35 = 2)  15 is equivalent to 3 in modulus 6 since 15 contains two full cycles of 6 and has 3 units left over (15 – 12 = 3)  59 is equivalent to 5 in modulus 9 since 59 contains six full cycles of 9 units and has 5 units left over (59 – 54 = 5) Essentially, then, the equivalent modulus value for a number can be obtained by dividing the number by the modulus and obtaining the integer (whole number) remainder; the integer remainder is the correct equivalent value in the selected modulus. Note: Mathematicians generally abbreviate the word modulus with the term mod and abbreviate the expression 2 in modulus 7 by 2(mod 7).

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Math Is Everywhere! Explore and Discover It! 5.1 Clock Systems: The Cycles of Life

Examples: The bulleted examples above may be more concisely stated as follows: 34/7 = 4 with a remainder of 6, so 34 is equivalent to 6 (mod 7) 37/7 = 5 with a remainder of 2, so 37 is equivalent to 2 (mod 7) 15/6 = 2 with a remainder of 3, so 15 is equivalent to 3 (mod 6) 59/9 = 6 with a remainder of 5, so 59 is equivalent to 5 (mod 9) Note: When we say “remainder,” we mean the integer remainder found with pencil and paper division that doesn’t involve the use of decimal places. The decimal part of a calculator quotient involves a fractional value for the remainder whereas we want to work with the actual whole number remainder.

15 ) on your calculator, you will obtain a result of 6 2.5. The remainder needed in this case is NOT 0.5! 0.5 is the decimal part of the quotient and represents a fractional value based on the number of decimal places shown. In this case, the fractional value is 5/10, and the complete quotient in the division of 15 by 6 is: 2 and 5/10. The decimal part of the quotient is a very different concept than that of a remainder as we normally consider it. It helps to think back to those long-ago days when you first learned to divide (without the use of a calculator, and before you learned about decimals) and you would think like this: 15/6 = 2 with a remainder (i.e., a whole number remainder) of 3. If you perform the division 15/6 (i.e.,

********************

Exercises and Projects for Fun and Profit Exercises in Analysis Some Modulus Practice: As a bit of exercise in this area, please convert these military clock hours into standard, conventional times based on a 12-hour clock system. In military time, the first two digits represent the number of hours past midnight; the second two digits represent the number of minutes past the hour.

1. 2. 3. 4. 5.

1700 hours 2100 hours 1400 hours 1930 hours 0915 hours

6. 0845 hours 7. 2245 hours 8. 2315 hours 9. 2030 hours 10. 1900 hours

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Math Is Everywhere! Explore and Discover It! 5.1 Clock Systems: The Cycles of Life

More Modulus Practice: Please convert the following numerical values into their equivalent values in each designated modulus (i.e., find the integer remainders after dividing by the modulus).

11. 12. 13. 14. 15.

27 in modulus 4 19 in modulus 6 52 in modulus 10 68 in modulus 7 35 in modulus 5

16. 17. 18. 19. 20.

38 in modulus 6 88 in modulus 7 80 in modulus 8 46 in modulus 4 71 in modulus 12

********************

Creative Project: Perpetual Calendar

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Math Is Everywhere! Explore and Discover It! 5.1 Clock Systems: The Cycles of Life

PERPETUAL CALENDAR Introduction A mathematical algorithm is a procedure consisting of a set of well-defined steps designed to carry out a specific objective. An algorithm for determining the day of the week for any given date is sometimes called a perpetual calendar. The one that follows is based on modular arithmetic and can be applied to any date between January 1, 1900, and December 31, 1999, inclusive. To use this calendar, you must also know that a year is a leap year if it is divisible by 4 unless it is a century year (i.e., 1700, 1800, 1900, or 2000), in which case it is generally NOT a leap year. However, if a century year is evenly divisible by 400 (e.g., 2000) then it IS a leap year. Consequently, in recent history there has been only one century year that was a leap year: 2000. The perpetual calendar algorithm is based on the fact that our normal calendar year consists of 365 days; hence our calendar is a modulus 365 system. In each normal year (not including leap years), there are exactly 52 seven-day weeks and one day left over. This leftover day causes the next year’s days to be “pushed ahead” by a day; e.g., January 1, 1900, was a Monday, but January 1, 1901, fell on a Tuesday. In addition, our seven-day week represents a modulus 7 system. This is quite important in determining the day of the week for a given date in the century as you shall see below. Using the fact that January 1, 1900, was a Monday, we will now construct an algorithm that will allow us to determine the day of the week for any given date in the twentieth century. The idea behind the algorithm is simply to determine how many days beyond Monday we should “push ahead” for a given date in the century.

The Beginnings of the Algorithm To begin, each normal full year of 365 days that passes after the starting date of January 1, 1900, causes the days of the week to be pushed ahead by one day. Obviously, then, it is important to include the number of years beyond 1900 in our determination. 

For the date March 12, 1911, this would mean that 11 full years had passed since January 1, 1900 (the extra 2 months and 12 days will be dealt with later).

In addition, each leap year that passes after January 1, 1900, causes the days of the week after February 29 to be pushed ahead by an extra day. Hence it is important to include the number of leap years beyond 1900 in our determination. A simple way to calculate this figure is to divide the last two digits of a given year by 4; the integer part of the quotient will indicate the number of leap years involved while the quotient’s remainder may be ignored.

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Math Is Everywhere! Explore and Discover It! 5.1 Clock Systems: The Cycles of Life   

For March 12, 1911, this would mean that 2 leap years had passed (i.e., 1904 and 1908; 1900 is not a leap year as noted above). To calculate this, we may divide: 11/4 = 2 with a remainder of 3. The integer quotient, 2, is the number of interest. For March 12, 1912, 12/4 = 3 indicates that 3 leap years have passed. For January 8, 1908, 8/4 = 2 indicates that 2 leap years have passed; however, since January 8 is before the extra day that occurs on February 29, we must somehow adjust our calculations to account for this fact. We will do this when we construct our modulus 7 month keys below. For June 10, 1947, 47/4 = 11 (with a remainder of 3) indicates that 11 leap years have passed. The quotient, 11, is the number of interest; we will ignore the remainder.

Constructing the Month Keys using Modulus 7 While it is true that each date in a year may be sequentially numbered from 1 to 365 (or 366), we often do not have access to this numbering system; most calendars that you might purchase in a bookstore do not include this information. Consequently, we will make use of our knowledge of modulus 7 arithmetic to devise a system for keeping track of how many full weeks have passed for each month of the year and how many “leftover” days remain. Example 1: Dates in April and the important month key for April

We’ll consider the first three months prior to April in a normal year: January February March

31 days 28 days 31 days

31/7 = 4 full weeks and 3 days left over = 3(mod 7) 28/7 = 4 full weeks and 0 days left over = 0(mod 7) 31/7 = 4 full weeks and 3 days left over = 3(mod 7)

To determine how many days we need to “push ahead” for a given date in April, we would need to know the total number of leftover days that the three preceding months had generated. From January through March, there were a total of 12 full weeks and a total of 3+0+3 = 6 days left over. We will now designate these 6 leftover days as the month key for April, i.e., the month key for dates that occur in the month of April.

Example 2: Dates in May and the important month key for May

We’ll consider the first four months prior to May in a normal year: January February March April

31 days 28 days 31 days 30 days

31/7 = 4 full weeks and 3 days left over = 3(mod 7) 28/7 = 4 full weeks and 0 days left over = 0(mod 7) 31/7 = 4 full weeks and 3 days left over = 3(mod 7) 30/7 = 4 full weeks and 2 days left over = 2(mod 7)

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Math Is Everywhere! Explore and Discover It! 5.1 Clock Systems: The Cycles of Life

To determine how many days we need to “push ahead” for a given date in May, we would need to know the total number of leftover days that the four preceding months had generated. From January through April, there were a total of 16 full weeks and a total of 3+0+3+2 = 8 days left over. Since 8 days represents 1 full week and 1 day left over, we may rewrite our total as 17 full weeks and 1 day left over. We will now designate this 1 leftover day as the month key for May, i.e., the month key for dates that occur in the month of May.

The following table contains the months and their respective month keys. Each month key represents the cumulative total of the leftover days from the preceding months where the cumulative total has been converted into a modulus 7 value. Note: Due to the fact that a leap year adds an extra day to a year and “pushes ahead” the days of the week, our calculator algorithm will calculate the pertinent number of leap years preceding a given date and include this number of “pushed ahead” days. However, due to the fact that this “pushing ahead” doesn’t begin until February 29, we must adjust the month keys for dates in January and February by reducing them by 1. The dates from January 1 through February 29 don’t get “pushed ahead” in a leap year. Note: Since January is the first month of the year, its dates don’t get “pushed ahead” by any other preceding months; consequently, its normal month key is 0. In a leap year, reducing the month key by 1 results in a month key that has a value of –1.

Month Keys month

leftover days at the end of the month January 3 February 0 March 3 April 2 May 3 June 2 July 3 August 3 September 2 October 3 November 2 December 3

total cumulative leftover days at the end of the month 3 3 6 8 11 13 16 19 21 24 26 29

total cumulative leftover days (mod 7)

3 (2 if leap year) 3 (2 if leap year) 6 1 4 6 2 5 0 3 5 1

month key (mod 7) Please use this month key in your calculations 0 (–1 if leap year) 3 (2 if leap year) 3 6 1 4 6 2 5 0 3 5

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Math Is Everywhere! Explore and Discover It! 5.1 Clock Systems: The Cycles of Life

The Effect of the Number of Days in a Date Lastly, we need to consider the actual number of days involved in a given date, since these will effectively “push ahead” the day of the week by as many days as are represented by the date. To determine the day of the week for April 23, for example, we would use the month key for April to find how many days were left over at the end of March, and then add the number of days represented by the date itself: month key for April: number of days in April: total:

These 29 days represent 4 full weeks and 1 day left over, i.e., 29 = 1 (mod 7), so the net effect is to “push ahead” one day into a new week.

The Day Key Since we are using January 1, 1900, as our reference point in this algorithm, and since this date fell on a Monday, we will construct a simple correspondence between the first week of this year and designated day keys. Each date beyond this first week of 1900 will then involve some number of “pushed ahead” days, i.e., leftover days that cause the corresponding day of the week to be “pushed ahead” in this reference week cycle. Once again, we are making use of modulus 7 in order to establish this correspondence; Sunday might also have been given the day key of 7, but since 7 is equivalent to 0 (mod 7), we are using the smaller and simpler value.

Day Keys day of the week Monday Tuesday Wednesday Thursday Friday Saturday Sunday

day key 1 2 3 4 5 6 0

The Completed Algorithm We have now established all of the necessary ingredients for a successful algorithm that will enable us to determine the day of the week for any given date from January 1, 1900, to December 31, 1999, inclusive.

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Math Is Everywhere! Explore and Discover It! 5.1 Clock Systems: The Cycles of Life

Example 3: Determine the day of the week for the date July 27, 1985.

1. 2. 3. 4. 5. 6. 7.

Write down the last two digits of the given year (1985) 85 Divide the last two digits of the year by 4 and ignore the remainder 21 Determine the month key from the Month Keys table for the given month (July) 6 Record the actual “day” part of the given date 27 Add the four above values together to form a sum 139 Find the value of this sum in modulus 7 (see pp.177–178); this is the day key 6 Find the day of the week that corresponds to this day key in the Day Keys table Saturday Hence, July 27, 1985 fell on a Saturday.

In review, the seven steps in the algorithm accomplish the following objectives, respectively: 1. 2. 3. 4. 5. 6.

Records the number of days pushed ahead due to the number of years passed since 1900 Records the number of days pushed ahead due to the number of leap years since 1900 Records the number of days pushed ahead due to the preceding months of the given year Records the number of days pushed ahead due to the actual “day” part of the given date Calculates the total number of days pushed ahead Converts this total into the effective number of leftover days via modulus 7; the modulus 7 value represents the effective number of leftover days and becomes the day key 7. Determines the correct day of the week based on the day key in the Day Keys table ******************** Assignments:

For each assignment, use the perpetual calendar algorithm described above to find the day of the week that corresponds to each of the given dates: # 1. 2. 3. 4. 5.

Assignment #1 November 22, 1963 September 1, 1939 February 29, 1988 June 28, 1914 January 3, 1959

Assignment #2 December 8, 1941 December 17, 1903 May 8, 1945 February 29, 1924 May 5, 1961

Assignment #3 October 29, 1929 June 6, 1944 August 21, 1959 July 20, 1969 February 29, 1984

Assignment #4 February 29, 1940 November 11, 1918 February 20, 1962 June 4, 1919 February 14, 1912

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Math Is Everywhere! Explore and Discover It! 5.2 The Modulus in Action: Check Digits and Error Detection

5.2 The Modulus in Action: Check Digits and Error Detection Introduction In this section, we will explore the use of modulus arithmetic in the field of coding theory and, in particular, in the construction of error-detecting schemes related to the transmission of data. Various numbers that we encounter in our daily lives use modulus arithmetic to ensure accuracy. Some examples are UPC numbers (Universal Product Code) found on items in retail stores, ISBNs (International Standard Book Numbers) found on books in print, ISMNs (International Standard Music Numbers) found on published music, and credit card numbers used by VISA, MasterCard, and some major retailers.

Error Detection: A Key to Accuracy The basic idea is that, when data is being transmitted electronically or in written form, there is the ever-present possibility that an error may accidentally be introduced through a power surge or outage, the miscopying of a digit, the transposition of two adjacent digits, or other similar errors. You can imagine the confusion and consternation that would result from someone accidentally using an erroneous credit card number while processing a transaction: the wrong person might get charged with the bill and that wrong person might be YOU!  To introduce the capability of automatic error detection into credit card numbers and other similar number codes, an extra digit called a check digit is often appended to the original number. This extra digit is calculated by means of a mathematical formula that uses the digits of the original number and modulus arithmetic to generate a special result that can easily be checked whenever the number is used or transmitted. If, in checking the transmitted number, the calculations do not produce the correct check digit, then the person or machine doing the checking knows that an error must have occurred and can ask for a retransmission of the number. As an example, we’ll consider the check digits involved with the twelve-digit universal product codes that appear along with bar codes on items of merchandise. The check digit is the rightmost digit in the UPC number and is based on an even/odd computation system in which each digit in

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Math Is Everywhere! Explore and Discover It! 5.2 The Modulus in Action: Check Digits and Error Detection

the UPC number that lies in an odd position (when numbered from left to right) is multiplied by three. The resulting products are then added together with the digits that lie in even positions to form a combined sum. Lastly, the check digit is chosen so that this combined sum plus the check digit is equivalent to 0 in modulus 10 (i.e., equivalent to 0(mod 10)).

x

x

Calculating a Check Digit (from the 11 original UPC digits) For example, let’s consider the UPC number 0-64144-04876-d where the rightmost digit, d, is the check digit. In terms of even/odd positions, the number has the following structure (when read from left to right): 0 6 4 1 4 4 0 4 8 7 6 d digit 2 3 4 5 6 7 8 9 10 11 12 position 1 odd even odd even odd even odd even odd even odd even parity Note: Parity is a mathematical term used to describe the even/odd nature of a number; in the above table, even indicates that the corresponding position number has an even parity and odd indicates that the corresponding position number has an odd parity. To begin to calculate the check digit based on the 11 original digits in the UPC number, the digits in odd positions are multiplied by 3 and the products are added together: Important Note: The digits in odd positions may themselves be either even or odd; it is only the parity of the position number that must be odd in this first set of calculations. odd position number 1 3 5 7 9 11

oddpositioned UPC digit 0 4 4 0 8 6

multiplied by 3 0·3=0 4·3=12 4·3=12 0·3=0 8·3=24 6·3=18 total:

product

0 12 12 0 24 18 66

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Math Is Everywhere! Explore and Discover It! 5.2 The Modulus in Action: Check Digits and Error Detection

Next, the digits in even positions, with the exception of position 12 (the check digit position), are added together and combined with the sum above: even position number 2 4 6 8 10 total:

UPC digit 6 1 4 4 7 22

66 (odd-position sum) + 22 (even-position sum) = 88 (combined sum) Lastly, we need to determine the digit that must be added to this combined sum to obtain a number that is equivalent to 0(mod 10), i.e., a number that is evenly divisible by 10 with no remainder (or a remainder of 0). In this case, the needed digit is 2 since 88+2=90 and 90=0(mod 10). This digit is known as the check digit for this UPC number and is placed in the rightmost position to form the complete 12-digit UPC number: 0-64144-04876-2 Another way to compute this check digit in relation to the combined sum is to find the next multiple of 10 that is larger than the combined sum and then subtract the combined sum from this multiple of 10. Examples: combined sum 88 41 73

next higher multiple of 10 90 50 80

check digit 90 – 88 = 2 50 – 41 = 9 80 – 73 = 7

Verifying a UPC Number When a twelve-digit UPC number, including the check digit, is transmitted, performing the above computation process on the leftmost eleven digits should result in the check digit, the rightmost digit in the UPC number. For more efficiency, the sum produced by the above computation on the leftmost eleven digits can be added to the transmitted check digit (the twelfth and rightmost digit) and the resulting sum can be examined to see whether it is equivalent to 0(mod 10). If so, then the transmission is most likely correct (unless there were multiple undetectable errors); if not, then an error has been made in transmitting the number and a retransmission can be requested. You’ve probably experienced this at some time when checking out at a grocery store; when a transmission error occurs, the cashier then needs to swipe your merchandise through the bar code reader again until it is read correctly.

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Math Is Everywhere! Explore and Discover It! 5.2 The Modulus in Action: Check Digits and Error Detection

A Narrow Escape: Modulus Arithmetic to the Rescue As an example, we’ll consider a scenario where someone purchases an item of merchandise and the bar code reader accidentally transposes two of the digits so that the erroneous UPC number represents a much more expensive item! Original UPC number (11 digits): Complete UPC number including the check digit (12 digits): The rightmost digit is the check digit.

0-10524-89673 0-10524-89673-7

Erroneous UPC number (first 11 digits): Notice the transposition of the digits 8 and 9. Erroneous UPC number with original check digit:

0-10524-98673 0-10524-98673-7

“Oh, dear!” and “Oh, my!” you think to yourself, “This could mean big trouble, headaches and aggravation!” Fortunately, retail stores use a computer program that is designed to verify each scanned bar code that represents a UPC number. Pretending for the moment that we are that verification program, we’ll use modulus arithmetic to test the electronically transmitted UPC number 0-10524-98673-7 to see whether it is a valid number. We begin by labeling the positions as before:

0 1 0 5 2 4 9 8 6 7 3 7 digit 2 3 4 5 6 7 8 9 10 11 12 position 1 odd even odd even odd even odd even odd even odd even parity

Then we’ll multiply all of the odd-positioned digits by 3 and find the sum of these products:

odd position

1 3 5 7 9 11

oddpositioned digit 0 0 2 9 6 3

multiplied by 3 0·3=0 0·3=0 2·3=6 9·3=27 6·3=18 3·3=9 total:

product

0 0 6 27 18 9 60

Next we add the even-positioned digits: 1 + 5 + 4 + 8 + 7 + 7 = 32 Adding the odd-position and even-position sums together yields a combined sum: 60 + 32 = 92.

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Math Is Everywhere! Explore and Discover It! 5.2 The Modulus in Action: Check Digits and Error Detection

Since 92 is NOT equivalent to 0(mod 10), i.e., 10 does not divide evenly into 92, we conclude that a transmission error must have occurred and we request a retransmission from the card reader. ********************

“Wonderful!” you exclaim, “I’ve just been spared an unwanted ordeal! Mathematics really does have some practical use, doesn’t it?”

********************

Exercises and Projects for Fun and Profit Exercises in Analysis Error Detection: The Modulus in Action You will now get some practice involving the use of modulus 10 in check digit schemes. The simple mathematical device known as the modulus turns out to be most useful and effective in error detection, among other things. “Clock arithmetic” most certainly has come into its own in these days of technology— you might say that it is a timely use of a very old concept. Have fun!  Missing Digits: Calculate the check digit, d, for each of these 12-digit UPC numbers. (See pp.186–187 for an example.) 1. 2. 3. 4. 5.

0-14800-71124-d 0-76808-28073-d 0-41415-01928-d 0-12587-00225-d 6-52729-10245-d

6. 6-41255-43128-d 7. 7-29148-53100-d 8. 0-71072-04212-d 9. 0-64144-04335-d 10. 0-74401-91041-d

Valid or Invalid? Determine whether each of the following UPC numbers is valid, i.e., determine whether its check digit is correct. (See pp.187–189 for an example.) 11. 12. 13. 14. 15.

7-16310-96210-4 0-41415-03214-6 0-11152-04046-2 0-39978-00937-1 0-74333-47688-6

16. 17. 18. 19. 20.

0-41415-07633-4 6-05663-60113-0 0-74333-47695-4 0-39059-72984-6 0-41196-10116-5

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Math Is Everywhere! Explore and Discover It! 5.2 The Modulus in Action: Check Digits and Error Detection

Further Investigations To Be Corrected or Not To Be Corrected The check digit error detection scheme for UPC numbers as described above can detect certain types of errors, like the transposition of two adjacent digits, but it cannot detect certain other types of errors. Using your analytical reasoning skills, describe one or two other ways that errors could occur in the copying or transmission of UPC numbers and determine whether the UPC check digit scheme can detect the errors that you describe. In particular, consider the cases of: 21. Replacing one digit with an incorrect digit 22. Transposing two digits that are separated by one digit e.g., ….458…. vs. ….854….. ********************

Creative Projects: ISBN Numbers and Modulus Arithmetic ISMN Numbers and Modulus Arithmetic ********************

ISBN NUMBERS AND MODULUS ARITHMETIC The simple device known as a mathematical modulus is used as a tool in the error detection scheme used for the International Standard Book Number (ISBN) for books in print. In this project, we will examine the process used to construct check digits for these book numbers and learn how modulus 10 plays a part. International Standard Book Number (ISBN) An ISBN is a thirteen-digit number that uniquely identifies each book in print. The book number contains five parts of varying length, where the parts signify the following: a three-digit prefix, the group identifier (often refers to the country in which the book is published), the publisher identifier, the title identifier, and a check digit (the rightmost digit in the number). Publishers and booksellers use ISBNs to place orders, keep track of inventories, etc., and hence it is important that communications involving these numbers are as accurate and as error-free as possible.

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Math Is Everywhere! Explore and Discover It! 5.2 The Modulus in Action: Check Digits and Error Detection

The check digit that serves as the thirteenth digit of the ISBN is the key to error detection and we will now take a look at the computational process that is involved in constructing one of these check digits. As an example, let’s suppose that we have found an ISBN that is missing its rightmost digit (the check digit): 978-0-310-92599-d. In this instance, the first twelve digits represent the various identifiers mentioned above and the letter d represents the missing check digit.

Constructing a Check Digit (from the first twelve digits of an ISBN) In order to determine the correct check digit for this number, we begin by assigning multipliers to each of the digits; the multipliers always consist of alternating values of 1 and 3, beginning with a 1 that is assigned to the check digit, d: digit multiplier

9 1

7 3

8 1

0 3

3 1

1 3

0 1

9 3

2 1

5 3

9 1

9 3

d 1

Reading the ISBN from left to right, the first three digits, 978, represent the prefix; the next digit, 0, represents the group identifier; the next three digits, 310, represent the publisher identifier; the next five digits, 92599 represent the book title identifier; and the final digit, d, represents the missing check digit. Next, we multiply each of the leftmost twelve digits by its multiplier: 9·1=9 7·3=21 8·1=8 0·3=0 3·1=3 1·3=3 0·1=0 9·3=27 2·1=2 5·3=15 9·1=9 9·3=27 Adding these values produces a sum: 9 + 21 + 8 + 0 + 3 + 3 + 0 + 27 + 2 + 15 + 9 + 27 = 124 The objective is to select a check digit so that the above sum plus the check digit is a number that is equivalent to 0(mod 10), i.e., the number is evenly divisible by 10 with no remainder. Since 124/10 = 12 with a remainder of 4, 124 is obviously not evenly divisible by 10, i.e., not equivalent to 0(mod 10), so we now choose a number to add to 124 so that the resulting sum is evenly divisible by 10. By the method of trial and error, 124 + 1 = 125, but 125 is not evenly divisible by 10 124 + 2 = 126, but 126 is not evenly divisible by 10 124 + 3 = 127, but 127 is not evenly divisible by 10 124 + 4 = 128, but 128 is not evenly divisible by 10 124 + 5 = 129, but 129 is not evenly divisible by 10

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Math Is Everywhere! Explore and Discover It! 5.2 The Modulus in Action: Check Digits and Error Detection

124 + 6 = 130 and Eureka! 130 is evenly divisible by 10, or equivalent to 0(mod 10). By trial and error, a method that is often tedious but can also be effective, we finally found the value that we had been searching for. 6 is the correct check digit for our ISBN and the complete number, therefore, is: 978-0-310-92599-6. We could have been a bit more efficient if we had simply found the next multiple of 10 that had a greater value than the partial sum, 124. Since 10·12=120, and this was a bit less than the sum that we had found, we could have calculated 10·13=130 to get the next higher multiple of 10 and then subtracted our partial sum, 124, from this new multiple to find the required check digit: 130–124=6.

******************** In any case, we can now rest assured that when this book number is copied or electronically transmitted in some manner the check digit will help to keep the ISBN free from accidental errors of the common varieties. Whew! I’m sure glad to know that! Once again, mathematics has come to the rescue and has provided us with a simple and effective means of detecting errors and ensuring accuracy. ******************** Assignments: For each assignment, please exercise your analytical reasoning skills and determine the check digits (d) for the given ISBNs: # 1. 2. 3. 4. 5.

Assignment #1 978-0-520-26719-d 978-0-553-21079-d 978-0-689-84224-d 978-0-553-21256-d 978-0-553-21158-d

Assignment #2 978-0-486-26113-d 978-0-812-96705-d 978-0-898-70268-d 978-0-679-44800-d 978-1-573-92039-d

Assignment #3 978-0-486-40664-d 978-0-553-21143-d 978-0-486-29279-d 978-0-060-51865-d 978-0-486-42459-d

Assignment #4 978-0-451-53110-d 978-0-684-82439-d 978-0-553-21195-d 978-0-520-24245-d 978-0-486-41426-d

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Math Is Everywhere! Explore and Discover It! 5.2 The Modulus in Action: Check Digits and Error Detection

ISMN NUMBERS AND MODULUS ARITHMETIC Recognizing the need for international standards for identifying published musical compositions, music publishers and librarians proposed a coded identifier scheme similar to the ISBN scheme used for books in print. This scheme was designed to meet the unique needs of music publishers and as a result has a slightly different structure. In this project, we will examine the process used to construct check digits for the International Standard Music Number (ISMN) and learn how modulus 10 plays a part. International Standard Music Number (ISMN) An ISMN consists of a thirteen-digit number that uniquely identifies each musical work in print. The thirteen-digit number contains four parts of various lengths, where the parts signify the following: a four-digit prefix, the publisher identifier, the item identifier, and a check digit (the rightmost digit in the number). Publishers and music sellers use ISMNs to place orders, keep track of inventories, etc., and hence it is important that communications involving these numbers are as accurate and as error-free as possible. The check digit that serves as the rightmost digit of the number is the key to error detection and we will now take a look at the computational process that is involved in constructing one of these check digits. As an example, let’s suppose that we have found an ISMN that is missing its rightmost digit (the check digit): 979-0-2018-0495-d. In this instance, the first twelve digits represent the various identifiers mentioned above while the letter d represents the missing check digit.

Constructing a Check Digit (from the first twelve digits of an ISMN) In order to determine the correct check digit for this number, we begin by assigning a numerical multiplier to each of the digits, working from right to left. These multipliers always consist of alternating values of 1 and 3, beginning with a 1 that is assigned to the check digit, d. digit multiplier

9 1

7 3

9 1

0 3

2 1

0 3

1 1

8 3

0 1

4 3

9 1

5 3

d 1

In this ISMN, there is a four-digit prefix, 979-0; the next four digits, 2018, represent the publisher identifier; the next four digits, 0495, represent the music item identifier; and the final digit, d, represents the missing check digit.

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Math Is Everywhere! Explore and Discover It! 5.2 The Modulus in Action: Check Digits and Error Detection

We then multiply each digit by its respective multiplier: 9·1=9 7·3=21 9·1=9 0·3=0 2·1=2 0·3=0 1·1=1 8·3=24 0·1=0 4·3=12 9·1=9 5·3=15 Adding these products produces a sum: 9 + 21 + 9 + 0 + 2 + 0 + 1 + 24 + 0 + 12 + 9 + 15 = 102 The objective is to select a check digit so that the above sum plus the check digit is a number that is equivalent to 0(mod 10), i.e., so that the resulting number is evenly divisible by 10 with no remainder. Since 102/10 = 10 with a remainder of 2, 102 is obviously not evenly divisible by 10, i.e., not equivalent to 0(mod 10), so we now choose a number to add to 102 so that the resulting sum is evenly divisible by 10. By trial and error, we can find the required value: 8 is the correct check digit for our ISMN since 102+8=110 and the complete ISMN therefore, is: 979-0-2018-0495-8. We can be a bit more efficient if we simply find the next multiple of 10 that has a greater value than the sum, 102. Since 10·10=100, and this is a bit less than 102, we could have calculated 10·11=110 to get the next higher multiple of 10 and then subtracted our sum, 102, from this new multiple to find the required check digit: 110 – 102 = 8. ******************** In any case, we can now rest assured that when this music number is copied or electronically transmitted in some manner the check digit will help to keep the ISMN free from accidental errors of the common varieties. Whew! I’m sure glad to know that! Once again, mathematics has come to the rescue and has provided us with a simple and effective means of detecting errors and ensuring accuracy. ******************** Assignments: For each assignment, please exercise your analytical reasoning skills and determine the check digits (d) for the given ISMNs: # 1. 2. 3. 4. 5.

Assignment #1 979-0-078-45976-d 979-0-2186-0738-d 979-0-3624-7051-d 979-0-43819-281-d 979-0-58732-597-d

Assignment #2 979-0-065-35681-d 979-0-2346-4639-d 979-0-5715-9084-d 979-0-42863-159-d 979-0-69342-763-d

Assignment #3 979-0-079-56934-d 979-0-2395-0652-d 979-0-3851-4093-d 979-0-44852-906-d 979-0-69719-872-d

Assignment #4 979-0-479-64508-d 979-0-7243-1076-d 979-0-3799-8426-d 979-0-44628-034-d 979-0-67539-684-d

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

5.3 Cryptography: The Mathematics of Privacy Introduction Privacy has become a national issue lately, especially with regard to personal financial information and the need to keep it safe from theft and Internet piracy. As a result, the study of cryptography—the process of writing in secret characters or codes—has become increasingly important. In this section, we will look at the nature of cryptography, examine the classical form of a substitution code, and learn some basic techniques for both coding and decoding messages. We will also take a brief look at the modern encryption techniques that are used today for protecting digital information. In the Exercises and Projects for Fun and Profit at the end of this section, you will have the opportunity to do some further investigations concerning cryptography and solve some cryptograms and cryptoquotes—puzzles that often appear in daily newspapers.

Cryptography The encoding of messages in order to keep them secret has most likely been going on since man first began to make written communications. Whether the intent is to keep information hidden from friends or from enemies, cryptography provides the writer with a means by which to do so. There is evidence that the Roman emperor Julius Caesar used codes (also referred to as ciphers) to communicate with the commanders in his armies; in modern times, cryptography has continued to play an important role in many war-related communications. In war-related situations, cryptanalysis—the science of analyzing and decoding encrypted messages—is equally as important as the process of encoding messages. One of the more famous examples of cryptanalysis involved the American success during World War II in breaking the German codes that had been constructed using a machine called the Enigma Machine. In the following descriptions of codes, we will use the standard convention of referring to the original message (before encoding) as plaintext; we will refer to the encoded message as ciphertext.

Classical Cryptography: The Substitution Cipher The two primary types of codes in early times were substitution ciphers and transposition ciphers. We will examine some simple substitution ciphers below; transposition ciphers will be the topic of a Creative Project in the Exercises and Projects for Fun and Profit at the end of this section. In a substitution cipher, each letter of the alphabet is encoded as a different letter of the

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

alphabet. The cipher used by Julius Caesar was a substitution cipher and was created by shifting the ciphertext alphabet to the right by a fixed number of places. In modern times, this type of shift cipher is referred to as a Caesar cipher. For example, a substitution cipher involving a shift in the ciphertext of three places to the right would look like this, where row #1 represents plaintext and row #2 represents ciphertext: 1 2

A X

B Y

C Z

D A

E B

F C

G D

H E

I F

J G

K H

L I

M J

N K

O L

P M

Q N

R O

S P

T Q

U R

V S

W T

X U

Y V

Z W

The plaintext message, WELCOME TO CRYPTOGRAPHY, would then be encoded in ciphertext as: TBIZLJB QL ZOVMQLDOXMEV To decode this message, the decoder would simply reverse the process to go from ciphertext to plaintext provided that he or she knew how many positions were involved in the alphabetic shift. If the shift number were unknown, then a decoder would begin by trying each of the 25 possible shifts (not including a shift of 26 places that essentially leaves the positions unchanged) and examining each result for an intelligible message as follows: 1) Use a one-letter shift and invert the rows so that the ciphertext is on top and the plaintext is on the bottom; then decode the message: 2 1

Z A

A B

B C

C D

D E

E F

F G

G H

H I

I J

J K

K L

L M

M N

N O

O P

P Q

Q R

R S

S T

T U

U V

V W

W X

X Y

Y Z

In this case, the decoded message would read: UCJAMKC RM APWNRMEPYNFW Since this is unintelligible, proceed to the next shift. 2) Use a two-letter shift and invert the rows so that the ciphertext is on top and the plaintext is on the bottom; then decode the message: 2 1

Y A

Z B

A C

B D

C E

D F

E G

F H

G I

H J

I K

J L

K M

L N

M O

N P

O Q

P R

Q S

R T

S U

T V

U W

V X

W Y

X Z

In this case, the decoded message would read: VDKBNLD SN BQXOSNFQZOGX Since this is still unintelligible, proceed to the next shift. 3) Use a three-letter shift and invert the rows so that the ciphertext is on top and the plaintext is on the bottom; then decode the message: 2 1

X A

Y B

Z C

A D

B E

C F

D G

E H

F I

G J

H K

I L

J M

K N

L O

M P

N Q

O R

P S

Q T

R U

S V

T W

In this case, the decoded message would read: WELCOME TO CRYPTOGRAPHY

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U X

V Y

W Z

Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

******************** Success at last! Good old-fashioned trial and error and a bit of persistence was all that was needed to decode the message. ********************

Arbitrary Substitution Ciphers In a more complicated version of a substitution cipher, letters of the alphabet are assigned to other arbitrarily selected letters of the alphabet in order to form a code. Since this assignment process does not necessarily involve a pattern, decoding these types of ciphers is decidedly more difficult and requires more extensive cryptanalysis. We will now examine this process. For example, an arbitrary substitution cipher might look like this, where row #1 represents plaintext and row #2 represents ciphertext: 1 2

A S

B A

C M

D P

E L

F E

G X

H O

I N

J B

K Y

L R

M C

N D

O W

P Q

Q F

R T

S G

T K

U H

V Z

W I

X V

Y J

Z U

Benjamin Franklin’s famous adage, “Early to bed, early to rise, makes a man healthy, wealthy and wise,” would be encoded as follows (without punctuation): LSTRJ KW ALP LSTRJ KW TNGL CSYLG S CSD OLSRKOJ ILSRKOJ SDP INGL Famous quotations that have been encoded using arbitrary substitution ciphers are referred to as cryptoquotes; these puzzles are popular pastimes that often appear in local newspapers.

Cryptanalysis Techniques To decode Franklin’s adage, we will need to employ a number of cryptanalysis techniques that are related to the structure of the English language:      

Frequency analysis for single letters Frequency analysis for letter pairs Double vowels and consonants Word length Word structure Sentence structure

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

These techniques are used either singly or in combination in order to shed light on the possible letter decodings and are interesting in and of themselves with respect to our English language. In order to use them effectively, you will need to employ your analytical reasoning skills and a bit of creative thinking. By now in this course of study, you should have developed a measure of confidence in your skills and I trust that the following application will be readily comprehensible.

Frequency Analysis and Word Length Foremost among cryptanalysis techniques is frequency analysis—the determination of the relative number of occurrences of each ciphertext character and comparison with the known relative frequencies of the occurrences of each letter of the alphabet in normal English composition. It is well established that certain letters of the alphabet occur much more frequently than others in normal composition. The approximate relative frequencies for the twelve most frequently used letters are given below: letter E T A O I N

frequency 13% 9% 8% 8% 7% 7%

letter S H R D L U

frequency 6% 6% 6% 4% 4% 3%

In a passage of some length, there is a good likelihood that the letter occurrences will follow a similar distribution. Even though Franklin’s adage is rather short, we can still use frequency analysis to help us with the decoding process. The ciphertext letter distributions for letters with more than one occurrence in the adage are: letter L S K R J G O

number of occurrences 8 8 4 4 4 3 3

letter T W N D P C I

number of occurrences 3 2 2 2 2 2 2

Frequency analysis would indicate that there is a good chance that some of the top five letters in the ciphertext distribution would correspond to some of the top five letters in the plaintext distribution, i.e., one or more of the letters L, S, K, R and J will likely be decoded to one or more

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

of the letters E, T, A, O, I or N. In fact, a reasonable conjecture would be that either L or S decodes to the letter E and that the other letter of the two decodes to T, A, O, I or N. We may then use word length to help us decide which choice might be correct. In the coded adage, there is one single-letter word, S. Since there are only two English words that consist of one letter, the words “a” and “I”, S must decode to one of these letters; consequently, our conjecture tells us that L most likely decodes to the letter E (the E’s may not appear to be directly below the L’s in the display below): LSTRJ KW ALP LSTRJ KW TNGL CSYLG S CSD OLSRKOJ ILSRKOJ SDP INGL E E E E E E E E

Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

Frequency Analysis and Word Structure Revisited With regard to frequency analysis for letter pairs, it has been established that the most common pairs are: th, he, in, en, nt, re, er, an, ti, es, on, at, se, nd, or, ar, al, te, co, de, to, ra, et, ed, it, sa, em, ro Given that the most common pairs in the ciphertext are KW and KO, it is likely that these are contained in this list. Focusing our attention on KO since it more constrained by the letters on either side of it, we may readily determine that th appears to be the only viable choice: LSTRJ KW ALP LSTRJ KW TNGL CSYLG S CSD OLSRKOJ ILSRKOJ SDP INGL EARLY T E EARLY T R E A E A A HEALTHY EALTHY A E Since the ciphertext word KW has only two letters and since we know that K decodes to the letter T, we may now use our knowledge of word structure and the fact that each English word must contain at least one vowel to deduce that W must decode to the letter O: LSTRJ KW ALP LSTRJ KW TNGL CSYLG S CSD OLSRKOJ ILSRKOJ SDP INGL EARLY TO E EARLY TO R E A E A A HEALTHY EALTHY A E At this point, you are probably ready to make a very educated guess about the decoded adage. ***** An Opportunity to Exercise your Analytical Reasoning Skills *****  If you didn’t already know what the adage was, what letter would you attempt to decode next? ********************

A Brief Look at Modern Encryption: The RSA Algorithm In today’s world of technology, computers are employed to decode secret messages. This led to the need to develop stronger forms of encoding that could not be “broken” even through the use of high-speed computers. In 1977, three mathematicians working at MIT, Ronald Rivest, Avi Shamir and Leonard Adleman, constructed a simple yet powerful algorithm involving very large prime numbers to encode and decode information. This algorithm was named in their honor and consists of the first letters of each of their last names: RSA. In order to break the code, factoring very large numbers is required. Since finding prime factors for very large numbers is extremely labor-intensive and since there are currently no known factorization methods that are fast enough to obtain the needed prime factors in any reasonable amount of time, the RSA algorithm is a very secure method for encoding information that needs 200 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

to remain secret. In fact, the keys needed to encode messages may be broadcast publicly without jeopardizing the secrecy of the message; hence, this type of cryptography is referred to as public key cryptography.

RSA in Action The RSA algorithm requires that a user first choose two prime numbers. After making those choices, the user then determines two keys, a public key and a private key, based on some simple mathematical relationships between the key numbers and the original two primes. The product of the two prime numbers and the public key are then made public so that anyone may use them to send the original user a coded message that only the original user can decode using the private key. The example below illustrates the algorithm using very small prime numbers for the sake of simplicity. 1. The code creator (i.e., original user) creates the public and private code keys: Step 1: Choose two prime numbers p=3 and q=11 and compute the product N = p·q = 33. Step 2: Calculate phi = (p – 1) · (q – 1) = (3 – 1) · (11 – 1) = 2 · 10 = 20. Step 3: Determine two keys, e (public key) and d (private key), such that:  1 < e < phi  e is relatively prime to phi (i.e., e and phi have no prime factors in common)  the product e·d is congruent to 1(mod phi), i.e., 1 (modulus phi) Since 21 is congruent to 1(mod 20) and 3 is relatively prime to 20, the code creator may choose e=3 and d=7 as the two keys; in this example, e will be the public key and d will be the private key. (Note: Since 81 is also congruent to 1 (mod 20) and 9 is relatively prime to 20, the code creator could also have chosen e=9 and d=9 as the two keys.) 2. The code creator sends the value for N and the public key, e, to someone who needs to send the code creator a secret message. In this case, N=33 and e=3. 3. The receiver of N and the public key may then use these values to encode a secret message and send it to the code creator. For the sake of simplicity, we’ll suppose that the message consists of a single number, 9. To encode this message: raise 9 to the power of e=3 (public key); determine this value in modulus N=33:

9 3 (mod 33) = 729(mod 33) = 3(mod 33)

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

The message character “9” has now been transformed into the encoded value “3”. 4. The encoded message is sent to the original code creator who is the only possessor of the private key, d=7, needed for decoding. The message is decoded as follows: raise the encoded value 3 to the power of d=7 (private key) and determine this value in modulus N=33: 37 (mod 33) = 2197(mod 33) = 9(mod 33) The encoded value of “3” has been successfully decoded to the original value of “9”. ********************

The Security Issue The security risk in this algorithm lies in the possibility that someone might be able to find the two prime factors, p and q, for the public value N. If these values were discovered, then the value for phi could easily be determined since phi=(p – 1)(q – 1). Adding 1 to the multiples of phi would produce the possible products e·d and, since e is also a public value, the possible values for d could then be determined and tested until the correct value for the private decoding key was found. Then the encoded message would no longer be secret and the code would be “broken.” Of course, the above example used very small primes for the sake of illustration; however, in real applications, the primes used are extremely large (consisting of hundreds of digits) and the product N cannot be factored in any reasonable length of time. As computers get faster and more powerful, the threat of possible factorization increases; on the other hand, larger and larger primes are being discovered that outstrip these increases in computer power so it appears that the public-key encryption process will continue to be secure for quite a while.

Conclusion In our brief excursion into the realm of cryptography, we examined the classic substitution cipher as a means of getting a basic feel for what is involved in encoding and decoding information. While these types of ciphers have been rendered inadequate by the advances in high-speed computing, they are still used in many recreational applications today. We also touched on the rudiments of public-key encryption methods that are employed today for data security. I hope that this brief survey has been of interest and that you enjoy the Exercises and Projects for Fun and Profit that follow; in particular, some of the Further Investigations are quite interesting and can lead you to many related topical areas of interest. Enjoy! 

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

Exercises and Projects for Fun and Profit Exercises in Analysis Note: See Helps and Hints for Exercises on pp.439–440 in Appendix A. Caesar ciphers 1. Use the cryptanalysis technique described in this section to encode the following plaintext by using a Caesar cipher that involves a shift of 10 places to the right (see top of p.196):

CRYPTOGRAPHY IS FUN 2. Use the cryptanalysis technique described in this section to encode the following plaintext by using a Caesar cipher that involves a shift of 7 places to the right (see top of p.196): SUBSTITUTION CIPHER 3. Use the cryptanalysis technique described in this section to decode the following ciphertext; the ciphertext was created by using a Caesar cipher (see p.196): ZLABP ZXK YB NRFQB

JVPQBOFLRP

4. Use the cryptanalysis technique described in this section to decode the following ciphertext; the ciphertext was created by using a Caesar cipher (see p.196): YWJ UKQ ZAYELDAN

PDEO?

Cryptograms 5. Use your analytical reasoning skills and the cryptanalysis techniques described in this section (pp.197–200) to decode the following cryptogram; in particular, notice word structure and sentence structure. Clue: V=F, i.e., the encoded V = F when decoded.

A G M ‘ C

C R A G

V E M?

6. Use your analytical reasoning skills and the cryptanalysis techniques described in this section (pp.197–200) to decode the following cryptogram; in particular, notice word length. Clue: F=S, i.e., the encoded F = S when decoded. MCXF XF K AKMCVAKMXGKY

GCKYYVPSV!

7. Use your analytical reasoning skills and the cryptanalysis techniques described in this section (pp.197–200) to decode the following cryptogram; in particular, notice sentence structure, word length, double letters, and the possible use of pronouns. Clues: G=D and B=H, i.e., the encoded G = D when decoded and the encoded B = H when decoded. GR

ORC

WTTG

M BMWG

KXEB

EBXD? 203

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

8. Use your analytical reasoning skills and the cryptanalysis techniques described in this section (pp.197–200) to decode the following cryptogram. Clues: M=C and D=S, i.e., the encoded M = C when decoded and the encoded D = S when decoded. DPLQ MPAQD MXE‘R FQ

DPKSQA

Q X D B K U.

Cryptoquotes 9. Use your analytical reasoning skills and the cryptanalysis techniques described in this section (pp.197–200) to decode the following cryptoquote; in particular, notice word length and sentence structure:

“C R

FX

RA

DRC

CR

F X. C M L C N Y

CMX

P H X Y C N R D.”

—Y M L U X Y E X L A X 10. Use your analytical reasoning skills and the cryptanalysis techniques described in this section (pp.197–200) to decode the following cryptoquote; in particular, notice letter frequency and word length: “A M L F L C H A X S U B WMC

LFLCH

AXLCL SK P KLPKMU PUJ P ASVL D E C D M K L E U J L C X L P F L U.” —L G G N L K S P K A L K

Public-key encryption 11. Using the example provided in the section entitled RSA in Action on pp.201–202 (especially note Step 4—this shows the decoding process that you will need to use below), decode the following secret message given that N=35 and d=5. In this case, the original message was a single word that was converted from letters to numbers, where A=1, B=2, C=3, D=4, E=5, and so on.

33 15 9 10 24 Example: To decode the first ciphertext term, 33, we would raise 33 to the 5th power (the private key) and convert it into a value based on modulus 35:

335 (mod 35) = 39,135,393(mod 35) = 3(mod 35) To convert 39,135,393 into an equivalent value in modulus 35: 1. Divide 39135393 by 35 on your calculator: 39135393/35=1118154.08571

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

2. Ignore the decimal part of the quotient and multiply the integer quotient 1118154 by 35 to get 39135390. This represents the number of times that 35 divides evenly into 39135393. 3. Subtract 39135390 from 39135393 to obtain the remainder 3 in modulus 35. This remainder represents the decoded value of 3. 4. Convert the decoded value of 3 into its alphabetic equivalent: 3=C. Note: If your calculator is not capable of displaying twelve places, you may use the calculator that is built into your computer’s Windows operating system. To access the calculator, click on the Start menu at the bottom left of your screen, click on Programs and then click on Accessories. In the dropdown menu, click on Calculator. In the Calculator dialog box, click on View at the top of the box and select Scientific in the dropdown menu. You may then click on the numerical values and the button labeled x^y to raise a number to a power, e.g., enter “33 x^y 5 = ” to obtain 39,135,393. You will need to press the “=” key in order to see the result.

12. Using the example provided in the section entitled RSA in Action on pp.201–202 (especially note Step 4—this shows the decoding process that you will need to use below), decode the following secret message given that N=39 and d=7. In this case, the original message was a single word that was converted from letters to numbers, where A=1, B=2, C=3, D=4, E=5, and so on. 7 8 3 21 8 32 Note: Please read the Example and Note that accompany the preceding exercise for helpful advice in decoding this message.

********************

Further Investigations 13. Cryptography on the Web There are numerous Web-based resources related to cryptography. Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on cryptography. Visit a few of these resources and write a brief report (minimum of one-half page) on your findings. Please include a webliography of the sites that you visited. 14. RSA Algorithm To learn more about cryptography and modern encryption techniques, use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on the RSA algorithm. Afterwards, write a brief report (minimum of one-half page) on your findings.

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

15. Kryptos—A CIA Enigma In the late 1980s, the United States Central Intelligence Agency (CIA) commissioned an artistic sculpture to enhance their new headquarters building as part of an effort to create an aesthetically pleasing work environment for its employees. Kryptos, as the sculpture was named, is a two-part sculpture in various types of stone; it contains four encoded sections of text that were written by the sculptor in collaboration with a retired CIA cryptographer. Up to the present, only three of the four sections have been successfully decoded; the fourth remains a mystery and is available for viewing by anyone interested in attempting to solve it. While the solution to this coded text is obviously very complicated, the story behind the sculpture is quite interesting. Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on the Kryptos sculpture and write a brief report (minimum of one-half page) on the features that interested or impressed you the most.

16. The Enigma Machine During World War II, the German military began using an electromechanical device consisting of rotors, spindles and electrical circuits to encode secret messages. The device was named the Enigma Machine and its cryptographic system was quite robust; however, due in part to user errors and other circumstances, the Allied forces were able to break the code of the Enigma Machine and help to bring the war to an end. Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on the Enigma Machine and write a brief report (minimum of one-half page) on your findings. Please include a webliography of the sites that you visited.

********************

Creative Projects: Cryptograms: A Popular Pastime Cryptoquotes: Another Favorite

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

CRYPTOGRAMS: A POPULAR PASTIME If you have ever attempted a word puzzle entitled Word Jumble, Word Scramble or something similar (often found in daily newspapers), then you have experienced a transposition cipher. A transposition cipher encodes plaintext by rearranging the placement of the plaintext without substituting other characters as in a substitution cipher. Example: The plaintext word CIPHER can be transposed and encoded as PREICH. Assignments:

For each assignment, decode the given six-letter cryptograms (each of the encoded words appears in Section 5.3 of the textbook); as an example, the cryptogram PREICH would decode to CIPHER.

# 1. 2. 3. 4. 5.

Assignment #1 MRLAON HREETI PRAAPE HSOECO VSYREU

Assignment #2 GICOND SACLPE PYOLME RTACFO CDNOES

Assignment #3 DRNEOM ASRECA DOCNEE RTLETE ECDDEO

Assignment #4 PLSMIY BUDOEL ZEPLZU LBUPIC GNEISL

********************

CRYPTOQUOTES: ANOTHER FAVORITE A type of cryptogram that is currently popular is a cryptoquote—a famous quotation that has been encoded via an arbitrary substitution cipher. Please see pp.197–200 for an explanation and an example of an arbitrary substitution cipher and the cryptanalysis techniques used in its decoding. In particular, it will help to analyze the letter frequencies of each letter in a given cryptoquote. The encoded letter with the most occurrences will usually decode to the letter E, the most commonly used letter in the alphabet. Examining word length will also be of assistance. In Assignment #1, for example, the encoded one-letter words, K and P, will decode to A and I but not necessarily in that order; specifically, either K or P will decode to the letter A. The encoded, repeated three-letter words PXT and WZR may provide some additional clues; often repeated three-letter words are “THE” and “AND”…

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Math Is Everywhere! Explore and Discover It! 5.3 Cryptography: The Mathematics of Privacy

Assignments:

Use the cryptanalysis techniques described in this section on pp.197–200 to decode the following cryptoquotes from well-known poets; these authors are often presented in English or Literature courses: Note: Please see Helps and Hints for Cryptoquotes on pp.440–442 in Appendix A. Assignment Assignment #1

Cryptoquote “WNB IBPTM TKJRIART KX P NBBT, PXT K — K WBBS WZR

BXR CRMM WIPJRCRT OH, PXT WZPW ZPM FPTR PCC WZR TKYYRIRXER.” —IBORIW YIBMW Assignment #2

“BQXACFM, BQXACFM, FXBBFM EBRG, SDQ X QDAYMG QSRB HDJ RGM, JK RLDNM BSM QDGFY ED SXOS, FXCM R YXRPDAY XA BSM ECH.” —TRAM BRHFDG

Assignment #3

“FRPXA! FRPXA! KDAEREP KARPWF, RE FWX SYAXCF YS FWX ERPWF, GWHF RJJYAFHM WHEN YA XQX TYDMN SAHJX FWQ SXHASDM CQJJXFAQ?” —GRMMRHJ KMHVX

Assignment #4

“ZCHERA, SB KPCZFORA, GAF BDJ HPGZZ PRGO DL EPR SCFACMPE OCFR DL NGJZ ORQROR, DA EPR RCMPERRAEP DL GNOCZ, CA HRQRAEB-LCQR; PGOFZB G SGA CH ADT GZCQR TPD ORSRSUROH EPGE LGSDJH FGB GAF BRGO.” —PRAOB ZDAMLRZZDT

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Math Is Everywhere! Explore and Discover It! 6.0 Introduction and Objectives

Chapter Six Mathematics and Music: Inseparable Partners 6.0 Introduction and Objectives The thumbnail sketches on this page represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

6.1 The Nature of Sound and Musical Scales: Good Vibrations The twelve-tone scale:

6.2 The Mathematics of Stringed Instruments: Structured Harmony

6.3 Digital Music and CD’s: Applied Mathematics … 1001000 0111001

1101100 …

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Math Is Everywhere! Explore and Discover It! 6.0 Introduction and Objectives

Learning Objectives for Chapter 6 1. To enable students to develop their analytical reasoning skills through pattern recognition and problem solving. 2. To enable students to develop their deductive reasoning skills through applying mathematical relationships to obtain additional data. 3. To allow students to exercise their creativity by conducting Internet research and writing reports on various mathematical topics. 4. To enable students to gain some familiarity with Web-based mathematical resources and to develop their critical thinking skills in assessing and evaluating these resources. 5. To demonstrate the usefulness and relevance of mathematics through an application involving digital music. 6. To engage students in active learning through the use of projects and activities. 7. To empower students to become lifelong learners.

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

6.1 The Nature of Sound and Musical Scales: Good Vibrations “Music is the pleasure the human soul experiences from counting without being aware that it is counting.” —Gottfried Wilhelm Leibniz

Introduction When and how the art of making music first began to be practiced is a matter of conjecture, but we do know that evidence of music appears in mankind’s earliest history. For example, archaeologists have discovered primitive flutes made from hollow bones that date back many thousands of years. In all cultures, music has played an important role in life for both individuals and their social groups. In this section, we will examine some of the connections between music and mathematics.

The Nature of Sound: Vibrations In order to understand music, we’ll first examine the basic nature of sound and hearing. In essence, sound is created by a vibration of some sort; without vibration, there is no sound. Let’s assume that you are in a room with a guitarist. When the guitarist plucks a guitar string, a transmission process is set in motion. The string begins to vibrate, pushing forward and backward on the air molecules surrounding it; these molecules in turn push forward and backward against neighboring air molecules. This chain reaction continues through the air across the room until the air molecules adjacent to your eardrum push against the eardrum itself, causing the eardrum to vibrate in that same forwardbackward motion. The eardrum transfers this vibration to small bones inside the middle ear that amplify the vibration and transfer it to the fluid in the inner ear in the form of waves. These waves cause the movement of tiny hairs attached to the cochlear nerve fibers; in turn, the cochlear nerve transmits these vibrations to the temporal lobe of the brain where the brain registers the vibration as a sound. Quite amazing, isn’t it? The original vibration, of course, is the prime mover in the sound/hearing process.

Musical Notes and Scales: The Building Blocks of Music Music is simply a collection of sounds, i.e., vibrations, that have been arranged in pleasing patterns. This arranging process involves selecting notes (tones), creating melodies, choosing

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

rhythms and creating harmonies to accompany the melodies. In this section, we will focus our attention on the notes, or tones, involved in musical construction. Each musical tone is intrinsically related to the number of vibrations per second that is required to produce the tone. Low notes have fewer vibrations per unit of time than do high notes. We measure these vibrations in cycles per second, i.e., the number of complete back-and-forth cycles that the source of the vibration makes in one second of time. This measurement is called the frequency of the vibration. A vibrating guitar string that makes 328 complete back-and-forth cycles per second is said to have a frequency of 328 cps (cycles per second). In order to distinguish between various tones, musicians decided to name each of the tones in an alphabetical manner. Recognizing that there was a very special relationship between tones whose frequencies were multiples of each other, and that these tones sounded quite similar, they decided to label tones such as these with the same alphabetic character. In particular, we’ll focus our attention on two tones (notes), the higher of which has a frequency that is exactly twice the frequency of the lower (i.e., a multiple of two). While various cultures use different numbers of notes from a given note up to and including the note having twice the frequency of the given note, early Western musicians conventionally used eight such notes. They labeled these notes as follows: A B C D E F G A Arriving at the higher A note meant arriving at a note with exactly twice the frequency of the lower A note. Since this higher ending note sounded very similar in pitch to the lower beginning note, the distance between these notes was called an octave due to the fact that eight notes were involved; however, there are only seven distinctly named notes within the octave. The differences in frequency between successive notes within an octave were established by the Greeks many centuries ago and Western musicians have continued to use these differences that consist of whole steps and half-steps. A musical scale is a specific group of tones, or notes, within an octave. There are many different types of scales: major, minor, chromatic, etc. One of the most commonly used groups of notes begins with the note C. In the C-major scale there is a half-step between the third and fourth notes, E and F, and between the seventh and eighth notes, B and C (in the diagrams below, four dashes represent a whole step while two dashes represent a half-step): C ---- D ---- E -- F ---- G ---- A ---- B -- C Each of these notes is represented by a white key on the standard piano keyboard. As another basic example, the following scale is known as the natural minor scale: A ---- B -- C ---- D ---- E -- F ---- G ---- A

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

This is the same set of notes as those in the C-major scale but the beginning note is A instead of C and the half-steps are in relatively different positions as a result. When playing these scales on an instrument, the effect is thus somewhat different. Johann Sebastian Bach (1685–1750), the famous German composer, began using a twelve-tone scale in his music. This scale consists entirely of half-steps and consequently has twelve distinctly named notes in an octave (thirteen notes altogether). To create this scale, a new note was inserted between each pair of notes that was separated by a whole step. These new notes were labeled with either sharps (#) or flats (b) depending on the emphasis desired. The resulting scale is as follows and represents both the black and white keys on a piano keyboard: C -- C# -- D -- Eb -- E -- F -- F# -- G -- G# -- A -- Bb -- B -- C This twelve-tone scale was widely accepted and came to be used by most Western musicians since Bach’s time; it is the scale currently used in our popular music of today. We’ll now take a closer look at the mathematics involved in this scale.

A Mathematical Supposition In a given octave, we know that the high C note has a frequency equal to twice that of the low C note. A very interesting and important question is: What are the frequencies of the notes that lie between the C notes?

******************** Analytical Moment: Using your growing analytical reasoning powers, you might suspect that these frequencies follow a specific mathematical pattern, much like some of the patterns that we investigated in Chapter 2. And you’re right! They do!  The associated question is: What type of pattern do they follow? ********************

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

To answer this, we might consider the types of sequences that we looked at in Chapter 2. As a memory refresher, two of the major types were:  

arithmetic sequences that involved a common difference between successive terms geometric sequences that involved a common ratio between successive terms

Each of these is a candidate for our musical pattern; it remains for us to determine which one might be correct. Using the octave beginning at the note called “middle C” on the piano keyboard, we have the following facts:   

the frequency of middle C (C4) is approximately 261.625 cps (cycles per second) the frequency of the C (C5) above middle C is approximately 523.25 cps there are thirteen notes altogether between C4 and C5, including the C notes themselves Note: The C note with a frequency of 261.625 cps is called “middle C” due to its location in the approximate middle of the standard piano keyboard; it is also referred to as “C4” since it is the fourth C note from the left end of the keyboard. The C note directly above middle C is referred to as C5 since it is the fifth C note from the left end of the keyboard.

The Arithmetic Pattern If we assume that the frequencies of the in-between notes follow an arithmetic pattern, then we can determine the common difference between terms. Since there are thirteen notes from the beginning of the octave to its end, there must be twelve intervals in between them. We may find the common difference as follows: 523.250 cps approximate frequency of C5 – 261.625 cps approximate frequency of C4 261.625 cps difference in frequency Dividing this difference by twelve will determine the common difference in frequency between each of the thirteen notes: 261.625 cps/12 intervals  21.80208 cps/interval Hence, successively adding approximately 21.80208 cps to the frequency of each note will produce an arithmetic sequence of frequencies from C4 to C5: note C (C4) C# D

frequency 261.625 283.42708 305.22916

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

Eb E F F# G G# A Bb B C (C5)

327.03124 348.83332 370.63540 392.43748 414.23956 436.04164 457.84372 479.64580 501.44788 523.25

The Geometric Pattern If, on the other hand, we assume that the frequencies of the in-between notes follow a geometric pattern, then we can determine the common ratio between terms. As noted earlier, since there are thirteen notes from the beginning of the octave to its end, there must be twelve intervals between them. We may find the common ratio, r, by solving the appropriate equation: frequency of C4 · r · r · r · r · r · r · r · r · r · r · r · r = frequency of C5 frequency of C4 · r 12 = frequency of C5 261.625 · r 12 = 523.25

Dividing by 261.625 produces the result: r 12 = 2

Taking twelfth roots provides the solution: r = the twelfth root of 2 =

12

2  1.059463

Hence, successively multiplying each note by the twelfth root of 2 will produce a geometric sequence of frequencies from C4 to C5:

note C (C4) C# D

frequency 261.625 277.18203 293.66413

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

Eb E F F# G G# A Bb B C (C5)

311.12631 329.62684 349.22748 369.99362 391.99459 415.30380 439.99905 466.16275 493.88223 523.25

Note: Due to a slight rounding off of the value for C4, the values in the table above are very slightly inaccurate; the standard tone for A above middle C is defined as 440 cps whereas the value in our table is 439.99905.

Will the Real Frequency Sequence Please Stand Up? Both the arithmetic and the geometric sequences appear to accomplish the purpose of spreading the notes “evenly” between C4 and C5, so the question still remains: Which sequence, if either, is the correct pattern?

******************** Analytical Moment: Here is an opportunity to make a mathematical conjecture. What do you think?  ********************

In order to find out, we’ll need to extend our reasoning a bit. The answer to our question may become clearer if we continue each of the sequences to the next higher C note, C6. This will give us a broader view of the situation and will provide further data on how the sequences progress. Since C6 is an octave above C5, it will have a frequency that is twice that of C5; i.e., C6 will have a frequency of approximately 1046.5 cps. Using this value, we may now construct the frequency sequences for the octave from C5 to C6. For the arithmetic sequence, we can determine the common difference as before: 1046.50 cps approximate frequency of C6 – 523.25 cps approximate frequency of C5 523.25 cps difference in frequency

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations 523.25 cps/12 intervals  43.60417 cps/interval Hence, successively adding approximately 43.60417 cps to the frequency of each note will produce an arithmetic sequence of frequencies. Note that this common difference is different than the one obtained for the octave from C4 to C5. We may also solve for the common ratio in the geometric sequence: frequency of C5 · r · r · r · r · r · r · r · r · r · r · r · r = frequency of C6 frequency of C5 · r 12 = frequency of C6 523.25 · r 12 = 1046.50 Dividing by 523.25 produces the result:

523.25  r 12 1046.50  523.25 523.25 r 12 = 2

Taking twelfth roots provides the solution:

12

r 12  12 2 1

r = the twelfth root of 2 =

12

2 = 2 12  1.059463

Note that this is the same common ratio that we found with regard to C4 and C5. The tables below contain the resulting arithmetic and geometric sequences between C5 and C6:

Arithmetic sequence d  43.60417 note frequency 523.25 C (C5) C# 566.85417 D 610.45834 Eb 654.06251 E 697.66668 F 741.27085 F# 784.87502 G 828.47919 G# 872.08336 A 915.68753 Bb 959.29170 B 1002.89587 1046.50 C (C6)

Geometric sequence r  1.059463 note frequency 523.25 C (C5) C# 554.36406 D 587.32827 Eb 622.25262 E 659.25369 F 698.45495 F# 739.98725 G 783.98918 G# 830.60760 A 879.99810 Bb 932.32551 B 987.76447 1046.50 C (C6) 217

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

The Bigger Picture While the notes in each of these sequences are “evenly” spread throughout the octave from C5 to C6, looking at a bigger picture will enable us to decide with certainty which of these is incorrect. The table below contains the common differences and common ratios for both octaves: octave

C4–C5 C5–C6

arithmetic common difference 21.80208 43.60417

geometric common ratio 1.059463 1.059463

As we noted earlier, the arithmetic common difference for each octave is different. As long as we stay within a single octave, this doesn’t present a problem; however, as soon as we attempt to transition from one octave to the next we are faced with a difficulty. Together, the two octaves form the sequence: C4 C# D Eb E F F# G G# A Bb B C5 C# D Eb E F F# G G# A Bb B C6

More specifically, the note progression B, C5, C# involves two different “common” differences: B  C5 represents a change in frequency of approximately 21.80208 cps C5  C# represents a change in frequency of approximately 43.60417 cps

This abrupt change in frequency means that there really isn’t a single “common” difference in the arithmetic frequencies from C4 to C6; in fact, it is impossible to find a single common arithmetic difference that will apply to the octaves C4–C5 and C5–C6 at the same time due to the requirement that the C notes have to double in frequency in each octave. The frequency values for C4, C5 and C6 force us to use different “common” differences within each octave and hence we may conclude that a single arithmetic pattern that could apply to the full range of notes on the piano keyboard simply doesn’t exist. On the other hand, the geometric common ratio is the same for both octaves and provides a uniform transition from one octave to the next. This is a much more likely candidate for the actual note progression pattern across the keyboard. To verify that the geometric sequence is the one actually used in the twelve-tone scale, we would have to compare the mathematical frequencies in our table of values with the actual frequencies of a properly tuned piano or other instrument. ******************** As it turns out, pianos are tuned in this manner, using the twelfth root of two as a common ratio between successive notes. Our supposition concerning the occurrence of a major type of mathematical pattern in the note frequencies has proved to be true!

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

A Powerful Generalization Using our analytical reasoning skills, we may now restate what we have discovered and make a sweeping generalization concerning the frequency of a given note. In order to do this, we’ll make use of the fact that the note A above middle C (also designated A4) has a frequency of exactly 440 cycles per second (based on the American standard definition): frequency of A above middle C = frequency of A4 = 440 cps We’ll also make use of our conclusion that the common ratio between successive notes is the twelfth root of two:

frequency of the note that is a half  step above the given note 12  2 frequency of a given note Rearranging this equation by multiplying by the frequency of a given note shows that: frequency of the note that is a half-step above a given note =

12

2 · frequency of a given note

In the octave that begins with middle C (C4), C4 -- C# -- D -- Eb -- E -- F -- F# -- G -- G# -- A4 -- Bb -- B -- C5

we have: frequency of Bb = frequency of B =

12

12

frequency of C5 =

2 · frequency of A4

2 · 12 2 · frequency of A4

12

2 · 12 2 · 12 2 · frequency of A4

As you can readily observe, there is a pattern emerging here. The number of occurrences of 12 2 in a given frequency equation is directly related to the given note’s number of half-steps above A4: note

Bb B C5

number of half-steps above A4 1 2 3

number of occurrences of 12 2 1 2 3

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

This leads directly to a formula for the frequency of any note above A4. To generalize, we’ll use the letter n as a variable to represent the number of half-steps above A4:

note

number of half-steps above A4 1 2 3 n

Bb B C5 note

number of occurrences of 12 2 1 2 3

What is the number of occurrences of 12 2 in the bottom row of the table? Clearly the answer is n. Since multiple occurrences of the factor 12 2 can be represented by the concept of “raising to a power,” we can construct the following formula: frequency of a note that is n half-steps above A4 =

 2  · frequency of A4 12

frequency of a note that is n half-steps above A4 =

n

 2  · 440 cps 12

n

This simple formula provides a convenient method for determining the frequency of any note above A4. ******************** This process of generalization is quite powerful and allows us to establish relationships that are both practical and efficient.

******************** If we need or want to know the frequency of a note that is twenty half-steps above A4, we no longer need to multiply 440 by 12 2 twenty times in succession; we can simply multiply 440 by one value:

 2 12

20

. Elegant, isn’t it? 

Conclusion In this section, we briefly examined the nature of sound and investigated the basic structure of the musical scale used in most Western popular music today. After making a conjecture that the frequencies of the notes in a scale followed a mathematical pattern, we constructed frequency patterns using two very prominent types of mathematical sequences: arithmetic and geometric.

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

Using our analytical reasoning skills, we observed that the arithmetic sequence did not allow for a smooth transition of frequency changes from one octave to another and then concluded that the geometric sequence was the most likely frequency pattern. In actuality, the twelve-tone musical scale is based on the geometric sequence of tonal frequencies.

********************

Exercises and Projects for Fun and Profit Exercises in Analysis Note: In answering some of the exercises, you will need to know how to find the root of a number on your calculator. For example, to find the twelfth root of 2, you may instead raise 2 to the 1/12 power:

On a scientific calculator, first type in the number 2, then press the “y^x” key (or the “x^y” key), type a left parenthesis, then type “1/12” (i.e., 1 divided by 12), type a right parenthesis, and press the “=” key. If you have a graphing calculator, please use the caret key, ^, in place of the yx or the xy key. To find numerical roots other than the twelfth root, simply replace the number 12 in the above example with the number of the root that you would like to find, e.g., use 4 for the fourth root, 7 for the seventh root, etc. Note: See Helps and Hints for Exercises on pp.442–443 in Appendix A. Unusual Scales 1. In the country of Spotsylvania, there are only four tonal steps (rather than our conventional twelve) between a given note and a note having exactly twice the frequency of the given note. Determine the common ratio between successive notes in this scale. Note: A given note means an arbitrarily selected note—any frequency will do. To make things more concrete, though, you may use the note with a frequency of 440 cycles per second as the "given note".

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

In this exercise, then, the "given note" would have a frequency of 440 cycles per second and the note having exactly twice the frequency of this note would have a frequency of 880 cycles per second. As stated, there are four tonal steps between these two notes; hence you may construct the following equation: 440 · r · r · r · r = 880 or 440 · r 4 = 880

Solving this equation for r will produce the common ratio (see p.217 for an example of solving a similar equation). 2. In the principality of Harmonium, there are only seven tonal steps (rather than our conventional twelve) between a given note and a note having exactly twice the frequency of the given note. Determine the common ratio between successive notes in this scale by solving the equation below for r (see p.217 for an example of solving a similar equation): 440 · r · r · r · r · r · r · r = 880 or 440 · r 7 = 880

3. In the land of Trivia, the concept of “three-ness” is employed whenever possible. As a result, the inhabitants designed their musical “octave” based on a given note and a higher note that had a frequency that was three times that of the given note (rather than our conventional two times). They also designed a musical standard in which “middle C” (C4) had a frequency of exactly 300 cycles per second. In this musical system, a. What is the frequency of C4? (Answer: 300 cycles per second.) b. What is the frequency of C5? c. What is the frequency of C6? 4. In the land of Trivia described in Exercise #3 above in which the highest note in an octave had a frequency that was three times that of the lowest note, the inhabitants also employed a twelve-tone scale based on a geometric progression of frequencies (just as we do in most Western music today). Determine the common ratio between successive notes in this scale by solving the equation below for r (see p.217 for an example of solving a similar equation): 300 · r · r · r · r · r · r · r · r · r · r · r · r = 900 or 300 · r 12 = 900 The Power of Generalization Revisited 5. As an opportunity to exercise your generalization powers, please construct a formula for the frequency of a note that is n half-steps below A4. This formula will be similar to the one developed in this section for the frequency of a note that is n half-steps above A4 (see pp.219– 220).

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

Creative Projects: Creating Good Vibrations Famous Math Quotes ********************

CREATING GOOD VIBRATIONS There are four basic types of vibrations:  Vibrations made by strings (guitar, violin, cello, etc.)  Vibrations made by reeds (clarinet, saxophone, harmonica, etc.)  Vibrations made by columns of air (flute, piccolo, recorder, etc.)  Vibrations made by percussions (drums, cymbals, triangles, vibraphones, etc.) For this hands-on project, get in a creative mood and create some simple musical instruments that will illustrate the four basic types of vibrations. In particular, create each of the following:    

a stringed instrument a wind instrument that uses a reed to create vibrations a wind instrument that uses a column of air to create vibrations a percussion instrument

All of these can be made from simple household items—nothing fancy is required, just the very basic capability of producing the various types of vibrations. You’ll need to put on your creative thinking caps and be imaginative. Assignment:

After you have succeeded, please write a brief report (minimum of one-half page) describing:  

each of your four homemade instruments the type of vibration that each embodies

Project Extension: “Equal-tempered” tuning: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on what is meant by “equal-tempered” tuning in relation to pianos. Write a brief report on your findings (minimum of one-half page).

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Math Is Everywhere! Explore and Discover It! 6.1 The Nature of Sound and Musical Scales: Good Vibrations

FAMOUS MATH QUOTES Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on famous math quotations and find three quotations from famous people that contain references to both mathematics AND music in the same quote (i.e., show a connection between mathematics and music). An example is the statement attributed to Pythagoras in which he states that, “All nature consists of harmony arising out of number.” In this case, “harmony” relates to music and “number” relates to math. The essential requirement is that each quote that you find must contain references to BOTH music AND mathematics in the SAME quote.

Assignment:

Please write a brief report containing for each quote:   

the quotation the name of the person being quoted the source in which you found it

********************

You may be surprised (and thoughtfully challenged) with some of the quotes that you discover. Enjoy! 

********************

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Math Is Everywhere! Explore and Discover It! 6.2 The Mathematics of Stringed Instruments: Structured Harmony

6.2 The Mathematics of Stringed Instruments: Structured Harmony “No one really understood music unless he was a scientist, her father had declared, and not just a scientist, either, oh, no, only the real ones, the theoreticians, whose language is mathematics. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships. And relationships, he had told her, contained the essential meaning of life.” —Pearl S. Buck

Introduction Stringed instruments offer some detailed insight into how mathematics and music are interrelated. The relative frequencies of the vibrations of each string on a stringed instrument are carefully designed and calculated to produce harmonious sounds when the strings are plucked or strummed together. As you might suspect, these harmonies are quite mathematical in nature. In the Exercises and Projects for Fun and Profit at the end of this section, you will have an opportunity to explore the nature of these relationships in a guitar, a well-known six-stringed instrument that has been popularized in recent decades by the rock-and-roll groups that have been a part of the music scene in America. The guitar is an interesting instrument to explore—its fret structure is very mathematically designed and, in the acoustic models, the body shape and internal structure are still the subject of much study and experimentation with regard to producing resonant and well-balanced sounds. Music is indeed a rich part of our heritage and seems to be a deep and abiding part of our nature. In fact, the latest research in Physics, known as String Theory, seems to indicate that at the core of all matter are tiny, vibrating bits of stuff called “strings.” These vibrating bits of stuff, of course, are producing some kind of musical tones since music is simply a collection of vibrations. Hence, it may well be that we are all musical at the very depths of our physical beings.

******************** What an inspiring thought! It’s quite delightful to think that the very fabric of our being is musical in nature!  ********************

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Math Is Everywhere! Explore and Discover It! 6.2 The Mathematics of Stringed Instruments: Structured Harmony

********************

Exercises and Projects for Fun and Profit

Creative Project: Guitar Analysis

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Math Is Everywhere! Explore and Discover It! 6.2 The Mathematics of Stringed Instruments: Structured Harmony

GUITAR ANALYSIS Music, as we all know, is a delightful part of life. It can affect our moods, attitudes, and countenance and elevate our spirits. As we examine music and its structures and relationships more closely, we find that mathematics is inherent in every aspect. In this project, we are going to analyze the structure of a guitar to discover the relationships between the notes produced and the physical location of the frets on the neck of the guitar.

The Guitar: A Most Delightful Instrument A guitar consists of a hollow body (to produce volume and resonance), a long, attached neck with frets (thin metal bars) spaced out along its length, and six strings that extend from a bridge on the body to tuning pegs at the far end of the neck. Plucking a string produces a vibrating motion in the string; the frequency of the vibration is called the fundamental frequency of the string and is measured in cycles per second. Pressing a string against one of the metal frets effectively shortens the length of the string and produces a correspondingly faster vibration and higher pitch, or sound. There is a mathematical relationship between the length of the string and the resulting frequency. Examples:  half the length of the string produces a frequency that is twice that of the fulllength string  one-third the length of the string produces a frequency that is three times that of the full-length string As you can see, this relationship is what we call “inverse”—when one of the related variables increases in value, the other variable decreases in value.

Musical Scale: The Frequency Staircase It is this inverse relationship that we are going to investigate in detail in this project. Since the seventeenth century, most musical instruments in Western cultures have been based on a twelvetone scale. In a twelve-tone scale, there are twelve notes in an octave and each note is separated from the next by a half-step. On a piano keyboard, each key (both black and white) is a half-step apart from its neighbors; on a guitar, the frets are positioned on the neck so that each fret represents a half-step from its neighbors.

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Math Is Everywhere! Explore and Discover It! 6.2 The Mathematics of Stringed Instruments: Structured Harmony

An octave is a series of notes beginning at a certain frequency and continuing to a frequency with twice the original value. There are actually 13 notes contained in an octave, 8 whole notes and 5 half-notes, and a resulting 12 half-steps in between them.

Example: notes: half-steps:

E F F# G G# A A# B C C# D D# E 1 2 3 4 5 6 7 8 9 10 11 12

The frequency of the vibration of each note is a constant multiple of the frequency of the preceding note. This one special multiplier, then, can be used from note to note to determine the proper frequencies as one progresses up the scale. Since there are twelve half-steps in an octave, and since the frequency of the highest note in the octave is exactly twice the frequency of the lowest note in the octave, it is clear that each half-step must somehow represent one-twelfth of this difference in frequency.

The Twelve-tone Scale Multiplier If we were dividing the octave into equal parts in an additive fashion, we could simply divide the difference in the high and low note frequencies by twelve, and then add this value successively to each note to obtain the next frequency. However, we are trying to divide the octave in a multiplicative manner rather than an additive manner so we need a different approach. We might think of it this way: begin with the frequency of the lowest note and multiply it repeatedly by some value (unknown at the moment) twelve times to reach the frequency of the highest note. At each step along the way, the frequency of one of the intermediate notes would be obtained, and after the twelfth multiplication the frequency of the highest note would be obtained. This highest note frequency is exactly twice the frequency of the lowest note and on this basis we can turn the above process into an equation that will help us determine the special multiplier that is needed. If we represent the lowest frequency by the variable “F” and the special multiplier by the constant “m”, we can write the following equation: F · m · m · m · m · m · m · m · m · m · m · m · m = 2F F  m12  2 F

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Math Is Everywhere! Explore and Discover It! 6.2 The Mathematics of Stringed Instruments: Structured Harmony

We can now solve this equation for m: F  m12 2 F  F F

divide the equation by F

m12  2

simplify

12

m12  12 2

m  12 2  1.059 This value for m,

12

take the twelfth root of both terms

simplify

2 , is the special multiplier that produces the twelve-tone scale.

Why Aren’t Guitar Frets Equally Spaced? To see how all of this comes together in a guitar, we need to look more closely at the physical dimensions involved with the placement of the frets on the neck. On the surface, it would be logical to assume that since each of the tonal half-steps are equally spaced, the frets on the neck of the guitar would also be equally spaced; however, that is not the case. In fact, the frets get progressively closer together as you move from the top of the guitar neck (where the tuning pegs are located) to the bottom of the neck (where the neck joins the guitar body). It is this seemingly illogical fret spacing that we will now investigate in detail to determine how it works. It is important to note that, as mentioned earlier, when you press a string against a fret, you effectively shorten the length of the string; this in turn raises the frequency of the string’s vibration, producing a higher note. The ratio of this effective string length to the original string length is used to compute the frequency ratio and the ultimate frequency of the note produced as shown below in the table.

Worksheet and Project Questions The following worksheet shows the actual measurements (in inches) of the fret placements on the neck of a guitar (as accurately as I could measure them). Although these placements are obviously the same for all six strings, we’ll focus on one of the E strings for the purpose of this project. The objective is to complete the entries on the following worksheet and then answer the questions that follow to discover the relationship between fret placement, string length and frequency.

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Math Is Everywhere! Explore and Discover It! 6.2 The Mathematics of Stringed Instruments: Structured Harmony

******************** Have fun as you explore the wonders of mathematics and music! You might even want to whistle while you work…. 

******************** WORKSHEET INSTRUCTIONS: Step 1: In the worksheet below, begin by converting all of the mixed numbers in the second column (distance to fret, D) to decimal values. Examples: 3  1 , or 1 and 3/8, has the decimal value of 1.3750 8 3  2 , or 2 and 3/4, has the decimal value of 2.7500 4 Please carry out decimals to four places to preserve accuracy in the measurements. Step 2: Next, find all of the values in the third column (effective length of string) by using the given formula, F–D or 25.5–D, to compute each value. Use the value for D from the previous column. Example: In the second row, the value for F–D is 25.5–1.3750=24.1250 Step 3: Then, find all of the values in the fourth column (length ratio) by using the given formula, (25.5–D)/25.5, to compute each value. You may use the value for 25.5–D from the previous column. Example: In the second row, the value for L=(25.5–D)/25.5 is (25.5–1.3750)/25.5 = 0.9461; using the value from the previous column saves a bit of work: 24.1250/25.5 = 0.9461 Step 4: Proceed to the fifth column (frequency ratio) and use the given formula, 1/L, to compute each value. Use the value for 1/L from the previous column. Example: In the second row the value for 1/L is 1/0.9461 = 1.0569

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Math Is Everywhere! Explore and Discover It! 6.2 The Mathematics of Stringed Instruments: Structured Harmony

Step 5: Finally, find all of the values in the last column (note frequency) by using the given formula, (1/L) ·328, to compute each value. Example: In the second row the value for (1/L) ·328 is 1.0569 · 328 = 347 (rounded off to the nearest whole number).

After completing the worksheet, you may then answer the questions that follow. WORKSHEET: To begin, fill in the entire table of values, using the formulas given and the examples provided. Please use four decimal places for accuracy! ******************** Formulas and Variables:

full length of the E string: 25.5 inches = K (Note: K is an arbitrarily chosen variable) fundamental frequency of the E string: 328 cps (cycles per second) distance from top of guitar neck to a particular fret: D effective length of E string = full length – distance to fret or effective length = K–D length ratio 

effective length of E string K D L full length of E string or K

frequency of note 

1 1  fundamental frequency or f   328 length ratio L

********************

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Math Is Everywhere! Explore and Discover It! 6.2 The Mathematics of Stringed Instruments: Structured Harmony

note

E F F# G G# A A# B C C# D D# E

distance to fret (mixed number): D 0 3 1 8 3 2 4 15 3 16 3 5 16 3 6 8 1 7 2 9 8 16 7 9 16 5 10 16 3 11 16 1 12 16 3 12 4

distance to fret (decimal): D 0.0000 1.3750

effective length of string: K–D= 25.5–D 25.5000 24.1250

2.7500

22.7500

length ratio:

frequency ratio:

note frequency:

L= (25.5–D)/25.5 1.0000 0.9461

1/L 1.0000 1.0569

f= (1/L) ·328 328 347

0.8922

1.1208

Assignment:

1. Complete the worksheet above by following the five steps listed on pp.230–231. 2. For each two successive notes, compare the frequency of the higher note (f) to the frequency of the lower note (f). (You will work your way down the list of notes in pairs, overlapping each pair as you go.)

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368

Math Is Everywhere! Explore and Discover It! 6.2 The Mathematics of Stringed Instruments: Structured Harmony

Examples:  For the first two notes, E and F, the frequencies are 328 and 347 cycles per second, respectively. The ratio of the higher to the lower is 347/328 = 1.0579

For the next two successive notes, F and F#, the frequencies are 347 and 368, respectively. The ratio of the higher to the lower is 368/347 = 1.0605

Now continue with the rest of the pairs of notes, and make a list of these ratios (in order). Please calculate all values to four decimal places. What do you notice about these values? (Bear in mind that there is some small amount of error in the measurements—I took these measurements from my wife’s guitar and while my measuring skills are pretty good they are not perfect….)

[Answer: You should notice that all of the ratios are approximately the same and quite close to the value of the special multiplier “m” as described earlier.] 3. For each two successive effective string lengths, compare the smaller effective length of string to the larger effective length of string. Examples:  For the first two notes, E and F, the effective string lengths are 25.5000 and 24.1250, respectively. The ratio of the smaller to the larger is 24.1250/25.5000 = 0.9461

For the next two successive notes, F and F#, the effective string lengths are 24.1250 and 22.7500, respectively. The ratio of the smaller to the larger is 22.7500/24.1250 = 0.9430

Now continue with the rest of the pairs of notes, and make a list of these ratios (in order). Please calculate all values to four decimal places. What do you notice about these values? (Ditto for the error in measurements…)

[Answer: You should notice that all of the ratios are approximately the same, but NOT equal to the value of the special multiplier “m” as described earlier.] 4. However, there is a definite relationship between the average frequency ratio found in Question #2 above and the average string length ratio in your answer to Question #3 above. Can you determine what that relationship is? [Answer: The average frequency ratio is the multiplicative inverse (or reciprocal) of the average string length ratio.] 5. Based on the above findings, explain why the fret spacings on a guitar are NOT equal (you might consider the effect on string lengths and note frequencies if the spacings were equal…). 233 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 6.3 Digital Music and CD’s: Applied Mathematics

6.3 Digital Music and CD’s: Applied Mathematics “Numbers are intellectual witnesses that belong only to mankind.” —Honoré de Balzac

Introduction In recent years, music recording has gone through a succession of remarkable changes. From Thomas Edison’s first gramophone with wax disks and a stylus made of bamboo to celluloid tape cartridges (remember the old 8-track tape players?) to music CD’s (formally known as compact disks) and MP-3 players, the recording process has come a long way. The advent of computer technology has made all of these advances possible and the binary system with its 0’s and 1’s has been instrumental in the process.

In this section, we will examine how mathematics has been applied to the field of music in order to create more effective and efficient ways of recording and transmitting musical works. To most people, a music CD simply looks like a shiny aluminum disk that has been covered with a plastic coating and a paper label; however, there is a lot of mathematics hidden beneath the surface (quite literally! ). Once again, we will see the power of the smallest of all number bases—the binary number system—as it is applied to a piece of continuous music to divide it into a huge number of discrete segments so that the segments may be easily transmitted electronically.

Music as a Collection of Sound Waves In essence, music is a collection of sound waves, each of which has a particular shape and amplitude. Although a wave is a continuous process, it may be broken up into tiny, discrete pieces, each of which has an amplitude (i.e., volume) that can be electrically measured in terms of its voltage. Each voltage measurement (in the base 10 system) may then be converted to binary numbers (in the base 2 system) and encoded as a string of 0’s and 1’s. These 0’s and 1’s may then be converted to a physical representation on a music CD and thus encode the particular voltage measurement. After millions of these voltage measurements have been converted and encoded on the CD, a special laser device is employed to “read” the encoded information and then an electronic converter is employed to change this information into audible sounds. While the process is a bit 234 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 6.3 Digital Music and CD’s: Applied Mathematics

technical, the concept of mathematically converting a wave into a series of binary numbers is at the heart of it all. In the Exercises and Projects for Fun and Profit that follow, you will have the opportunity to explore some Web-based resources concerning digital music and mathematics in the form of how CD’s work and how analog (i.e., continuous) music is converted through applied mathematics to digital (i.e., discrete) music. Once again, you will see that the simplest of all number systems has quite amazing and far-reaching applications!

******************** There seems to be a life-principle that appears in all disciplines and is embodied by the old adage, “Big things come in little packages.” In the case of the binary number system this certainly holds true—the lowliest of integers, 0 and 1, are proving to have enormous impact and application in this modern age of technology that we have entered.

********************

Exercises and Projects for Fun and Profit Creative Project: Digital Music and Mathematics

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Math Is Everywhere! Explore and Discover It! 6.3 Digital Music and CD’s: Applied Mathematics

DIGITAL MUSIC AND MATHEMATICS During the past few decades, developments in technology have made possible the conversion of musical sounds into digital formats. With the advent of digital storage and retrieval devices such as the compact disk (CD), substantial collections of sounds can easily be stored—hence the revolution in the music industry and the switch from cassette tapes to music CDs as the medium of choice. The conversion of musical sounds into digital format involves transforming the music into something a computer can understand. Here is another example of where number bases are used, viz., the binary system of 0’s and 1’s. The objective of this project is to discover how this process actually takes place.

******************** It’s quite a fascinating process involving the binary number system and some high technology. Hope you enjoy the project!  ********************

Assignment: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on:   

how CDs are made how CDs work how music is converted from sound waves into a binary format

Afterwards, write a report that describes your findings. Reports should be a minimum of two pages in length (double-spaced with a maximum of 12pt font size and 1-inch margins). Your complete report should include a minimum of one and one-half pages of text, not including pictures and diagrams. With pictures and diagrams included, your report should be a minimum of two pages in length.

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Math Is Everywhere! Explore and Discover It! 6.3 Digital Music and CD’s: Applied Mathematics

******************** After this project, you’ll be able to impress your friends and family with how much you know about modern technology!  ********************

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Math Is Everywhere! Explore and Discover It! 6.3 Digital Music and CD’s: Applied Mathematics

Notes:

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Math Is Everywhere! Explore and Discover It! 7.0 Introduction and Objectives

Chapter Seven Mathematics in Art, Architecture and Nature 7.0 Introduction and Objectives The thumbnail sketches on this page represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

7.1 Mathematical Perspective in Art: A Renaissance Breakthrough

Horizon line

Vanishing point

7.2 Symmetry and Tilings: Mathematical Beauty and Artistry

7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci x

1 5  1.618 2

7.4 Long-term Projects 239 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 7.0 Introduction and Objectives

Learning Objectives for Chapter 7 1. To enable students to develop their analytical reasoning skills through pattern recognition and problem solving; in particular, students will calculate the Golden Ratio. 2. To enable students to develop their deductive reasoning skills through applying mathematical relationships and constructing the Fibonacci sequence. 3. To enable students to identify the various types of mathematical symmetry and explain their uses in tilings of the plane. 4. To enable students to identify spiral growth patterns in nature; in particular, students will construct the logarithmic spiral found in seashells. 5. To allow students to investigate occurrences of the Golden Ratio and the Fibonacci sequence in their homes and yards. 6. To enable students to apply their knowledge of symmetry through a practical application involving textile designs. 7. To allow students to exercise their creativity by conducting Internet research and writing reports on various mathematical topics involving the connections between mathematics and other fields as well as mathematics and nature. 8. To enable students to gain some familiarity with Web-based mathematical resources and to develop their critical thinking skills in assessing and evaluating these resources. 9. To introduce some long-term projects that will involve advance planning and preparation. 10. To demonstrate the pervasiveness of mathematics in the world around us. 11. To engage students in active learning through the use of interactive and hands-on projects and activities. 12. To empower students to become lifelong learners.

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Math Is Everywhere! Explore and Discover It! 7.1 Mathematical Perspective in Art: A Renaissance Breakthrough

7.1 Mathematical Perspective in Art: A Renaissance Breakthrough “The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.” —Galileo Galilei

Introduction Since the beginning of recorded history, man’s artistic expressions have been evident. Cave drawings found in various parts of the world are among the first of these known expressions and generally depict basic events of life via stick-figure drawings. In comparison to modern artistic expressions, these stick-figure drawings appear rather primitive. Down through the centuries, artistic expression has undergone various stages of development. Both the medium and the style have changed with time, from the stick-figure drawings of cave dwellers to drawings and designs of greater complexity in various media such as stone, wood, clay, ivory, bits of colored stone and glass, and more.

Mathematical Perspective in Art Up until the time of the Renaissance in the fifteenth and sixteenth centuries, most art was rather “flat” in appearance and seemed to lack the three-dimensional depth that we are used to seeing today. During the Renaissance period, artists became interested in how to create more realistic paintings in terms of visual depth and perspective. After experimenting with various techniques, they began to use a mathematical approach to accomplishing this objective and developed a theory of perspective based on geometric vanishing points. This provided them with a very effective way to produce realistic drawings that appeared to have the proper depth and perspective. Mathematical perspective is based on how we normally perceive the world around us—an object that is close to us appears larger than that same object when it is at a distance from us. For example, when you stand next to your car in the parking lot, your car looks to be its normal size; however, if, while standing next to your car, you observe the same model car at the far end of the parking lot, the more distant car appears to be much smaller. An illustrative example of this difference in perception involves a pair of long, straight railroad tracks. An observer standing between the two tracks perceives them to be their actual distance apart when looking directly down at them from above; however, the tracks appear to get closer together as the observer looks farther down the tracks until, at some considerable distance away from the observer, the tracks appear to converge into a single point. This is the point that Renaissance artists named the vanishing point.

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Math Is Everywhere! Explore and Discover It! 7.1 Mathematical Perspective in Art: A Renaissance Breakthrough

Vanishing Points and the Horizon Line In order to create the illusion of distance within a drawing or painting, Renaissance artists employed the concept of a vanishing point to great advantage. A vanishing point is an imaginary point in a drawing where all lines that are parallel to a given line converge. A single vanishing point was employed for drawing objects that were viewed head-on; two vanishing points were used for objects that were viewed from an angle. We will examine one-point perspective for the sake of simplicity and leave the two-point perspective as a Further Investigation in the Exercises and Projects for Fun and Profit at the end of this section. In the case of the railroad tracks, the one-point perspective views these head-on as though the railroad tracks were perpendicular (or orthogonal) to the plane of the drawing. A single vanishing point (V) is chosen somewhere in the drawing, usually somewhat above the position of the beginning of the tracks; the horizontal line through the vanishing point is known as the horizon line.

V

horizon line

Figure 1 This single vanishing point may also be used to view the railroad tracks from the side provided that the tracks are intended to be perpendicular to the plane of the drawing.

V

horizon line

Figure 2 242 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 7.1 Mathematical Perspective in Art: A Renaissance Breakthrough

Auxiliary Vanishing Point The proper placement of railroad crossties in the above drawings (see Figures 1 and 2) is a bit more complicated since these crossties are parallel to the plane of the drawing instead of perpendicular to it. In order to obtain the correct perspective, an additional point called an auxiliary vanishing point is needed; this point, A, is placed on the horizon line at some distance away from the principal vanishing point (see Figure 3 below). Afterwards, a line is drawn that connects A with the foremost point on the railroad track that is farthest away from A. This line intersects the near side of the tracks at point B. A horizontal line segment is then constructed at B that joins the two tracks; it intersects the track on the right at C. Next, a line is drawn that connects A and C. This line intersects the near side of the tracks at D. As before, a horizontal line segment is then constructed at D that joins the two tracks; it intersects the track on the right at E. This process is continued until all of the crossties have been drawn. Each of the lines beginning at A is then erased, leaving only the crossties. The result is a drawing in which true mathematical perspective is achieved.

A

V

D B

horizon line

E C

Figure 3

Conclusion This approach to creating the illusion of three-dimensional perspective on a flat piece of canvas or other drawing medium was quite effective and allowed the Renaissance artists to achieve very realistic effects in their drawings and paintings.

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Math Is Everywhere! Explore and Discover It! 7.1 Mathematical Perspective in Art: A Renaissance Breakthrough

In the Exercises and Projects for Fun and Profit at the end of this section, you will have the opportunity to try your hand at constructing a true perspective drawing of a room with a hardwood floor. When you are finished, you will see how orderly and realistic it looks (if you follow the instructions carefully  ).

********************

Exercises and Projects for Fun and Profit Further Investigation 1. Two-point Perspective Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on two-point perspective for drawing objects viewed at an angle and write a short report (minimum of one page) on how this is done. Please include a webliography of the websites that you visit. ********************

Creative Project: Mathematical Perspective

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Math Is Everywhere! Explore and Discover It! 7.1 Mathematical Perspective in Art: A Renaissance Breakthrough

MATHEMATICAL PERSPECTIVE CONSTRUCTION PROJECT During the Renaissance, artists first began to employ mathematics in their rendering of artistic images. The Italian artist, architect and engineer Filippo Brunelleschi (1377–1446) is generally credited with developing a geometrical approach to three-dimensional realism. Other artists of the period such as Masaccio, daVinci, Michelangelo, Dürer and Holbein also began to use this technique in their paintings. In this project, we will construct a simple scene using the geometric principles of the Renaissance to illustrate the effect of true mathematical perspective. Assignment: We will begin with a blank 8 ½ x11 inch sheet of typing paper placed horizontally on a work surface. Note: It’s important to either “lightly draw” or “draw dark, solid lines” according to the directions—the lightly drawn lines will be erased at the end of the project. . 1. Lightly draw a horizontal line 2 and ½ inches (2 ½ inches) from the bottom of the paper and all the way across the paper. Label this line L1. 2. Measure 2 ½ inches in from the left end of this line and mark a point, A. 3. Continuing from point A and moving to the right, mark 12 additional points ½ inch apart along the original horizontal line L1. These points are B, C, D, E, F, G, H, I, J, K, L and M, respectively. There should be an additional 2 ½-inch space remaining on line L1 to the right of point M. 4. Connect points A and M with a dark solid line. This will form the foreground base of the finished drawing. 5. Draw a dark, solid, vertical line upward from point A for a distance of 4 inches; label the point at the top of this segment with the letter N. 6. Draw a dark, solid, vertical line upward from point M for a distance of 4 inches; label the point at the top of this segment with the letter P. 7. Connect the tops of these new vertical lines with a dark, solid line segment so as to form a rectangle with line segment AM as the base and NP as the top. This will form the foreground rectangle AMPN of the finished drawing. 8. Lightly draw a horizontal line 5 ½ inches from the bottom of the paper and all the way across the paper. This is called the horizon line; label this line L2.

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Math Is Everywhere! Explore and Discover It! 7.1 Mathematical Perspective in Art: A Renaissance Breakthrough

9. Mark a point at the middle of line L2; this is point Q and is known as the principal vanishing point. 10. Lightly connect points A and Q. 11. Lightly connect points M and Q. The new line segments AQ and MQ now form a triangle together with the base, AM. 12. Draw a dark, solid, SHORT horizontal line segment 4 inches from the bottom of the paper connecting line segments AQ and MQ. This new line segment serves as the crossbar in the letter A formed by AQ, MQ and this new line segment. 13. Label the left end of this SHORT horizontal line segment R and label the right end of it S. These are the two ends of the crossbar on the letter A formed by AQ, MQ and the short horizontal line segment. 14. Darken the line segments AR and MS. These will form the “wall-to-floor” joints in the completed drawing. 15. Lightly draw a line segment connecting points N and Q. 16. Lightly draw a line segment connecting points P and Q. 17. Draw a dark, solid line segment vertically upward from point R to connect with line segment NQ; label this new endpoint T. 18. Draw a dark, solid line segment vertically upward from point S to connect with line segment PQ; label this new endpoint U. 19. Connect the points U and T with a dark, solid line segment. The rectangle RSUT will form the background rectangle of the finished drawing. 20. Darken the line segments NT and PU. These will form the “wall-to-ceiling” joints in the completed drawing. 21. Lightly draw line segments connecting each of the points B, C, D, E, F, G, H, I, J, K and L with point Q. 22. Darken the portions of these line segments between the line L1 and the line segment RS. These lines will form the hardwood flooring of the room in the drawing. 23. You may now erase any line segments that have not been darkened.

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Math Is Everywhere! Explore and Discover It! 7.1 Mathematical Perspective in Art: A Renaissance Breakthrough

24. You may also draw short, horizontal line segments at random places on the hardwood floor to indicate places where the floorboards meet at a joint if you wish.

******************** And now you have done it! Congratulations! What remains is a true mathematical perspective drawing of a rectangular room with walls, ceiling and a hardwood floor. As a finishing touch, you might attempt to draw an object in the room, perhaps a framed painting on the wall or some freestanding object in the middle somewhere. ********************

And now you see how exacting and precise perspective drawing actually is. In the days of the Renaissance, this was a breakthrough in the field of art and it transformed flat, two-dimensional drawings with out-of-kilter perspective into realistic, lifelike images. It was quite revolutionary for its time. In our modern culture today, computer graphics have taken the principles of mathematical perspective and enhanced them with the ability to rotate, reflect and undergo morphing transformations that also represent quite a breakthrough.

******************** We might well wonder what forms artistic expression will take in the next few centuries (or even in the next 50 years).  ??? ********************

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Math Is Everywhere! Explore and Discover It! 7.2 Symmetry and Tilings: Mathematical Beauty and Artistry

7.2 Symmetry and Tilings: Mathematical Beauty and Artistry “The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.” —Aristotle

Introduction As was the case with music, artistic designs have long been a part of mankind’s history, dating back to the days of cave drawings and beyond. In this section, we will investigate a type of design known as a tiling or tessellation in which a particular shape is repeated over and over again in various directions until it completely fills the space around it. A common example of this is a tiled floor in which squares of ceramic tile are placed adjacent to one another in a repeated manner until the entire floor is covered with tiles; another example is found in the tiled structure of a honeycomb in which hexagonal wax cells are constructed adjacent to one another in a repeated manner until the entire honeycomb is formed. In the process of investigating tilings such as these, you’ll learn about the inner geometric workings of a tiling—translations, reflections, and rotations—and about the prototiles and generating sets that form the basis of a tiling. You’ll also learn to recognize tilings in the world around you and come to appreciate the prevalence of tilings in the world of design. Tilings are truly a design science, filled with geometry, color, symmetry and more. In addition, we will investigate the concept of mathematical symmetry and discover how it is related to the inner workings of a tiling. This section will provide opportunities for you to use some basic math concepts to create mathematical “works of art”—you may even want to use them to decorate your home or possessions. 

Examples of Symmetry and Tiling Symmetry and tilings have a rich and colorful history—so much so that entire books have been written on the subject. While we don’t have time and space here to survey their entire development, we’ll take a brief look at some examples of tilings and tessellations in various cultures to whet your appetite for further exploration. Each example listed below is accompanied by a Web-based resource for further exploration. In the Exercises and Projects for Fun and Profit at the end of this section, you will have an opportunity to investigate some of these cultural examples in greater detail. Examples Persian and Oriental rugs are famous for their quality and durability as well as for their artistic designs. Many of these incorporate tilings in the border around the edges and

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Math Is Everywhere! Explore and Discover It! 7.2 Symmetry and Tilings: Mathematical Beauty and Artistry

exhibit types of symmetry in the central portion of the rug. The rich cultural history that is associated with these rugs adds to their intrinsic value. Moorish architecture is full of examples of tilings that have been incorporated into the overall design of buildings and structures. The Alhambra Palace in Spain is a classic example of this type of architecture and it is an excellent place in which to investigate tilings and tessellations. In fact, one of the goals of the architects of this building was to cover all surfaces with decorative motifs. The Navajo Indians are famous for their woven blankets and rugs, some of which incorporate tilings and many of which use geometric designs. The twentieth century art of M.C. Escher is perhaps the most famous example of tilings and tessellations in art. His works include some very unusual and creative tiling designs and are well worth investigating. One of the Creative Projects in the Exercises and Projects for Fun and Profit at the end of this section will provide you with an opportunity to investigate his work in greater depth. Note: Many Internet resources are available for each of these examples; you are welcome to explore further if you are interested.

Tiling Construction Basics: Translations, Reflections, and Rotations As amazing as it may seem, even the most complicated-looking tiling is constructed using only three basic geometric motions: translations, rotations, and reflections. These geometric motions are technically called transformations and are more specifically known as isometries: rigid motions of an object in a plane (flat surface) that preserve both the object’s size and the object’s shape. 

A translation simply moves an object from one location to another: object

B 

translated object (moved to a new location)

B

A rotation rotates an object about a particular point. The following example demonstrates a 90-degree rotation about the center of the letter in a counterclockwise direction:

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Math Is Everywhere! Explore and Discover It! 7.2 Symmetry and Tilings: Mathematical Beauty and Artistry

rotated object (in original location)

A

object

A

A reflection creates a mirror image of an object about a particular axis. The following example demonstrates a reflection about the axis to its right: object

reflected object (in a new location)

T

T

axis of reflection

Glide Reflections: A Favorite Combination A number of tilings and tessellations make use of a translation combined with a reflection to produce what is known as a glide reflection: translated and reflected object (moved to a new location)

B

object

___ ___ ___ ___ ___ ___ ___ ___ ___

B Symmetry The four basic ways of moving an object around in the plane described above correspond to the four types of mathematical symmetry:    

translational symmetry (all objects have this symmetry since they look the same after they have been translated) rotational symmetry (measured in degrees of rotation) reflection symmetry (mirror-image symmetry about some axis) glide reflection symmetry (or glide symmetry; a combination of translation and reflection symmetries)

Some objects or designs can have more than one type of symmetry as in the following examples: 250 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 7.2 Symmetry and Tilings: Mathematical Beauty and Artistry  

the letter A has: o translational symmetry o reflective symmetry about a vertical line through the center of the A the letter H has: o translational symmetry o rotational symmetry of 180 degrees o reflection symmetry about a vertical line through the center of the H o reflection symmetry about a horizontal line through the center of the H o glide reflection symmetry

Symmetry has a natural visual appeal and has been used by artists, architects and designers throughout the ages.

Combination of Motions: Functional and Artistic Composition Now that you have become cognizant of the three basic geometric isometries and corresponding symmetries, you may begin to combine these motions to create more interesting effects such as the glide reflection described above. The combinations are called compositions and allow you to become quite artistic in your expressions. Each of the three isometries may be applied to itself (this composition is technically called an iteration) or to either of the other two isometries. ***** An Opportunity to Exercise Your Analytical Reasoning Skills *****  A reflection applied to a translation would be an example of a composition using two isometries; in fact, this is the glide reflection described above. Using your analytical thinking skills, determine how many compositions are possible using only two of the isometries at a time. ******************** Very good! That’s it! There are nine possible compositions. The reasoning process that you probably used to determine this total involves what is known as the fundamental counting principle. A version of this principle that is applicable to our situation is as follows: If an isometry can be selected in M different ways and a second isometry can be selected in N different ways, then a combination of the two isometries may be selected in M·N ways. Since there are only three isometries from which to choose, it is clear that M=N=3; consequently, there are 3·3=9 possible compositions. In practice, it turns out that some of the effects of these nine are either duplicates of other isometry effects or compositional effects or are somewhat uninteresting from an artistic viewpoint. The isometries and compositions that are most used in tessellations are:  translations, rotations and reflections (the basic isometries—rigid motions in the plane)

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Math Is Everywhere! Explore and Discover It! 7.2 Symmetry and Tilings: Mathematical Beauty and Artistry  

a translation applied to a reflection (glide reflection) a rotation applied to a reflection

This makes life considerably simpler when constructing tessellations. Aren’t you glad? 

Conclusion In the Exercises and Projects for Fun and Profit below, you will have the opportunity to use these isometries and compositions to construct your own creative tessellations and designs. In addition, the projects are designed to make you aware of the many connections between mathematics and art, both in the professional world and in your daily lives and environments. I trust that you will find these to be enjoyable and that you will be inspired to be artistically creative in the process. ********************

Exercises and Projects for Fun and Profit Creative Projects: Tilings in Your Home Symmetry and Tilings The Mathematical Art of M.C. Escher

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Math Is Everywhere! Explore and Discover It! 7.2 Symmetry and Tilings: Mathematical Beauty and Artistry

TILINGS IN YOUR HOME In this project, you are to explore your home or living space for geometric designs and locate three examples of tilings in the structure or décor. As mentioned in class, geometry is an integral and often beautiful part of our human experience and has been put to many commercial uses. There are many instances of geometric applications in your home as well as in the world around you—you just need to tune in to them. Assignment: Please submit a brief report (minimum of one-half page) that describes the tilings that you discover. You may sketch diagrams as illustrations if you’d like. Have fun! Math Is Everywhere!

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Math Is Everywhere! Explore and Discover It! 7.2 Symmetry and Tilings: Mathematical Beauty and Artistry

SYMMETRY AND TILINGS This project involves some reading and learning about the mathematical concepts of symmetry and tilings. Both of these topics have long and illustrious histories; symmetry, in particular, has been used in art and architecture in all major civilizations and cultures. The purpose of this project is to allow you to gain some knowledge and familiarity with these fundamental aspects of geometry.

Assignment: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on symmetry and tilings. Afterwards, please answer the following questions: 1. 2. 3. 4. 5.

Describe the four kinds of symmetry (isometries). Which two symmetries are combined in a glide reflection? Can a tiling be constructed entirely from equilateral triangles? How many different wallpaper tilings are there? What basic geometric shape is used in a Penrose tiling?

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Math Is Everywhere! Explore and Discover It! 7.2 Symmetry and Tilings: Mathematical Beauty and Artistry

THE MATHEMATICAL ART OF M.C. ESCHER M.C. Escher (1898–1972) was a self-taught Dutch artist and mathematician who pioneered the creative use of tilings and tessellations in his art work. He was greatly influenced by the many tessellations that he discovered in the decorative motifs used in the construction of the Alhambra, a fourteenth-century Moorish palace near the town of Granada, Spain. A very original and creative thinker, Escher developed his own categorization schemes for tessellations and used these as a basis for a number of his artistic expressions. In addition, he used mathematics to generate very unusual depictions based on hyperbolic geometry. Altogether, Escher’s work represents some of the best in the twentieth century and has been popularized through the sale of postcards and other items that depict his work. In this project, you will have the opportunity to explore his life and background as well as his many artistic creations. ******************** Assignment: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on Escher (or M.C. Escher). Afterwards, please write a brief report (minimum of one-half page) on those features that interested you most.

******************** Escher’s work is quite fascinating to say the least. His use of metamorphoses and impossible constructions is most unusual and I trust that you’ll find them to be enjoyable as well.  ********************

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Math Is Everywhere! Explore and Discover It! 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci

7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci “The human mind has first to construct forms, independently, before we can find them in things.” —Albert Einstein

Introduction From the aesthetic design of the Parthenon in ancient Greece to the Gothic arches in European cathedrals to the structural strength of the geodesic domes of modern times, mathematics has played an important role in architecture throughout the centuries. In fact, it is safe to say that all architecture involves mathematics of some sort, whether it is in the design or in the structural properties of a building.

The Fibonacci Sequence In this section, we will take a look at a few of the many connections between mathematics and architecture and investigate the Fibonacci sequence, a very unusual sequence of numbers that underlies some of these connections. In addition, in the Exercises and Projects for Fun and Profit at the end of this section, we will see how this sequence of numbers occurs in nature’s architecture in various ways, from seedpods to flower petals to plant structures and more. Fibonacci was a thirteenth-century mathematician who helped to transform the mathematics of his time by promoting the use of the Hindu-Arabic numeration system with its place values instead of the cumbersome Roman numerals of his day. This transition away from Roman numerals opened the door for more advanced mathematical studies to be done. Fibonacci’s book entitled Liber Abaci described methods of calculating in the decimal place-value system and was instrumental in effecting this transition. Liber Abaci is now considered a landmark in the field of mathematics.

The Golden Ratio Another topic that we will investigate in this section is a mathematical relationship that has in more recent times become known as the Golden Ratio and was widely used by the ancient Greeks in their architecture. This ratio originated in the division of a line segment into two smaller line segments of differing lengths. When a line segment is divided into two segments of differing lengths, the result is a total of three line segments: the original segment, AC, and the two smaller segments, AB and BC, where AB is longer than BC as shown below. A B C

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Math Is Everywhere! Explore and Discover It! 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci

To facilitate the explanation of the golden ratio, we will assign lengths to each of these three segments such that:  segment AC has a length of 1 unit (longest segment)  segment AB has a length of x units (middle-length segment)  segment BC has a length of 1 – x units (shortest segment) With respect to these lengths, we may construct the ratio of the longest segment to the longer of the two other segments and we may also construct the ratio of the longer of the two other segments to the shorter of those two segments:

1 x AC AB and and , or using assigned lengths, x 1 x AB BC This relationship was referred to by the Greeks as a division based on the extremes (the longest and shortest lengths) and the means (the middle length, used twice): While they didn’t call it the golden ratio in those days, the Greeks believed that a division in which these two ratios were equal was a very special division that was more pleasing in its interrelationships than any other. The terms divine proportion and golden ratio came into usage many centuries later. The divine proportion, or golden ratio, is defined by the following proportional equation involving the two ratios:

1 x AC AB  , or using assigned lengths,  x 1 x AB BC

Phi and phi: An Unusual Relationship In the above equation involving the lengths of the three line segments, we may use our algebraic skills and the Quadratic Formula to determine which values of x satisfy the equation and produce the golden ratio: 1 x  x 1 x x2  1 x

x2  x 1  0 x

original proportion by cross-multiplication by rearranging terms

2  b  b 2  4ac  1  (1)  4(1)(1)  1  5   by the Quadratic Formula 2a 2(1) 2

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Math Is Everywhere! Explore and Discover It! 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci

This yields the solutions: x

1 5 1 5  1.618  0.618 and x  2 2

Since we are interested only in positive values for the lengths of our line segments, we can ignore the negative value, –1.618; hence, the appropriate solution for x in the golden ratio equation is: x  0.618 . This value was later given the name phi. We could have set out to determine the golden ratio in a similar but slightly different manner, i.e., we could have assigned lengths to each of the original line segments as described below: A

  

segment BC has a length of 1 unit segment AB has a length of x units segment AC has a length of 1+x units

B

C

(shortest segment) (middle-length segment) (longest segment)

In this case, the golden ratio calculation would be slightly different as well and would produce the following proportion: 1 x x  x 1

***** An Opportunity to Exercise your Analytical Reasoning Skills *****  Following the example for the original proportion, solve this second proportion and determine the solutions to the resulting quadratic equation. Observe the similarities between these solutions and the solutions to the original proportion found above. Interesting, isn’t it? ****************************** Somewhat surprisingly, the solutions to this second quadratic equation were additive inverses of the solutions to the first quadratic equation. Since we are interested only in positive values for the lengths of our line segments, the appropriate solution for x in the second golden ratio proportion is: x  1.618 ; this value was given the name Phi to distinguish it from its numerical cousin, phi. As a result, there are two related golden ratios:

phi 

width  0.618 and length

Phi 

length  1.618 width

You will have an opportunity to learn more about their interesting interrelationships in the Exercises and Projects for Fun and Profit at the end of this section.

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Math Is Everywhere! Explore and Discover It! 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci

The Golden Rectangle Using the longest line segment as the length and the longer of the other two segments as the width, we may construct a rectangle that measures 1 unit by 0.618 units.

0.618 units

1 unit This ratio of the width to the length of this rectangle is phi=0.618 and the ratio of the length to the width is Phi=1.618; as a result, this rectangle became known as the golden rectangle. This rectangle has been used by many architects in their building designs, from the time of ancient Greece (and perhaps even earlier) to the present. A recent example is the former Bank of America building in Tampa, Florida, in which the window openings were constructed in the shape of golden rectangles.

******************** This is a most fascinating area of study and I trust that you will learn much here that will be both enlightening and intriguing. Some of my former students have reported that they have returned to these topics even after completing this course of study.

********************

Exercises and Projects for Fun and Profit Creative Projects: The Amazing Rabbit Population Golden Rectangles in your Home Fibonacci Numbers in Nature and the Arts Logarithmic Spirals—Seashell Design

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Math Is Everywhere! Explore and Discover It! 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci

THE AMAZING RABBIT POPULATION Long, long ago, in a land far away, a mathematician once presented the following scenario: Suppose that a colony of rabbits has the following breeding habits:  A pair of baby rabbits (one male and one female) requires one month to reach maturity from birth.  In the second month, when the rabbits have matured, they mate.  In the third month, they produce a new pair of baby rabbits (one male and one female).  In every month thereafter, they also produce a new pair of baby rabbits (one male and one female). Suppose also that each new pair of baby rabbits follows the same breeding pattern as their parents. Assignments:

Please answer the following questions: 1. How many pairs of rabbits will there be at the end of one year? 2. Once you have established the sequence of numbers of pairs of rabbits in each month of the first year, for each pair of successive months make a ratio of the larger number of rabbits to the smaller number of rabbits. Record these eleven ratios in a list and calculate their decimal values to four decimal places: Examples:

number in month #2 number in month #1

The first ratio would be:

The second ratio would be:

number in month #3 number in month #2

3. What numerical value do these successive ratios (in decimal form) seem to be approaching? 4. How many pairs of rabbits will there be after a given number of years? To answer this easily, you will need to determine the general pattern for how many pairs of rabbits there will be at any given time based on your answer to #1 above, and then extend the pattern to the given number of years in each assignment below: assignment pairs of rabbits after:

Assignment #1 2 years

Assignment #2 3 years

Assignment #3 4 years

Assignment #4 5 years

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Math Is Everywhere! Explore and Discover It! 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci

GOLDEN RECTANGLES IN YOUR HOME The Greeks popularized the notion of a golden rectangle and used this idea in their art and architecture. A golden rectangle is one in which the ratio of the length to the width is 1 5 . approximately equal to the golden ratio, Phi, where Phi = 2 Given the golden ratio, Phi ≈ 1.61803, we can construct a golden rectangle of any desired proportions. If 11 inches is the desired length, l, of a golden rectangle, determine the corresponding width, w; i.e., solve for w in the equation: l 11   1.61803 w w

Using a blank sheet of typing paper, 8.5 x 11 inches, mark the corresponding width, w, on the shorter side and cut off the excess width, thus creating a golden rectangle measuring 11 inches by w inches. Using this rectangle as a model, you may now find other golden rectangles or close approximations to golden rectangles in and around your home. Assignment:

Please locate a minimum of three approximately golden rectangular objects and, for each one that you find, record its dimensions (length and width) and calculate the ratio of the larger dimension to the smaller dimension. These ratios should be between 1.3 and 1.9; ratios outside this range are not considered golden rectangular for the purposes of this project. Write a brief report (minimum of one-half page) that includes:   

a brief description of each of the objects the length and width for each object the approximate “golden” ratio (the larger dimension divided by the smaller dimension) for each object

As an example, a 3"x5" file card has a ratio of 5/3=1.667.   

length = 5 inches width = 3 inches approximate “golden” ratio = 5/3 = 1.667

While you may not find any object that has the exact ratio, Phi, you’ll surely find some fairly close approximations. Have fun! Math is Everywhere!

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Math Is Everywhere! Explore and Discover It! 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci

FIBONACCI NUMBERS IN NATURE & THE ARTS The Fibonacci number sequence begins like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … This should sound quite familiar if you have already completed the Amazing Rabbit Population project. This sequence is a most fascinating mathematical structure. In fact, it turns out that this simple and unassuming sequence of numbers is related to many occurrences in nature, from seedpods to flower petals to plant structures and much more. Some stock market analysts use the associated golden ratio in their efforts to analyze and predict the market’s behaviors.

Assignment:

1. Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on Fibonacci numbers in nature. Afterwards, write a brief report (minimum of one-half page) on what you find to be of most interest. Note: Please find at least one example of an occurrence of a Fibonacci number in a plant or flower in your yard or neighborhood and describe this occurrence(s) in your report.

2. Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on Fibonacci numbers in art. Afterwards, write a brief report (minimum of one-half page) on the features that interested you most. ******************** Happy exploring! 

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Math Is Everywhere! Explore and Discover It! 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci

LOGARITHMIC SPIRALS—SEASHELL DESIGN Construction Instructions Note: A completed version of this project may be found on p.166 as well as on the cover of our textbook, Math Is Everywhere! Explore and Discover It! You may use this as a guide while following the instructions below.

Assignment: Note: Each of the small division markings on a standard 12-inch ruler calibrated in the U.S. system of units indicates one-sixteenth of an inch; there are sixteen of these markings within every inch on the ruler.

The longer markings within each inch represent even numbers of sixteenths, e.g., twosixteenths (or one-eighth), four-sixteenths (or one-quarter), etc.; the shorter markings within each inch represent odd numbers of sixteenths, e.g., one-sixteenth, threesixteenths, etc.

Step 1: Beginning with the golden rectangle that you constructed in our previous project entitled Golden Rectangles in your Home, mark off a square on the left-hand side of the rectangle (when placed horizontally) using the width of the rectangle, 6.8 inches, as the measure of the side of the square. Note: As indicated at the bottom of p.264, 6.8 inches is approximately equivalent to 6 and 13/16 inches; since the smallest division marker on a standard U.S. ruler is 1/16 of an inch, you will need to count 13 of these division markers in order to make the correct measurement.

To do this, position the golden rectangle horizontally, and then measure and mark 6 and 13/16 inches on both the top and bottom edges of the golden rectangle beginning from the left edge of the rectangle and proceeding toward the right edge for each of these measurements. Connect these two marks with a vertical straight line. This will produce a square measuring 6 and 13/16 inches on each edge on the LEFT side of the original rectangle and will also automatically produce a smaller golden rectangle (aligned vertically) on the right-hand side of the original rectangle.

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Math Is Everywhere! Explore and Discover It! 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci

Step 2: Using the same technique as outlined above, mark off a square on the TOP of this smaller rectangle, thus creating an even smaller golden rectangle (aligned horizontally) below it. The length of the sides of this square may be found by subtracting 6 and 13/16 inches from 11 inches to obtain 4 and 3/16 inches.

Step 3: Mark off a square on the RIGHT of this new rectangle, thus creating an even smaller golden rectangle (aligned vertically) to the left of it.

Step 4: Mark off a square on the BOTTOM of this new rectangle, thus creating an even smaller golden rectangle above it.

Step 5: Beginning in the lower left-hand corner of the original golden rectangle, draw a smooth curve upwards and over to the opposite corner of the first and largest square; continue onwards to the opposite corner of the second square, and onwards to the opposite corner of the third square, and so on, until you run out of squares. In doing so, you will be creating a spiral curve that is becoming smaller as you go. (See a completed version of this project on the back cover of our textbook.)

******************** The result is a continuous curve called a logarithmic spiral. This spiral shape is very much like that of the seashell known as the chambered nautilus; nature has made some marvelous uses of this spiral and it’s all so mathematical. 

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Note: 6.8 inches is equal to six and eight-tenths of an inch. To convert eight-tenths into sixteenths of an inch, the common U.S. ruler division, use a proportion and solve for x:

x 8  10 16

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Math Is Everywhere! Explore and Discover It! 7.3 Mathematics in Architecture and Nature: The Golden Section and Fibonacci

This yields 10x=128, or x=12.8; i.e., the measurement you need is 6 and 12.8 sixteenths. To make things a little simpler, round 12.8 to 13 to obtain an approximate measure of 6 and thirteen sixteenths, or 6 and 13/16 inches. To get good results on this project you’ll need to be very accurate with your measurements…

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Math Is Everywhere! Explore and Discover It! 7.4 Long-term Projects

7.4 Long-term Projects “To be nobody-but-yourself in a world which is doing its best, night and day, to make you everybody else means to fight the hardest battle which any human being can fight; and never stop fighting.” —e. e. cummings Three of our course projects represent major assignments and require more time than usual to complete. They are presented here so that you may begin to plan accordingly. The Webliography Report involves selecting the ten best websites from among the ones that you visited during this course of study and evaluating and reporting on their content. The Term Report is a major assignment involving Chapters 6 through 8. We will cover a number of topics in those chapters that will be of interest and you may find it advantageous to be mindful of the need to choose a report topic as we cover these materials. The Stock Market Investment Report requires an extended period of time to complete since it involves tracking some stocks and observing their changes in value.

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Exercises and Projects for Fun and Profit

Creative Projects: Webliography Report Term Report Stock Market Investment Report

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Math Is Everywhere! Explore and Discover It! 7.4 Long-term Projects

Math topic: Numeration in Other Cultures Website address (URL): http://www-history.mcs.st-and.ac.uk/ Site rating: Excellent! Description: This site…

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Math Is Everywhere! Explore and Discover It! 7.4 Long-term Projects

TERM REPORT Assignment Guidelines: 1. In order to obtain maximum credit, reports must be SUBSTANTIAL (i.e., must investigate a topic in some depth and detail—not a brief surface description). This report takes the place of a final exam and as such requires a substantial effort. 2. Length of reports may vary—some will benefit from illustrations, diagrams, graphics, etc. The determining factor in length is whatever is needed in order to produce a report that is substantial; in general, length will correlate with amount of credit awarded. A length of four pages is required as a minimum in order to receive full credit; reports of shorter length will receive only partial credit. Your report must be typewritten and must use:     

one-inch margins all around 12-point font size double-spacing for the body of the report single-spacing for your name and date at the top of the first page single-spacing for the webliography at the end of the report

Your complete report should include a minimum of three pages of text, not including pictures and diagrams and not including your webliography. With pictures and diagrams and webliography included, your report should be a minimum of four pages in length. 3. Reports should include a webliography of at least three websites. For each site, please give a brief description of the contents of the site and an evaluation of the site’s value. 4. In general, copying and pasting Internet content into your report is not allowed unless the source is directly acknowledged. For the purposes of this report, I will allow you to copy and paste website content directly into your report but only if you enclose the statement(s) with quotation marks and cite the source of the statement by indicating which website the quote was taken from. The total amount of copied and pasted (quoted, with source acknowledged) textual material may not exceed 50% of your textual content. Example of acceptable “copied and pasted” material with cited source: “M.C. Escher, during his lifetime, made 448 lithographs, woodcuts and wood engravings and over 2000 drawings and sketches. Like some of his famous predecessors, Michelangelo, Leonardo da Vinci, Dürer and Holbein, M.C. Escher was left-handed.” (source: http://www.mcescher.com/)

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Math Is Everywhere! Explore and Discover It! 7.4 Long-term Projects

REPORT TOPIC SUGGESTIONS These topics are related to material that we will cover in Chapters 6 through 8 of Math Is Everywhere! Explore and Discover It! You may suggest something else if so desired. Mathematics and music—music theory The mathematics of a particular musical instrument Mathematical perspective in Renaissance art Mathematical perspective in the works of M.C. Escher Mathematics and the works of Leonardo daVinci Symmetry in art Tilings and tessellations Mathematical designs in textiles Geometric patterns in quilting Computers and graphic art Use of Geometry in the architecture of European cathedrals Geometry in ancient Chinese art and architecture Geometry of Stonehenge Applications of Geometry in biology or medicine Mathematics in art and architecture: the golden rectangle The Fibonacci sequence and its occurrences in nature Fractal occurrences in nature Fractal art and/or music

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Math Is Everywhere! Explore and Discover It! 7.4 Long-term Projects

STOCK MARKET INVESTMENT REPORT Investing your money, whether in a savings account, a government bond, or a company’s stock, is an important part of financial planning. In this project, we are going to simulate an investment in the stock market and record the progress of the investment over the course of the second half of the semester (approximately eight weeks in Fall and Spring; five weeks in Summer). In the Business & Finance section of your local newspaper you will find a daily listing of the companies included in the New York Stock Exchange (NYSE) and a number of values associated with their stocks. You can also find this information on the Internet; a number of financial sites provide an option to “Get Quotes”. Use the search engine found at www.google.com to conduct some Internet research on stock quotes. The most important items in a stock quote are: name of the company (abbreviated, e.g., WMT stands for WalMart) opening value of the stock (per share) for the day (Open) closing value (per share) for the day (Close) Note: Some sites list only the Open value, others list only the Close value, while some refer to the current value of the stock as Value. The most important value for this project is the opening (Open) or closing (Close) value for the day; this is sometimes referred to as Value. You may choose either of these values to track and record for the duration of this project. The opening value (Open) represents the price of one share of a company’s stock at the opening of the stock market on Wall Street in New York City on a given day at 9:30 a.m.; the closing value (Close) corresponds to the value at 4:30 p.m. when the stock market closes. The opening or closing value for the day is the value that you will be recording once a week and tracking for the duration of the project. If the value of your stock at some time after you begin this project is higher than it was when you started, then your stock has increased in value and your investment is worth more than when you started.

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Math Is Everywhere! Explore and Discover It! 7.4 Long-term Projects

Assignment: You will have \$1000 of “pretend” money to invest (or as reasonably close to \$1000 as possible) in this project. The objectives are: 1. 2. 3. 4.

choose three stocks in which to invest, either at random or by personal choice “purchase” an appropriate number of shares of each stock track their progress weekly (once a week on approximately the same day each week) report the results and give a brief analysis Note: You may use the search engine at google.com and enter the search expression get quotes for stocks to find stock share prices.

Your final report should include: 1. 2. 3. 4.

the names of your selected stocks the initial value of each stock (open or close value when you began the project) the number of shares purchased for each stock a table of the opening (Open) or closing (Close) values for each stock on a weekly basis (record them once a week on approximately the same day each week) 5. a line graph for each of the stocks (based on the table of values above) showing a visual picture of its growth 6. a final evaluation of the gains or losses of each of your stocks including possible reasons for the gains or losses based on items and events in the news, both national and international. As an example, the recent upswing in the outsourcing of certain manufacturing jobs may cause a rise in some stock values and a decline in others. Note: To calculate the gain or loss for a particular stock, find the difference in value between the initial cost of a share of the stock (i.e., the Open value in the first week of the project) and the final cost of a share of the stock (i.e., the Open value in the last week of the project) and then multiply this difference by the number of shares that you purchased. 7. a conclusion stating how much total net gain or loss you experienced for all of your stocks combined

******************** Happy investing! Since this is just pretend money, you don’t have to worry about whether you might lose money on your investments—just enjoy the process and learn something about the stock market at the same time.  ********************

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Math Is Everywhere! Explore and Discover It! 7.4 Long-term Projects

Notes:

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Math Is Everywhere! Explore and Discover It! 8.0 Introduction and Objectives

Chapter Eight Fractals: New Structures in Mathematics 8.0 Introduction and Objectives The thumbnail sketches on this page represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

8.1 Mathematical Fractals: A Natural Geometry

Geometric fractal:

8.2 Fractal Applications

Natural fractal:

8.3 Famous Triangles: Sierpinski Meets Pascal 1 1 1 1 1

1 2

3 4

1 3

6

1 4

1

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Math Is Everywhere! Explore and Discover It! 8.0 Introduction and Objectives

Learning Objectives for Chapter 8 1. To enable students to develop their analytical reasoning skills through pattern recognition and problem solving. 2. To enable students to develop their deductive reasoning skills through identification of fractal structures involving self-similarity. 3. To allow students to investigate occurrences of mathematics in their homes and yards; in particular, students will investigate fractal structures in plants. 4. To allow students to exercise their creativity by conducting Internet research and writing reports on various mathematical topics involving the application of fractal mathematics to other fields. 5. To enable students to gain some familiarity with Web-based mathematical resources and to develop their critical thinking skills in assessing and evaluating these resources. 6. To demonstrate the pervasiveness of mathematics in the world around us. 7. To engage students in active learning through the use of interactive and hands-on projects and activities. 8. To empower students to become lifelong learners.

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Math Is Everywhere! Explore and Discover It! 8.1 Mathematical Fractals: A Natural Geometry

8.1 Mathematical Fractals: A Natural Geometry “Abstractness, sometimes hurled as a reproach at mathematics, is its chief glory and its surest title to practical usefulness. It is also the source of such beauty as may spring from mathematics.” —E. T. Bell

Introduction Fractals are a relatively recent development in the field of mathematics and have only recently gained attention as a result of scientific applications of the mathematical theory behind them. They are a simple and at the same time complex mathematical entity and have produced many surprising results during their short history. In this section, we will examine the basic mathematics that lies behind fractal shapes and images and, in the Exercises and Projects for Fun and Profit at the end of this section, explore a very special fractal known as the Mandelbrot set, named in honor of one of the pioneers in this field. Later in this chapter, we will investigate some fractal applications.

Fractals and Fractal Geometry While fractals were first discovered and described in purely mathematical terms as sets of values related to the output of simple polynomial functions, their properties turned out to be rather interesting from a geometrical point of view. The following definition of a fractal focuses on this geometric aspect: A fractal is a geometric structure that contains a number of parts, each of which has the same basic structure (and shape) as the fractal as a whole but on a smaller scale. A familiar example of a fractal is a tree and its branches. A tree consists of a main trunk and a number of attached branches. Each branch, when considered as a stand-alone object, has the same structure as the parent tree: a branch consists of a main trunk and a number of attached branches. Each of these attached branches, in turn, has the same structure as the parent branch but on a somewhat smaller scale. And on it goes, until we reach the very smallest twigs, each of which has the same basic structure once again but on a very much smaller scale. These reduced-sized copies of the whole typify the geometric nature of fractals. This is referred to as self-similarity. A head of broccoli, a cloud, and the human body’s circulatory system are other examples of objects that are classified as natural fractals and whose parts exhibit selfsimilarity. You will have the opportunity to view other natural fractals in the Exercises and Projects for Fun and Profit at the end of this section.

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Math Is Everywhere! Explore and Discover It! 8.1 Mathematical Fractals: A Natural Geometry

Sierpinski’s Triangle We will also encounter some purely mathematical fractals in this chapter, one famous example of which is the Sierpinski Triangle. This triangular structure begins with a single equilateral triangle that is successively subdivided into smaller equilateral triangles. It is a fascinating piece of mathematics and exhibits the self-similarity that is so characteristic of fractals. The first few iterations (repetitions) are shown below; the rule for each iteration is: locate the midpoints of each side of each upward-oriented triangle and connect these midpoints with line segments: Initial triangle:

First iteration of the rule:

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Math Is Everywhere! Explore and Discover It! 8.1 Mathematical Fractals: A Natural Geometry

Second iteration of the rule:

The triangle resulting from the second iteration of the rule contains three copies of the triangle resulting from the first iteration. These three copies are smaller in size but retain the same structure found in the previous iteration—self-similarity once again. We will now take a look at the beginnings of fractal development and the basic mathematics involved. While the terminology used here may be a bit unfamiliar, this brief survey of the underlying mathematics will be helpful in our investigation of other purely mathematical fractals at the end of this section.

Mathematical Recursion In the early 1900s, mathematicians began investigating polynomial functions together with the concept of recursion, or iteration. The basic idea was to choose a polynomial function f(x) and then:  assign an initial value to the variable x; this is often called the seed value  compute the value of f(x) for this value of x  assign this value of f(x) to the variable x  compute the value of f(x) for this value of x  and so on… This process of continually reassigning the output value of f(x) to the next input value of x is called recursion, or iteration. It is a simple process but often has surprising results. As an example, we will demonstrate this process for f(x) = x 2 :

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Math Is Everywhere! Explore and Discover It! 8.1 Mathematical Fractals: A Natural Geometry 

assign the initial value of ½ to the variable x: x = ½ 2

1 compute f ( ½ ) =   = ¼ 2 assign ¼ to the variable x: x = ¼ 2

  

1 compute f ( ¼ ) =   = 1/16 4 1 1 assign to the variable x: x = 16 16 2 1 1 1 compute f   =   = 256  16   16  and so on…

1 1 1 1 , , , ,… 2 4 16 256 This sequence is known as a recursive sequence and each term in the sequence depends on the previous term for its value. As you can see, each successive term in this sequence is becoming progressively smaller and will eventually approach 0. This sequence represents the orbit of the seed value 0.5. The successive outputs for each iteration may be put into a sequence:

If we had begun with an initial seed value of x=2, the resulting sequence would have been: 2, 4, 16, 256, ... In this case, each successive term in the sequence becomes progressively larger and will eventually approach infinity. We say that the orbit of the seed value x=2 escapes to infinity. Changing the function naturally affects the results of the iteration process. As another example, we will demonstrate the recursion process for f(x) = x 2  0.5 and the original seed value of x = 1/2 = 0.5:         

assign the initial value of 0.5 to the variable x: x = 0.5 compute f ( 0.5 ) = 0.5 2 – 0.5 = 0.25 – 0.5 = –0.25 assign –0.25 to the variable x: x = –0.25 compute f (–0.25 ) = (0.25) 2  0.5 = 0.0625 – 0.5 = –0.4375 assign –0.4375 to the variable x: x = –0.4375 compute f (–0.4375 ) = (0.4375) 2  0.5 = –0.3086 assign –0.3086 to the variable x: x = –0.3086 compute f (–0.3086 ) = (0.3086) 2  0.5 = –0.4048 and so on…

In this case, the successive results are negative numbers and begin to fluctuate in value; as it turns out, this behavior continues and the orbit values hover between 0 and –1. As a result, the orbit of the seed value x=0.5 does not escape to infinity. 278

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Math Is Everywhere! Explore and Discover It! 8.1 Mathematical Fractals: A Natural Geometry

Orbit Behaviors, Complex Numbers and the Mandelbrot Set Thus far, we have seen three types of orbits: those that approach 0, those that approach infinity, and those that remain within some finite boundaries. For the purposes of analyzing mathematical fractals, we will partition the three types of orbit behaviors into two categories: A. Those orbits that escape to infinity B. Those orbits that do NOT escape to infinity, i.e., those that approach 0 or remain within some finite boundaries The mathematical recursion process may also be used with complex numbers of the form a + bi, where a and b are real numbers and i is the imaginary number equal to the square root of negative one, i   1 . The sequence of complex numbers that is produced by a given polynomial function and an initial seed value will have a behavior that can be categorized in much the same way that we categorized real number sequence behaviors (orbits). The fractal image known as the Mandelbrot set always uses the same initial seed value, 0, and will instead vary the complex number, c, that is part of the Mandelbrot polynomial function, f ( x)  x 2  c . Each value for c=a+bi will produce a different orbit behavior for the seed value 0. These orbit behaviors will be categorized in the same way as for real number orbits. Since complex numbers can be graphed on a two-dimensional plane where the horizontal axis represents the real numbers and the vertical axis represents the imaginary numbers, we can create a visual picture of the orbit behaviors connected to each complex number c used in the Mandelbrot function, f ( x)  x 2  c , as follows:  

If the initial value for c=a+bi results in behavior A, then we will assign the color red to the complex number c=a+bi on the graph; If the initial value for c=a+bi results in behavior B, then we will assign the color black to the complex number c=a+bi on the graph.

This means that each complex number in the plane will be colored either red or black in the Mandelbrot set. As it turns out, many of the complex numbers associated with behavior B are clustered together near the origin in a very unusual shape known as a cardioid. The boundaries of this shape provide the most interest and they exhibit an absolutely amazing structure: intricate, beautiful and self-similar. In the Creative Project entitled Introduction to Fractals, you will have an opportunity to explore this set through the use of some Java applets that allow you to zoom in and observe this detail in high-color resolution. The results are quite surprising and very impressive!

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Math Is Everywhere! Explore and Discover It! 8.1 Mathematical Fractals: A Natural Geometry

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Exercises and Projects for Fun and Profit

Creative Project: Introduction to Fractals

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Math Is Everywhere! Explore and Discover It! 8.1 Mathematical Fractals: A Natural Geometry

INTRODUCTION TO FRACTALS The study of fractals is a relatively new branch of mathematics, originating with mathematical research carried out in the early 1900s. In recent decades, fractals have taken on importance as more and more instances and applications of their structure were discovered. In recent years, fractal images have taken their place in the field of art while fractal music has gained a following in some music circles.

So the immediate question is: “What is a fractal?” This is not a simple question to answer due to the highly technical background out of which fractals come. To respond that a fractal is a mathematical set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension is less than satisfying for most people.  Consequently, for our purposes we will settle for a less rigorous and more approximate definition: A fractal is a geometric structure that contains a number of parts, each of which has the same basic structure (and shape) as the fractal as a whole but on a smaller scale.

There are many mathematical structures that are classified as fractals; for examples:     

the Sierpinski triangle the Koch snowflake the Peano curve the Mandelbrot set Lorenz attractors

We will encounter some of these later in this section and in other sections in this chapter. Fractal structures also exist in:  clouds  mountains  coastlines  plants and trees  the human circulatory system  various other natural phenomena

The Mandelbrot Set One of the most famous fractals, the Mandelbrot set, is generated by iterating a simple mathematical function over the set of complex numbers. Complex numbers are of the form a + bi, where a and b are real numbers and i is the imaginary number equal to the square root of 281

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Math Is Everywhere! Explore and Discover It! 8.1 Mathematical Fractals: A Natural Geometry

negative one, i   1 . In graphing complex numbers, the x-axis represents the real number axis and the y-axis represents the imaginary number axis. To generate a graph of the Mandelbrot set, the function f ( x)  x 2  c is iterated beginning with a specified complex number as the value for c:    

the seed value x=0 is squared and added to c=a+bi. This results in 0 2 +c = c = a+bi this complex number a+bi is then squared and added to c=a+bi, producing a new complex number: e+fi = (a+bi) 2 + c = (a+bi) 2 + (a+bi) this new complex number e+fi is then squared and added to c=a+bi, producing a third complex number: g+hi = (e+fi) 2 + c = (e+fi) 2 + (a+bi) and so on.

The sequence of resulting values is known as the orbit of the seed value 0. For each complex number, c, in the plane, this process is repeated so that each complex number is associated with a particular orbit of the seed value 0.

A Computer-assisted Visualization of the Mandelbrot Set A computer screen is actually composed of a large rectangular array of picture elements (pixels) which can be set to either on or off; when they are on, they can take on any of a number of colors depending on the capacity of the computer’s video card. To generate the Mandelbrot set on a computer screen, each pixel is assigned to a complex number, a+bi, as if the complex number were an ordered pair on a graph. Then for each pixel, the behavior of the orbit associated with the complex number corresponding to the pixel is examined.

If the orbit approaches infinity, then the pixel is assigned a color that is dependent on how fast this sequence begins to grow; red is generally used for slowly growing sequences, followed by the colors of the rainbow in order (red, orange, yellow, green, blue, and violet) for increasingly faster sequences. On the other hand, if the orbit does NOT approach infinity, i.e., the sizes of the numbers in this ongoing sequence stabilize and/or converge to some finite value, the pixel is assigned the color black and belongs to the Mandelbrot set. The end result is the famous Mandelbrot set—a black cardioid shape with spherical globular attachments and further intricacies along its boundary. The boundary is the most interesting part of the set and reveals mathematical and artistic detail that is quite astounding!

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Math Is Everywhere! Explore and Discover It! 8.1 Mathematical Fractals: A Natural Geometry

Assignment:

1. Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on fractals. Afterwards, please write a mini-report (one or two paragraphs) on your observations concerning fractal images. 2. Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on the Mandelbrot set and visit one of the websites that allow you to explore the Mandelbrot set by zooming in on a particular part of the boundary. Afterwards, please write a mini-report (one or two paragraphs) on what you observed while zooming in on the image.

******************** Have fun! 

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Math Is Everywhere! Explore and Discover It! 8.2 Fractal Applications

8.2 Fractal Applications “The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.” —Albert Einstein

Introduction In this section, we will explore some applications of fractals that have arisen during recent years and we will also explore fractal structures closer to home. More and more instances of fractal structures are being discovered and it is fascinating to see how this relatively new mathematical concept is being employed to describe a number of natural phenomena. One example is a very recent discovery made by biologists: the rods and cones at the back of the retina in a human eye possess a fractal structure and display self-similarity in their various shapes and sizes. Another example has to do with plant structures and the discovery that many plants with complex appearances have very simple fractal structures. One of the Exercises and Projects for Fun and Profit at the end of this section will lead you to discover this for yourself. It is rather exciting when you have an opportunity to see more deeply into nature and its underlying principles!

Fractal Structures: A Study in Weeds Before studying this topic some years ago, I was unaware of how many plants exhibit fractal structure. Since that time I have made a number of discoveries in this vein (please pardon the play on words ), in county parks, flower shops, and elsewhere and will now share one of these as a means of demonstrating this concept. The following incident took place several years ago when my family and I became busy with life’s issues and neglected to take care of the small garden that we had established in our backyard. By late summer, some of the weedy plants had grown to be quite large; one in particular was approximately five feet tall. As I was looking out of the window one morning, I was struck by the fact that this tall weed appeared to have a sort of repetitious structure, i.e., the branches seemed to all have the same look about them although the branches at the top of the plant were distinctly smaller than the branches at the bottom. I decided to investigate further and went outside to have a closer look at the weed. As I began to observe it, I noticed that each branch was indeed a smaller replica of its parent branch. In addition, I also noticed that each stem would grow upwards for a certain distance and then form two additional stems, a longer one and a shorter one. Each of these two stems repeated this behavior, i.e., the longer stem grew upwards for a certain distance and then formed two additional stems, a longer one and a shorter one, and the original shorter stem also grew upwards for a certain distance and then formed two additional stems, a longer one and a shorter one.

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Math Is Everywhere! Explore and Discover It! 8.2 Fractal Applications

In fact, it turned out that every stem in the entire plant acted in the same manner until finally, at the very top, the smallest stems formed flowers at their tips. I was completely astounded and so excited that I had my entire family come out to see this new discovery. They weren’t quite as impressed as I was (and my children thought I was a little, well, you know….), but nonetheless it was an enlightening experience and one that I will not easily forget.

******************** The personal discoveries that you make in life are far more memorable than the discoveries that others have made and then explained to you. They are akin to having a Eureka! moment while attempting to solve a problem or puzzle and are to be highly valued. One of the purposes of this course is to lead you to make your own personal discoveries so that you, too, can experience the joy and delight that accompanies them. Best wishes for many such discoveries!  ********************

Fractal Growth Patterns: Complexity from Simplicity I’ll describe the basic rule of growth, or growth pattern, for the fractal weed in my yard below and then sketch some successive iterations of the rule so that you can get an idea of what the weed structure was like. I will also include a digital photo of the weed that I took at the time even though it is difficult to see all of the details clearly when printed in black and white. Basic rule or pattern for each stem’s growth: grow upwards for a certain distance and then form two new stems, a longer one and a shorter one. The original stem enters the stem junction and two new stems exit the stem junction. (Each of these two new stems will be somewhat shorter than the original stem. The place where the two new stems begin to grow is called a node or stem junction). Note: It would be interesting to study the ratios involved in the lengths of each successive set of stems, from the bottom of the plant to the top. I suspect that these lengths get smaller in an interesting mathematical progression; however, I didn’t take the time to pursue these measurements at the time.

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Math Is Everywhere! Explore and Discover It! 8.2 Fractal Applications

Basic growth pattern—first iteration of the rule:

Second iteration of the growth rule:

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Math Is Everywhere! Explore and Discover It! 8.2 Fractal Applications

Third iteration of the growth rule:

Even at this beginning stage of the weed’s development, I trust that you can see how intricate and complex the plant will look when it has gone through many successive iterations in all directions from the central stem. The two-dimensional representation can give only an indication of the three-dimensional reality; the actual photos below will hopefully give a better indication of this structure.

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Math Is Everywhere! Explore and Discover It! 8.2 Fractal Applications

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Math Is Everywhere! Explore and Discover It! 8.2 Fractal Applications

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Math Is Everywhere! Explore and Discover It! 8.2 Fractal Applications

Conclusion As you can see, this common weed is rather complicated looking; however, it is completely defined and determined by the very simple growth rule above, i.e., each stem grows upwards for a certain distance and then forms two new stems, a longer one and a shorter one. The original stem enters the stem junction and two new stems exit the stem junction. The parts of the plant are clearly self-similar and satisfy the geometric definition of a fractal. In the Exercises and Projects for Fun and Profit at the end of this section, you will have the opportunity to discover your own backyard fractals and more. ********************

I trust that you will enjoy our explorations in this section—they are a bit off the beaten path but quite interesting nonetheless!

********************

Exercises and Projects for Fun and Profit Creative Projects: Fractals in Nature Fractal Music Fractal Art ********************

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Math Is Everywhere! Explore and Discover It! 8.2 Fractal Applications

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Math Is Everywhere! Explore and Discover It! 8.2 Fractal Applications

FRACTAL MUSIC Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on fractal music, a relatively recent development in the field of musical composition. There are a number of websites devoted to this topic and many of them provide fractal music interludes for your listening pleasure (???). This music is quite different in nature than the traditional or contemporary music to which we are accustomed. Assignment: Write a mini-report (one or two paragraphs) on your reactions to fractal music. Also, state whether you think that fractal music should be given the same status as other more traditional categories such as classical, jazz, blues, rock, etc. In other words, do you think that it is really “music”? ********************

FRACTAL ART Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on fractal art, a relatively recent development in the field of artistic composition. There are a number of websites devoted to this topic and many of them provide examples of fractal art for your viewing pleasure (???). This type of art is quite different in nature than the traditional or contemporary art to which we are accustomed, and yet for some it has an interesting appeal. Assignment: Write a mini-report (one or two paragraphs) on your reactions to fractal art. Also, state whether you think that fractal art should be given the same status as other more traditional categories such as realism, impressionism, pointillism, etc. In other words, do you think that it is really “art”?

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Math Is Everywhere! Explore and Discover It! 8.3 Famous Triangles: Sierpinski Meets Pascal

8.3 Famous Triangles: Sierpinski Meets Pascal “In the pure mathematics, we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven.” —Edward Everett, American statesman

Introduction In mathematics, two of the most famous triangular structures are Pascal’s Triangle, named after the French mathematician, Blaise Pascal (1623–1662), and Sierpinski’s Triangle, named after the Polish mathematician, Waclaw Sierpinski (1882–1969). Note: Pascal’s Triangle was actually used by mathematicians long before Pascal’s time but his extensive writing about the triangle and its structure linked him forever to the triangle and was responsible for it being named in his honor. Each of these structures was originally related to different fields and applications in mathematics. Pascal was interested in the study of probability with regard to the odds of winning games of chance while Sierpinski was involved in mathematical set theory. Neither of them was aware at the time that their work with triangle constructions would become so widely used and admired or that the two triangles would turn out to be related to each other in some ways. In the twentieth century, mathematicians discovered an interesting connection between the two triangular structures. We will examine this connection and then you will have an opportunity to use your analytical reasoning skills in the Exercises and Projects for Fun and Profit at the end of this section.

Pascal’s Triangle Pascal’s Triangle is a remarkable structure that embodies a wealth of mathematical information including the binomial coefficients, powers of 2, triangular numbers, and more. The first several rows are shown below: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1

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Math Is Everywhere! Explore and Discover It! 8.3 Famous Triangles: Sierpinski Meets Pascal

This triangular structure is constructed by simply adding two adjacent entries together and recording the sum in the row below, directly between the two adjacent entries, and then appending the number 1 to each end of the newly constructed row.

Sierpinski’s Triangle Sierpinski’s Triangle is the structure that we encountered in a previous section of this chapter; it is a mathematical fractal and can be described at various levels of iteration. A triangle that we encountered earlier represented the second iteration of the original construction rule and is shown below:

The Triangle Connection In Pascal’s Triangle, please note the numbers 4, 6 and 4 in the fourth row, the repeated numbers 10 and 10 in the fifth row, and the number 20 in the sixth row. Note: For purposes of simplicity and convenience, the top row of Pascal’s Triangle is denoted as the 0th row, or row #0. These six numbers form the outline of a small equilateral triangle inside Pascal’s Triangle. If you were to color this entire triangular region gray, it would correspond to coloring the topmost of the smaller, downward-oriented equilateral triangles in Sierpinski’s Triangle as shown below:

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Math Is Everywhere! Explore and Discover It! 8.3 Famous Triangles: Sierpinski Meets Pascal

If you were to do a similar coloring of the triangular region in Pascal’s Triangle that was associated with the next group of even numbers (beginning with the numbers 8, 28, 56, 70, 56, 28 and 8 in the eighth row and proceeding downwards), this coloring would correspond to coloring the large, central equilateral triangle in Sierpinski’s Triangle as shown below:

In essence, then, even numbers that are adjacent to each other in Pascal’s Triangle correspond to downward pointing equilateral triangles in Sierpinski’s Triangle. This is a most unusual connection between the two triangles.

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Math Is Everywhere! Explore and Discover It! 8.3 Famous Triangles: Sierpinski Meets Pascal

******************** Mathematics is full of interesting (and sometimes puzzling) connections. Although you may not think so, discovering new connections is well within the reach of the average mathematics student. The best way to find these connections is to keep an open and inquiring mind and ask a lot of questions! Even when there is no one there to answer them…  ********************

Exercises and Projects for Fun and Profit Creative Project: Triangle Connections

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Math Is Everywhere! Explore and Discover It! 8.3 Famous Triangles: Sierpinski Meets Pascal

TRIANGLE CONNECTIONS Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on Pascal’s Triangle and, in particular, explore sites that involve coloring the triangle based on a specified modulus value in order to see the Pascal-Sierpinski connection.

Assignment: Observe the different coloring effects when using prime numbers as the modulus value versus composite (nonprime) numbers. Write a brief report (minimum of one-half page) on your discoveries.

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Math Is Everywhere! Explore and Discover It! 8.3 Famous Triangles: Sierpinski Meets Pascal

Notes:

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Math Is Everywhere! Explore and Discover It! 9.0 Introduction and Objectives

Chapter Nine Linear Relationships: Straight as an Arrow 9.0 Introduction and Objectives The thumbnail sketches on this page represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

9.1 Mathematical Relationships and Functions: “Vary-ables” in Action The Rule of Four Golden Puffs—A Shocking Discovery

9.2 Linear Functions: A Deeper Look

9.3 Linear Models: Analytic and Predictive Linear Regression—The Line of Best Fit

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Math Is Everywhere! Explore and Discover It! 9.0 Introduction and Objectives

Learning Objectives for Chapter 9 1. To enable students to develop their analytical reasoning skills through pattern recognition and problem solving. 2. To enable students to develop their inductive and deductive reasoning skills while constructing linear models and predicting future outcomes. 3. To enable students to create, interpret and analyze a scattergram. 4. To enable students to use linear regression and correlation analysis and their critical thinking skills to determine whether a linear relationship exists and, if so, the strength of the correlation. 5. To allow students to explore and investigate mathematical relationships that pertain to daily life. 6. To allow students to use Web-based mathematical resources in order to locate relevant data for mathematical investigations. 7. To demonstrate the practicality of mathematics through linear modeling. 8. To engage students in active learning through the use of projects and activities.

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Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

9.1 Mathematical Relationships and Functions: “Vary-ables” in Action “Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.” —John W. N. Sullivan

Introduction Relationships are prevalent in every aspect of life, both academic and personal. In fact, one could very reasonably argue that relationships are at the very heart of life itself. Down through the centuries, mathematicians and scientists have invested much time and energy in defining relationships of various sorts in an attempt to help clearly depict and understand the world around them. A mathematical relationship is simply a precise description of how two measurable quantities are related.

“Vary-ables”: The Stuff of Relationships As an example, we’ll consider the relationship between the sales tax on an item and the price of the item, a relationship that we encounter in our daily lives. This relationship is a simple one and can easily be described in mathematical terms: sales tax = sales tax rate · price of the item Since both sales tax and price take on different values, we call them mathematical variables. You might even call them “vary-ables” since that is their essence: a variable is a quantity that is able to vary. In a relationship between two measurable quantities, we will identify a dependent “vary-able” and an independent “vary-able.” While in some cases these roles are interchangeable, in most cases it is clear which quantity naturally depends on the other for its value. In the relationship between sales tax and price, the sales tax naturally depends on the price of the item for its value. In this case, we say that the sales tax is the dependent variable and, on the other hand, that the price is the independent variable.

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Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

******************** Identifying the dependent variable is an important issue; it determines the way in which we construct a graph or visual depiction of the relationship. ********************

Those Unchanging Constants While variables are central to relationships, there are other values called constants that are often involved in relationships. A constant is exactly that: a value that cannot vary within the relationship and hence remains constant. In the above example concerning sales tax, the sales tax rate would be considered a constant since it does not change (at least not very often). In this relationship, the sales tax and item price are variables whereas the sales tax rate is a constant. As an example, in Pinellas County, Florida, the current sales tax rate is 7% and the complete relationship may be written as: sales tax = 7% of the item price sales tax = 0.07 · item price

The Rule of Four Once a relationship between two measurable quantities is established, it is often helpful to describe the relationship in various ways. Four types of descriptions have been found to be most useful; the “Rule of Four” is an expression that is used to conveniently describe these four types of relationship descriptions:    

written description (often in sentence form) table of corresponding values of the variables algebraic description—an equation graphical description—a visual picture of the relationship

The most precise and concise of these is the algebraic description of a relationship in the form of an equation. An equation can be easily manipulated and used to generate corresponding values of the variables. The graphical description, while not as precise, is the most effective way to communicate the nature of the relationship and depict the general behavior of the dependent variable. Using our sales tax example, the Rule of Four descriptions would be as follows: written description: The sales tax on an item is found by multiplying the price of the item by the sales tax rate of 7%.

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Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

table of values: item price \$10 \$20 \$30

sales tax \$0.70 \$1.40 \$2.10

algebraic description: sales tax = 0.07 · item price This can be written more concisely as: tax = 0.07 · price and can be abbreviated even further by using single letters to represent the variables: T = 0.07p graphical description:

As can be observed in the graphical description, the sales tax rises slowly but steadily as the item price increases. The steady rise is indicative of what is called a linear relationship, a very fundamental type of mathematical relationship that we shall investigate in more detail in the next section of this chapter.

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Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

Relationships Are Everywhere! Now that we have examined the basic ingredients in a mathematical relationship, variables and constants, and the descriptive forms that may be used to communicate the nature of a relationship, we can begin to apply this knowledge to the world around us. ******************** Establishing and quantifying relationships can be a very eye-opening and revealing endeavor. In the process of doing so, hitherto unnoticed facts or features often come to light and serve to enlighten us in some way concerning the relationship. Discoveries such as these are personally rewarding and can serve as milestones along our way. ******************** As an example, I’ll relate a true story involving my younger son that took place several years ago but that is often remembered in our family. It all began one day when we were sitting at the breakfast table and my son, mostly from lack of anything else of interest to do while eating, happened to read the various pieces of information printed on the cereal box from which he had obtained his breakfast. He pointed out that there were 15 grams of sugar per serving listed in the cereal’s nutrition facts and noted that the cereal (Golden Puffs) was certainly very sweet. We then inspected each of the other cereal boxes at the table and found that each listed a different amount of sugar per serving with values ranging from 0 grams to 17 grams. Naturally, we suspected that the higher amounts of sugar corresponded to the sweeter cereals; however, much to our surprise, the highest amount of sugar was contained in Raisin Bran—a cereal that we all felt was much less sweet than Golden Puffs. This realization raised some interesting questions, e.g., “How can this be possible?” and led us deeper into our investigation.

cereal Golden Puffs Raisin Bran

sugar content (by weight) 15 g 17 g

perceived sweetness very sweet moderately sweet

******************** Analytical Moment: Can you determine what was at the root of our cognitive dissonance? ********************

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Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

A Surprising Discovery We made several conjectures before arriving at the realization that there was another nutritional fact that was central to our investigation; as you may have guessed, the weight of the serving size was a critical factor in the relationship. The serving size for Golden Puffs was 30 grams while the serving size for Raisin Bran was 54 grams. When considered together with the sugar content, these serving size values changed our perspective. By determining for each cereal the ratio between its sugar content and its serving size, we were able to construct a measurement of the relative amount of sugar contained in a serving, i.e., the percentage of sugar in the serving. The results are shown below: cereal

sugar content

serving size

Golden Puffs

15 g

30 g

Raisin Bran

17 g

54 g

sugar serving size 15 = 0.50 30 17 = 0.31 54

percentage of sugar

50% 31%

Our collective mental light bulbs lit up with the excitement of a new discovery! Of course! The sugar ratio (or sugar percentage) was the key feature in understanding why Golden Puffs tasted so sweet. We then marveled at the shocking fact that each spoonful of Golden Puffs was effectively half sugar!! And that a bowlful of Golden Puffs was half sugar as well!

******************** And thus it was that our family came into a new realization of the effectiveness of mathematics in quantifying simple, everyday relationships. Not only was this a simple process, but it was quite instructive and illuminating. None of us had ever considered how much sugar was contained in cereals other than the recognition that some were quite sweet; now, with the help of some basic mathematics, we knew just how sweet those cereals really were. In addition, we had made a personal discovery that would be fondly remembered in our family for years to come.

********************

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Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

You might think that we were a bit extreme, but we went on to investigate every cereal box in our pantry and others at the grocery store. Before long, we knew which cereals had the highest percentages of sugar and which had the lowest and, given that we believed that limiting sugar intake was a good thing, we began purchasing those cereals with lower sugar percentages. Note: For the record, Golden Puffs topped the list in terms of sugar percentage (along with some other cereal brands). We didn’t find any cereals that had more than 50% sugar, and we found a few that had 0% sugar, e.g., the original Shredded Wheat. In the years since then, I have noticed that the number of cereals containing 50% sugar seems to have dropped, perhaps a reflection of a more health-conscious public.

My son went on to construct a graph of the sugar percentages in a number of the cereals that we had investigated. Using the amount of sugar and the serving size as dependent variables and the cereal brand as the independent variable, he made a bar graph that showed the amount of sugar compared to the serving size for each cereal. When the two bars for sugar amount and serving size were considered together for an individual cereal, you could “see” a picture of the sugar ratio. Done in different colored markers, the graph was quite effective and immediately became a family focal point.

Extending the Concept The percentage of sugar is just one example of how mathematics can be used to quantify relationships in everyday experiences. I would strongly encourage you to try this out at some opportune moment—you may make a wonderful discovery of your own! Some suggestions for investigations:          

How much cheese is in a cheese pizza (i.e., what is the ratio of cheese to dough)? This could be investigated for several brands of pizza. Which pizzas have the best (highest) ratio of topping-covered area compared to uncovered crust area? What percentage of a television show time period is filled with commercials? What percentage of M&M’s in a bag of M&M’s is green? What percentage of an hour of AM radio broadcasting is actually music (rather than news, weather, traffic reports or other announcements)? What percentage of an hour of FM radio broadcasting is actually music (rather than news, weather, traffic reports or other announcements)? What percentage of an hour of public (i.e., noncommercial) radio broadcasting is actually music (rather than news, weather, traffic reports or other announcements)? How many miles per gallon does your automobile get in city driving? How many miles per gallon does your automobile get in highway driving? How many watts of power does it require to run your home computer per hour? Per month?

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Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action 

How much does it cost per month to operate your home computer?

The possibilities are endless. Each of the above relationships may be determined by using simple data collection methods and basic mathematics; for those topics that interest you personally, the results are often revealing and rewarding. You’ll have opportunities to analyze some of these relationships in the Exercises and Projects for Fun and Profit at the end of this section. Meanwhile, we’ll move on to a related and very important topic in mathematics.

Functions: A Special Kind of Relationship Mathematicians have a special interest in relationships in which each value of the independent variable has exactly one corresponding value for the dependent variable. In this respect, these relationships are clear and completely unambiguous. These types of relationships are called functions and are fundamental features of many areas of study in mathematics. To illustrate the concept of a function, we’ll consider the tables of values for two relationships, A and B; in each of them, x is the independent variable and y is the dependent variable. As is customary, the independent variable is listed in the left-hand columns of the tables:

relationship A: a function x independent 1 2 3 4

relationship B: NOT a function

y dependent 2 4 6 8

x independent 1 1 3 4

y dependent 2 3 6 8

******************** Discovery Moment: Judging by the tables of values above, what do you think is the deciding factor in whether a relationship is a function?

******************** As you may have observed, relationship A follows a simple and obvious pattern: each value of the y-variable is exactly two times greater than the corresponding value of the x-variable. In fact, we may construct a very simple algebraic description of relationship A: y=2·x

or

y = 2x

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Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

Although relationship B has almost the same set of values, there is a distinctly unusual occurrence in that the value x = 1 has two different corresponding y-values. This brings an element of uncertainty and ambiguity into the relationship. If we were to try to construct an algebraic description for relationship B, we would have to qualify matters a bit; i.e., we might say something like: y = 2x except that when x=1, y can equal 2 (i.e., y=2x) OR y can equal 3 (i.e., y=3x)

As you can see, this is awkward at best. In fact, this duplicity is the determining factor that prevents relationship B from being a function. Relationships that are also functions must have only one corresponding y-value for each x-value of the x-variable. Multiple corresponding y-values for a given x-value are the indicators that a relationship is NOT a function. relationship A: a function Clear and unambiguous, i.e., each x-value has exactly one corresponding y-value

x 1 2 3 4

y 2 4 6 8

relationship B: NOT a function Ambiguous concerning x=1, i.e., there are two different y-values for x=1

x 1 1 2 3

y 2 3 4 6

Functions, then, are quite deterministic and dependable; given a particular value of the xvariable, there is no doubt about the value of its corresponding y-value because there can be only one corresponding value.

Some Useful Function Terminology In order to apply any new mathematical concept, it is most useful to develop some appropriate terminology with which to describe the various facets or features of the concept. To review, a working definition of a function may be stated as follows: A function is a mathematical relationship in which each value of the independent x-variable has exactly one corresponding (and dependent) value for the y-variable. Some related definitions and terminology are: The set of possible values for the independent x-variable is called the domain of the function. 308 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

The set of the corresponding values for the dependent y-variable is called the range of the function. The algebraic relationship between x and y is called the rule of the function. In most functions that describe real-world situations, there is a naturally proscribed set of values for the domain. In the sales tax example, the set of values for the item price would begin at \$0.00 (in the case where an item was being given away for free, as in retail sales where you can “buy one, get one free”) and continue to the highest priced item in the store under consideration. In the case of a hypothetical Florida store whose highest priced item is \$100.00, the domain would consist of the set of the values from \$0.00 through \$100.00, inclusive. As previously noted, the rule of this function (the algebraic relationship between p and T) may be written as: T = 0.07p The range of this particular function can found by substituting each of the smallest and the largest values of the domain into the rule: lowest tax = 0.07 · smallest domain value lowest tax = 0.07 · smallest value of p lowest T = 0.07 · \$0.00 = \$0.00 highest tax = 0.07 · highest domain value highest tax = 0.07 · highest value of p highest T = 0.07 · \$100.00 = \$7.00 Hence the range of this function consists of all whole dollars-and-cents values from \$0.00 through \$7.00, inclusive. Note: This method of finding the range works only for very limited classes of functions (e.g., linear and exponential); alternatively, the range may be determined by inspecting a graphical view of the function.

The Famous Function Notation: f(x) The rule in the earlier example of a function, y=2x, may be formalized a bit further by introducing the standard notation for functions in the form of f(x). As you may recall from previous algebra classes, f(x) is simply a formal way of describing the y-variable in a function: y = a function of x or, in an abbreviated fashion, y=f(x)

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Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

This indicates that the dependent variable y is actually a function of the independent variable x. In other words, y depends on x for its value, and the rule that describes the relationship between x and y is given by the expression for f(x) itself. Consequently, we can describe the rule for the function as follows: y=2x may also be written as f(x)=2x

With respect to the rule expressed in the tax example, T=0.07p, we may describe this more formally as follows: f(p)=0.07p

Another Look at the Concept of Function: The Function as a Machine A simple machine is a very effective model of a function due the deterministic nature of a machine. Machines act according to specific rules and will always provide the same output for a given input. This is equivalent to saying that each input to the machine will have exactly one output value and, therefore, the machine naturally embodies the concept of a function. As an example, we’ll consider an ordinary toaster that you might find in someone’s kitchen (even yours). The toaster is an electromechanical machine that accepts items as input, follows a simple rule, and provides items as output. Using the standard mathematical convention, we’ll represent the input items with the variable x (the independent variable) and the output items with the variable y (the dependent variable). The rule that the toaster always follows is: toast whatever is inserted into the toaster slot. Using our formal notation for functions, we can write this in more algebraic terms as follows: y = toast(x) or, in a more abbreviated fashion, y=t(x)

The following table illustrates some input and output items based on the simple rule that the toaster will toast anything that is inserted into its slot:

input item: rule: output item: x y=toast(x) y bread toast(bread) toasted bread bagel toast(bagel) toasted bagel waffle toast(waffle) toasted waffle pop-tart toast(pop-tart) toasted pop-tart rice cake* toast(rice cake) toasted rice cake * More details in The Rice Cake Episode below.

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When you insert a bagel and push down on the toaster’s lever, the nature of the toaster guarantees that you will always get a toasted bagel as output—never a toasted waffle or anything else. Consequently, a toaster acts like a function should: each input value has one and only one output value.

The Domain of a Function: A Very Important Consideration Continuing with our illustration of a toaster function machine, we can apply the function terminology that we reviewed earlier in this section. The natural domain of the toaster consists of all items that are normally “toastable” without damaging the toaster. Bread, bagels, waffles, etc., are elements of the domain whereas kitchen knives, glasses of water, cheese, etc., are not elements of the domain. The range of the toaster consists of all of the toasted versions of the domain elements. The rule of the toaster is simply to toast the domain elements. As you can see from the above domain counterexamples, there are some items that you definitely don’t want to put in your toaster; these items represent restrictions on the domain. ******************** These restrictions are of great importance in mathematics and are generally intended to avoid undefined quantities and/or catastrophic results when performing calculations. ********************

Needless to say, inserting a kitchen knife into your toaster is a hazardous undertaking!

The Rice Cake Episode: A Cautionary Tale The rice cake in the list of input items in the table above has been marked with an asterisk due to its unusual characteristics; while rice cakes are not restricted from the domain of the toaster, they must be treated very carefully as the following true story illustrates. Some years ago, a friend of ours told us of a delightful food discovery that she had made: toasted rice cakes. She had purchased some rice cakes at a health food store, toasted them in her toaster, and had found that the toaster had transformed the bland and tasteless rice cakes into a gastronomic delight—the flavor was entirely different and, with a little added butter, almost

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heavenly. As you can imagine, my wife and I could hardly wait to try one and purchased some rice cakes at the next opportunity. Having put two rice cakes into our toaster, we went into the living room and sat down to wait for a few minutes while they toasted. Before very long, though, we smelled something that was distinctly like smoke. Upon entering the kitchen, we discovered to our great dismay that the rice cakes had caught fire and that flames were leaping up out of the toaster and scorching the bottom of the kitchen cabinet above.  Fortunately, we were able to extinguish the flames before much damage had been done; we then realized our error: we had overlooked the fact that rice cakes are very light and papery and that a lower temperature setting was required in order not to set them on fire. After this, we were very careful to watch over them as they toasted and have enjoyed many wonderfully delectable toasted rice cakes in the years since that episode. ******************** The moral of this short story: the domain of a function and the accompanying restrictions on the domain are VERY IMPORTANT! 

********************

Conclusion Relationships are prevalent in all aspects of life and in some cases it is most useful to quantify these relationships in mathematical terms. In this section, we examined the role of constants and variables, both independent and dependent, in several relationship examples. We made use of the Rule of Four in describing relationships and noted that each of the four descriptions has its particular value.

Through the recounting of a true story involving breakfast cereals, we saw how mathematics can be used to analyze relationships in very simple yet revealing ways. We then went on to examine mathematical functions, a special kind of relationship. We reviewed function terminology including domain, range and rule, and the function notation, f(x). We ended by looking at the concept of a function as a machine and used the example of a kitchen toaster to illustrate the connection. As a final consideration, we considered the importance of restrictions on the domain of a function in order to prevent undefined quantities and/or catastrophic results when performing calculations.

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Exercises and Projects for Fun and Profit Exercises in Analysis Note: Please see the Helps and Hints for Exercises on p.443 in Appendix A. Which Depends on What? In #1–4, the pairs of measurable quantities are related. Please identify the dependent variable in each pair (i.e., the variable that more naturally depends on the other for its value):

1. altitude of a commercial airplane; temperature outside the airplane 2. average outdoor temperature in July (near sea level); geographical latitude 3. number of calories burned; hours walked (at 3 mph) 4. speed of an automobile; time required to brake and come to a stop Mutifaceted descriptions: In #5–8, use the Rule of Four (see p.302) to describe each of the functions based on the written descriptions given in each exercise, i.e., for each written description of a function: a) construct an algebraic relationship b) construct a table of sample values that contains three typical ordered pairs c) draw a graphical representation (i.e., plot the ordered pairs on a graph)

5. Income tax is equal to 15% of a person’s adjusted income. 6. Miles traveled is equal to 30 miles per gallon times the number of gallons of gasoline used. 7. Monthly phone charge is equal to \$40 plus the number of long-distance minutes times \$0.10 per minute. 8. Monthly water bill is equal to \$55 for the first 6000 gallons (or any portion thereof) plus the number of gallons used in excess of 6000 gallons times \$0.20 per gallon. To Be or Not to Be: In #9–12, identify which of the following are functions (see pp.307–308):

9.

y = 4x + 3

10. y = 0.5x – 12

(Hint: Make and examine a brief table of values for x and y.) (Hint: Make and examine a brief table of values for x and y.)

11. {(5, 7), (9, 11), (13, 15), (13, 19)} (Hint: Make and examine the table of values for x and y.) 12. {(2, 3), (4, 5), (6, 8), (8, 11)} (Hint: Make and examine the table of values for x and y.) 313 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

A Picture is Worth 1000 Words: In #13–16, identify the dependent variable (y-variable) in the relationship and then sketch a graph of the relationship using the data approximations in the table below; be sure to put the independent variable on the horizontal axis of your graph. 1. altitude temp 2. temp latitude 3. calories hours 4. speed stopping time (feet) (°F) (°F) burned walked (mph) (feet) 0 60 66 51°N 140 0.5 20 40 10,000 25 76 41°N 280 1.0 30 75 20,000 –12 83 13°N 420 1.5 60 240

13. altitude of a commercial airplane (feet); temperature outside the airplane (°F) 14. average outdoor temperature in July (near sea level) for London, Rome and Bangkok, respectively (°F); geographical latitude 15. number of calories burned; hours walked (at 3 mph) 16. speed of an automobile (mph); time required to brake and come to a stop (feet)

Further Investigations Investigations of the First Kind: For each of the exercises in #17–20, investigate the indicated relationship. Write a report that describes:

a) a chart or table of the relevant data b) a graphical picture of the data (bar graph or circle graph) c) your analysis and conclusions 17. Which pizzas have the highest ratio of topping-covered area to uncovered crust area? (Please try a few different brands of pizza—minimum of two required.) 18. How much cheese is in a cheese pizza (i.e., what is the ratio of the volume of cheese to the total volume of the pizza)? This investigation requires rearranging the contents of a pizza into a measuring device like a one-quart container; naturally, this results in a rather mangled pizza, but if you don’t mind that, the investigation may be of some interest. This could be investigated for several brands of pizza, but only one brand of pizza is required for this project. 19. What percentage of green M&M’s are in a bag (any size) of M&M’s? Please measure this for at least three different bags of M&M’s and then find the average percentage for the three bags. To find the average percentage of green M&M's, first find the percentage for each of the three bags individually; afterwards, add these three values together and divide by 3. 20. What percentage of brown M&M’s are in a bag (any size) of M&M’s? Please measure this for at least three different bags of M&M’s and then find the average percentage for the three 314 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 9.1 Mathematical Relationships: “Vary-ables” in Action

bags. To find the average percentage of brown M&M's, first find the percentage for each of the three bags individually; afterwards, add these three values together and divide by 3. Investigations of the Second Kind: For each of the exercises in #21–24, investigate the indicated relationship. Write a report that describes:

a) b) c) d)

a chart or table of the relevant data a graphical picture of the data (bar graph or circle graph) your analysis and conclusions the call letters of the radio or television station that you used

Investigations of the Third Kind: For each of the exercises in #25–28, investigate the indicated relationship. Write a report that describes:

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28. How much does it cost per month to operate your home computer? Note: The average computer, including its monitor, uses 200–300 watts of power during each hour of use; you’ll need to check your monthly electric bill to determine the cost of a kilowatt-hour (1000 watt-hours).

29. Functionality in your Home: Find and describe three function machines in your home (appliances are a good place to begin; see pp.310–311 for an example). Write a brief report that includes clear and precise descriptions of: a) the natural domain of each function b) the rule of each function c) the range of each function 30. Investigations of the Fourth Kind: Decide upon a relationship in your daily life that you would like to investigate. Write a report that describes: a) b) c) d) e)

the nature of the relationship how and where you collected your data a chart or table of the relevant data a graphical picture of the data your analysis and conclusions ********************

Creative Project: The Shape of Daylight

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THE SHAPE OF DAYLIGHT A Modeling Project Assignments: For each assignment, please find and record the length of day for the twenty-first day of each of the 12 months of the year. You may find this data at the U.S. Naval Observatory website (see below). This set of data will allow you to determine twelve data points, i.e., twelve ordered pairs of the form (date, length of day). Length of day is measured from sunrise to sunset. Afterwards, construct a graph that shows the relationship between the length of day (the dependent variable; vertical axis) and the month of the year (the independent variable; horizontal axis) based on actual data from the indicated location. Connect the data points with a smooth curve. The result will be a “picture” of the amount of daylight that is received from the sun over the course of a calendar year; it will provide a visual sense of the shape of daylight and how it varies throughout the year. assignment location: city and state Assignment #1 St. Petersburg, Florida Denver, Colorado Assignment #2 Seattle, Washington Assignment #3 Anchorage, Alaska Assignment #4 Note: The U.S. Naval Observatory provides data for the length of day for many locations in the world: http://aa.usno.navy.mil/data/docs/Dur_OneYear.php

Enter the indicated state and city into interactive Form A and then click on the button labeled Compute Table. The lengths of day are given in the form hh:mm, where the first pair of digits represents the number of hours and the second pair of digits represents the number of minutes. It will be helpful to convert the hours and minutes into minutes only to make graphing the data a bit simpler. For example, 6 hours and 51 minutes is equivalent to: (6 hours · 60 minutes per hour) + 51 minutes = 360 + 51 minutes = 411 minutes ******************** Mathematics is a wonderful tool for representing real-world phenomena in order to see visual pictures of the relationships involved. 

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Math Is Everywhere! Explore and Discover It! 9.2 Linear Functions: A Deeper Look

9.2 Linear Functions: A Deeper Look “Do not worry about your difficulties in mathematics. I assure you that mine are greater.” —Albert Einstein

Introduction In their efforts to describe and quantify relationships in the world around us, mathematicians have discovered that many of these relationships fall into two basic categories: linear and exponential. In this section, we will examine linear functions and endeavor to deepen our understanding of this basic type of relationship; in the next chapter, we will similarly examine exponential functions and then compare and contrast these two basic types of relationships. Both of these functions are quite useful in describing natural phenomena and in providing a means by which to predict behaviors. These functions are known as mathematical models.

Slope: A Very Important Measurement Concept When two “vary-ables” are in relationship with each other, we often look to the table of x- and yvalues in order to see which x-values are connected to which y-values, i.e., we examine the ordered pairs in the relationship. In addition, it is often very useful to know exactly how fast the dependent y-values are changing with respect to the corresponding independent x-values. In order to measure this rate of change, mathematicians compare a change in two specific values of the dependent variable with the change in the two corresponding values of the independent variable; these values are obtained from two ordered pairs in the relationship. This comparison takes the form of a ratio:

change in two specific values of the dependent y  var iable change in the two corresponding values of the x  var iable This ratio was given the name slope and, in a practical sense, measures the steepness of the incline of the line segment that connects the two specified ordered pairs. For reasons that are lost in history, mathematicians began using the letter “m” rather than the expected “s” to more concisely refer to this ratio known as slope. Other terms have been invented and used in the ensuing years; some of these are shown below: slope  m 

change in y  y vertical change rise    change in x  x horizontal change run

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To expand on the definition listed above, we can label the x- and y-variables of the very general ordered pair, (x, y), with suitable subscripts so as to create two specific yet general ordered pairs that represent any two ordered pairs in the relationship: ( x1 , y1 ) and ( x 2 , y 2 )

We can then apply the definition of slope to these two general ordered pairs: slope  m 

change in y  y vertical change rise y 2  y1 y1  y 2      change in x  x horizontal change run x 2  x1 x1  x 2

The last two entries in this definition provide the familiar formulas for the slope of a line: m

y 2  y1 y  y2 or m  1 x 2  x1 x1  x 2

How to Read a Graph During my years of teaching, I have often found that students benefit from some basic instruction in how to read a graph, or graph-story. Reading a graph is very much like reading a book since a graph contains some of the same elements as a book. Both a graph and a book contain:   

a main character a story plot a set of numbered pages

In addition, we read a graph from left to right, just as we read a book. In a graph of a function, the main character is the y-variable. Just as a reader of a book is interested in the actions of the main character (hero or heroine), a reader of a graph is primarily interested in the behavior of the values of the y-variable: are the values high or low, positive or negative, increasing or decreasing, etc. The values of the x-variable, on the other hand, serve simply as page numbers in the graphstory; for example, page 1 occurs where x=1, page 2 occurs where x=2, and so on. In an algebra graph-story, there also are pages numbered 1.1, 1.5, 2.7, etc. In summary, a reader of a graph-story is interested in the value of the y-variable on a specific page in the story. The values of the y-variable may go up and/or down as the reader progresses through the successive pages of the story and it is this up-and-down behavior of the y-values that comprise the graph-story plot. Some graph-story plots are rather uninteresting (as are some

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books) while others are revealing or dramatic; we shall examine examples of both in ensuing sections. Digging a bit deeper into the graph-story, we find that while the overall shape of a graph, i.e., the shape created by the behavior of the y-values, gives us a general indication of the nature of the relationship between the two variable quantities being measured, the slope at a given point on the graph enables us to quantify how fast the two variable quantities are changing with respect to each other at the given point. Note: Actually, we can’t algebraically find the slope at only one point (one ordered pair) on a graph due to the need to measure the change in two specific y-values and the change in the two corresponding x-values; however, we can choose two points (two ordered pairs) that are very close to each other. For example, if we choose the ordered pair in which x=1 and also choose the ordered pair in which x=1.1, we will be able to calculate an approximation of the slope at the single ordered pair in which x=1. (The further refining of this approximation comprises a major topic in the study of Calculus.)

In the graph shown below, each of the slopes of the three line segments between the indicated ordered pairs has a different value: 30  30 1

For the leftmost line segment between (0, 0) and (1, 30): slope  m 

For the middle line segment between (1, 30) and (2, 90): :

For the rightmost line segment between (2, 90) and (3, 105): : slope  m 

slope  m 

60  60 1 15  15 1

(3, 105) (2, 90)

(1, 30)

(0, 0)

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As you can see, the calculations of the slope values enable us to quantify the different rates of change in the y-values relative to the x-values. If this graph represented the relationship between distance traveled in miles (dependent variable) and elapsed time in hours (independent variable), then during each hour the rate of change of distance traveled with respect to elapsed time would be as follows: 

For the leftmost line segment: slope  m 

30 miles traveled  30 miles per hour 1 hour elapsed

For the middle line segment:

slope  m 

60 miles traveled  60 miles per hour 1 hour elapsed

For the rightmost line segment: slope  m 

15 miles traveled  15 miles per hour 1 hour elapsed

In this case, the value of the slope measurement for each line segment quantifies the speed of travel and provides very useful information about the scenario. ******************** Since we often like to know how fast quantities change, the slope is a very useful piece of information, one that is often of great value in investigations in business, sciences such as mathematics, physics, chemistry and biology, and social sciences such as sociology and economics.

********************

Linear Functions and their Graphs: Keeping Things Straight and Steady Now that we have discussed the concept of slope and how to read a graph, we may begin our investigation of linear functions—one of the basic types of relationship models for the world around us. A graph of a linear function has a very predictable shape: a straight line (you guessed it!). This straight-line nature is directly related to the slope at a given point.

******************** Discovery Moment: Can you infer the fundamental characteristic of the slope of a linear function from the straight-line nature of its graph?

********************

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Math Is Everywhere! Explore and Discover It! 9.2 Linear Functions: A Deeper Look

In the graph of the linear function shown below, the vertical changes and corresponding horizontal changes have been indicated for three of the points (ordered pairs) on the graph. Notice how regular and repetitious these are! In fact, the horizontal changes were intentionally made equal so that it would be appropriate to compare the corresponding vertical changes and, as you can see, both vertical changes have the same value. (10, 20)

 10 units

(5, 10)

 10 units

(0, 0)

This means that each line segment that connects two successive ordered pairs has the same slope as the other line segments; in other words, the slope of the line is everywhere the same.

******************** This is an inherent characteristic of a linear function: its slope is constant.

********************

Examples of Linear Functions Many real-world relationships exhibit this property; some examples are: salary = \$6.00 · number of hours worked sales tax = 0.07 · sales price annual interest on savings account = 0.02 · amount in savings account monthly telephone bill = \$24.95 + \$0.10 · number of long-distance minutes used

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In each of these examples, the slope, or rate of change of the dependent variable with respect to the corresponding change in the independent variable, is constant. This feature ensures a steady change in the values of the y-variable, the main character in the graph-story of a linear function, just like the steady changes in height experienced while climbing or descending a staircase; we often refer to this change as steady growth (or decline). In the salary example, the slope is given by the constant rate of change of \$6 per hour, or

\$6.00 . hour

In the sales tax and interest examples, the slopes are given by the constant rates of change of: \$7.00 7 \$2.00 2   0.07 and   0.02 , respectively. \$100.00 100 \$100.00 100

In the telephone bill example, the slope is given by: \$0.10 min ute Note: The \$24.95 in this relationship is known as the initial value, i.e., the value of the y-variable when the value of the x-variable is 0; it is also known as the y-intercept since it represents the y-value at the point where the graph intersects the y-axis. More on this will follow.

A General Equation (or Model) for a Linear Function Using the above examples of linear functions, we can now construct a general equation that describes all such functions. In the first three examples, the observable pattern is: dependent variable = slope · independent variable

Since, by agreed upon convention, the y-variable is considered as the dependent variable and the slope is represented by m, we can rewrite this as: y=m·x or y = mx

In the telephone bill example, the value for the y-intercept is conventionally represented by the letter b and we may write: y = b + mx or y = mx + b

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This should be very familiar from your previous algebra experiences.

A Conceptual Understanding of the Model While you may be familiar with the above equation, y = mx + b, many students have difficulty explaining the inner workings of this relationship, i.e., WHY does y = mx + b. So let’s investigate this relationship in more detail and come to a deeper conceptual understanding; then you’ll be able to easily explain the “Why” of the relationship model. First of all, we need to recognize that the x- and the y-variables in the equation relationship represent specific x- and y-values that are related, i.e., that form ordered pairs in the relationship. For example, in the equation y=2x+3, the x- and y-variables represent the following pairs of values (along with the infinite number of other pairs that satisfy the equation):

y

=

2

·

x

+

3

3 5 7 9

= = = =

2 2 2 2

· · · ·

0 1 2 3

+ + + +

3 3 3 3

(x,y) ordered pair (0, 3) (1, 5) (2, 7) (3, 9)

The equation, then, is focused on the y-value of an ordered pair and states that this y-value of an ordered pair may be found by multiplying a given x-value times the slope value of the line and afterwards adding the value of the y-intercept, b, to this product. There are three important observations that will lead us to a deeper conceptual understanding:

The first of these is to recognize the special importance of the key reference point (ordered pair) in which the x-value is 0. This point lies on the y-axis of the graph and every change in x is measured from the vertical line in which x=0 (i.e., the y-axis). o Consequently, an x-value of an ordered pair tells us how many units of change have occurred in the x-direction (from the vertical line where x=0) in order to arrive at the ordered pair on the graph.

The second is to recognize that the slope ratio provides a very important fact about the relationship (equation). o In particular, the slope ratio tells us how many units of change in the y-direction correspond to ONE unit of change in the x-direction.

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The third is to observe that when the slope ratio is multiplied by the x-value of an ordered pair, i.e., the number of units of change in the x-direction, the product tells us how many corresponding units of change will occur in the y-direction. o

In other words, the product m  x tells us how many total units of change will occur in the y-direction for a given x-value of an ordered pair.

The Simplified Linear Model: y=mx Some examples will illustrate these principles. To begin, we’ll focus our attention on the simplified linear equation, y = mx, in which the y-intercept is equal to 0 (i.e., b=0); the key reference point in this equation is the ordered pair (0,0). In this case, not only is a change in the x-direction measured from the line x=0 (i.e., the y-axis) but the corresponding change in the y-direction is also measured from the line y=0 (i.e., the x-axis). This is the simplest type of linear equation. Example 1: y=2x or, emphasizing the slope ratio, y 

2 x. 1

In this example, the slope ratio tells us that there will be 2 units of change in the ydirection for every 1 unit of change in the x-direction. When we choose an x-value such as x=3, we are choosing to establish an ordered pair in which there has been 3 units of change in the x-direction rather than the 1 unit of change indicated in the slope ratio. Since each unit of change in the x-direction produces 2 units of change in the y-direction, our chosen 3 units of change in the x-direction will consequently produce three times as much change in the y-direction:

y  m x 

2 units in y  direction  3 units in x  direction  6 units in y  direction 1 unit in x  direction

Consequently, we have established that the ordered pair, (x, y) = (3, 6), belongs to the relationship y=2x. (3, 6)

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Example 2: If we choose x=5, then we are choosing to establish an ordered pair in which there has been 5 units of change in the x-direction rather than the 1 unit of change indicated in the slope ratio. Consequently, these five units of change in the x-direction will produce five times as much change in the y-direction:

y  m x 

2 units in y  direction  5 units in x  direction  10 units in y  direction 1 unit in x  direction

Consequently, we have established that the ordered pair, (x, y) = (5, 10), belongs to the relationship y=2x.

The Complete Linear Model: y=mx+b We will now expand our investigation to the more general equation of a line: y=mx+b. In this case, the key reference point in which x=0 is the point (0, b); this is the point on the graph at which the graph intercepts the y-axis. As usual, changes in the x-direction will be measured from the line x=0; however, the corresponding changes in the y-direction will now be measured from the value y=b rather than from the value y=0. After calculating the corresponding change in y-value for a given x-value by multiplying the slope times the given x-value, we must then add this corresponding change in y-value to the initial y-value of b in order to determine the correct y-value for the ordered pair. Example 3: y=2x+ 3 or, emphasizing the slope ratio, y 

2 x 3. 1

In this example, the slope ratio tells us that there will be 2 units of change in the ydirection for every 1 unit of change in the x-direction. When we choose an x-value such as x=4, we are choosing to establish an ordered pair in which there has been 4 units of change in the x-direction rather than the 1 unit of change indicated in the slope ratio. Since each unit of change in the x-direction produces 2 units of change in the y-direction, our chosen 4 units of change in the x-direction will consequently produce four times as much change in the y-direction:

y  m x 

2 units in y  direction  4 units in x  direction  8 units in y  direction 1 unit in x  direction

Important Note: These eight units of change in the y-direction begin at the key reference point in which x=0, i.e., at the point (0, b) = (0, 3) rather than at the point (0, 0); hence we need to add the initial y-value of b=3 units to these eight units of change to obtain the total y-value for the ordered pair.

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Consequently, the total y-value is 8+3=11 and we have established that the ordered pair, (x, y) = (4, 11), belongs to the relationship y=2x+3.

(4, 11)

 8 units

(0, 3)  3 units

The Power of Generalization Once Again In the above diagram, notice how the formula for the equation of a line naturally comes into 2 being. Since the slope value, 2, may be considered as a ratio, , and since the denominator 1 representing the number of units of change in the x-direction is 1, multiplying this ratio by the chosen x-value, i.e., m · x, will naturally result in the corresponding change in the y-direction. The four units of change in the x-direction produce:

m·x=

2 · 4 = 8 units of change in the y-direction. 1

Adding these eight units to the initial three y-units in the key reference point (0,3) produces the final y-value of the ordered pair (4, 11) in the relationship. 11 =

2 ·4+3 1

This procedure to determine a y-value of an ordered pair in a linear relationship may be carried out for any chosen x-value and any initial key reference point.

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******************** Hence we may conclude that, in general, the following equation represents all linear functions: y = mx + b where m represents the slope of the line, b represents the value of the y-intercept, and the x- and y-variables represent those x- and y-values that form valid ordered pairs in the linear relationship.

********************

What About Slopes that Involve Rational and/or Negative Numbers? In the previous examples, the slope value was an integer and the understood denominator in the slope ratio of 1 unit of change in the x-direction was most convenient. If the slope value is a 2 rational number (i.e., a fraction such as ), then there is already a denominator involved in the 3 slope ratio. However, it is quite helpful to ignore the denominator in a rational slope value and instead introduce a denominator of 1 as we did earlier with integer slope values. Given the 2 rational slope of m  , we may consider it as follows: 3 2 unit change in y  direction 2 3 m  3 1 unit change in x  direction Viewed in this manner, the slope ratio tells us that there is a

2 unit change in the y-direction for 3

every ONE unit of change in the x-direction. ******************** Introducing the denominator of 1 for rational slope values allows us to view ALL slope values as measurements in the y-direction with an understood corresponding change of 1 unit in the x-direction.

********************

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3 , we may apply the same method as described 4 above and introduce denominators of 1 to obtain:

With regard to negative slopes such as –2 and 

2 m and 1

3 m 4 1 

In both cases, the introduced denominators are given as positive values while the numerators retain their negative values. When we consider these slopes, then, it is natural to interpret the negative values as downward changes in the y-direction rather than the upward changes in the y-direction associated with positive slope values. ******************** Consequently, all slopes may be interpreted as the amount of upward or downward change in the y-direction corresponding to ONE unit of change, moving from left to right, in the x-direction.

********************

Retrospection: The Importance of the Slope In the linear relationship y=mx+b, there are only two defining constants—m, the slope, and b, the y-intercept. Note: While m and b may be considered as variables in the above equation, for a given line they are considered as constants; the only variables in a specific linear relationship are the x- and y-variables.

The y-intercept simply tells us where the graph crosses the y-axis, i.e., it tells us the value of the y-variable when the value of the x-variable is 0. The slope, however, conveys much more interesting information about the relationship; it tells us how fast the y-values are changing with respect to the x-values. In other words, the slope tells us how much change occurs in the ydirection for each unit of change in the positive x-direction (i.e., from left to right).

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relationship y=½x+3

slope value ½

y=¾x–5

¾

y = –3x – 7

–3

y = 10x + 20

10

slope information There is ½ unit of change in the y-direction for each 1 unit of change in the positive x-direction There is ¾ unit of change in the y-direction for each 1 unit of change in the positive x-direction There are –3 units of change in the y-direction for each 1 unit of change in the positive x-direction There are 10 units of change in the y-direction for each 1 unit of change in the positive x-direction

The slope information is of interest and importance due to our natural desire to know how fast quantities are changing in a relationship. If, for example, you happen to have a leaky faucet in your home, it is important to know how fast the water is leaking: 1 ounce per day, 1 cup per day, 1 quart per day, 1 gallon per day, etc. This knowledge will then determine the urgency of making repairs. The amount of water leakage per day is the information provided by the slope value in the relationship between amount of leakage and elapsed time amount of water leakage = amount of water leakage per day · number of days y = m · x

As another example, if you have money invested in an interest-bearing account, the most important issue is the interest rate on your account. Again, this information is provided by the slope value in the relationship between the annual interest earned on your account and the amount of your investment: amount of earned annual interest = rate of interest · amount of investment y = m · x

******************** Ultimately, then, the slope is the most important feature of a linear relationship since it conveys the rate of change between the two variable quantities; knowing this rate of change is most valuable in terms of assessing the effects of the relationship.

********************

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Conclusion In this section, we investigated the particular type of mathematical relationship known as a linear function in order to deepen our understanding of the concepts involved. After a review of the concept of slope, we drew comparisons between reading a graph and reading a book. We then went on to examine linear functions and determined that their identifying feature is a constant slope, or constant rate of change. We also developed a general model for linear functions and endeavored to answer the question, “WHY does y=mx+b?” We examined:  

the key reference point (i.e., the y-intercept) the role of the slope ratio

Introducing a denominator of 1 to all slope values, whether integer, rational or negative, was seen as helpful in understanding the meaning of slope in a relationship. We concluded with some remarks on the importance and value of the slope in a linear relationship.

********************

Exercises and Projects for Fun and Profit Exercises in Analysis Note: See the Example Exercises at the end of this section, pp.334–335.

Finding the Slope: In #1–4, use the slope formula involving two ordered pairs to find the slope of the line containing the given points. (See Helps and Hints for Exercises #1–4 on p.334 at the end of this section.)

1. 2. 3. 4.

(0, 1) and (2, 9) (4, –2) and (7, –11) (–5, 2) and (4, –4) (1, –3) and (11, 5)

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More about Slope: In #5–8, find the slope of the line whose y-intercept is 5 and that contains the given point. (See Helps and Hints for Exercises #5–8 on p.334 at the end of this section.)

5. 6. 7. 8.

(2, 9) (4, –7) (6, 1) (12, –4)

Finding the Equation of a Line: In #9–12, find an equation of the line whose y-intercept is 3 and that contains the given point. (See Helps and Hints for Exercises #9–12 on p.334 at the end of this section.)

9. 10. 11. 12.

(3, –3) (2, 15) (4, 5) (8, 1)

Finding the Equation of a Line: In #13–16, find an equation of the line that contains the given points. (See Helps and Hints for Exercises #13–16 on p.334 at the end of this section.)

13. 14. 15. 16.

(1, 6) and (–2, –6) (–2, 13) and (3, –2) (–3, 12) and (1, –8) (4, 27) and (2, 11)

Identifying the Slope: In #17–20, identify the slope and write it in ratio form; please include the appropriate quantifiers such as dollars, days, gallons, hours, etc. (See Helps and Hints for Exercises #17–20 on p.335 at the end of this section.) Hint: The slope will be a constant value in the relationship and will be represented by a ratio involving the preposition “per”.

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17. 18. 19. 20.

Total cost = number of items · price per item Distance traveled = miles per gallon · number of gallons (of fuel) Production cost = production cost per item · number of items + initial machinery cost Salary for second year = salary for first year + percent salary increase · salary for first year

********************

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EXAMPLE EXERCISES #1–4 Finding the Slope Use the formula for the slope of a line given at the beginning of this section to determine the slope. Exercise #2: m =

9  2  (11)   3 47 3

#5–8 More About Slope The y-intercept is part of the ordered pair, (0, 5). Using this point together with the point given in each of the exercises will provide two points with which you may find the slope as in #1–4 above. Exercise #6: Use the points (4, –7) and (0, 5) and the formula for the slope as in #2.

 7  5  12   3 40 4 #9–12 Finding the Equation of a Line The y-intercept is part of the ordered pair, (0, 3). Use this point together with the point given in each of the exercises to find the slope of the line. Then use the standard equation of a line, y = mx + b, and substitute the value of the slope for m and substitute the value 3 for b to write the appropriate equation. Exercise #10: Use the points (2, 15) and (0, 3) to find the slope: m = 6. Then use the values m=6 and b=3 to write the equation: y = 6x + 3. #13–16 Finding the Equation of a Line Use the two given points to find the slope of the line as in #1–4. Then use the standard equation of a line, y = mx + b, and substitute the value of the slope for m and substitute the coordinates of one of the given points for x and for y. This will allow you to solve the resulting equation for b. Then rewrite the equation, y = mx + b, using the values for m and for b.

13  (2) 15   3 , so m = –3. Using the point (3, –2) and 23 5 its values for x and y (x = 3 and y = –2), we can now substitute these along with –3 for m into the equation of a line: Exercise #14: The slope is

–2 = (–3)(3) + b and –2 = –9 + b and 7 = b Then rewrite the equation using values for m and b: y = –3x + 7

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#17–20 Identifying the Slope The slope is always described as a ratio. Look for the English expressions that involve ratios in order to identify the slope. Exercise #18: The only English expression that involves a ratio is "miles per gallon". miles We can write it as a ratio as follows: gallon

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9.3 Linear Models: Analytic and Predictive “Many secrets of art and nature are thought by the unlearned to be magical.” —Sir Francis Bacon

Introduction The world around us is full of relationships waiting to be discovered and explored. Mathematics provides both a technical language and a set of analytic tools that are wonderfully suited to describe these relationships; mathematicians and others spend a good deal of time constructing models (equations or formulas) for relationships that are of interest. In this section, we will learn how to construct and how to use some of these mathematical models.

Simple Linear Models Now that we have an understanding of the concept of slope and the role that it plays in linear functions, or linear models, we may begin to use this understanding to construct models of relationships that we encounter in the world around us. We have already mentioned some of these models in the previous section:

\$6.00 · number of hours worked 1 hour \$0.07 sales tax = · sales price \$1.00 \$0.02 annual interest on savings account = · amount in savings account \$1.00 \$0.10 monthly telephone bill = \$24.95 + · number of long-distance minutes used min ute amount of leakage total amount of water leakage = · number of days 1 day salary =

In each of the above models, a relationship (more specifically, a function) between two measurable quantities is established. While these are rather basic models, they exhibit two features that are of importance:  

a measurement of the rate of change between the variables (i.e., the slope) a means of calculating or predicting the value of the dependent variable for any value of the independent variable that is within the domain of the function

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The rate of change between the variables is the single most important feature of a linear relationship and is often of practical importance as well:  

the rate of water leakage mentioned in the example above is of crucial interest in determining what action needs to be taken the cost per minute of long-distance minutes is of significant interest in determining how long you may want to talk long-distance on the telephone

In addition, we are often interested in predicting what will happen if we increase the value of the independent variable, e.g., how much water will be wasted if we wait to call the plumber until Monday (Murphy’s Law dictates that water leaks mostly occur on Friday night or Saturday…), or how much will our phone call cost if we talk for sixty minutes? This predictive capability of a model contributes greatly to its usefulness and practicality. Not all relationships in the world around us are linear in nature but, for those that are, a linear model is a valuable tool in analyzing the relationship and predicting outcomes. You’ll have opportunities to construct some linear models in the Exercises and Projects for Fun and Profit at the end of this section.

More Involved Linear Models While the models mentioned above are very simple and completely linear, many relationships in the world around us are not as clear-cut and straightforward. Some of these more complicated relationships, however, exhibit distinctly linear tendencies and may be represented by a linear model approximation. In other words, some relationships contain data points that do not lie on a perfectly straight line but are relatively close to a theoretical straight line that passes through the “middle” of the data points; this straight line may be conveniently used as an approximation to the actual set of data points. As an example, consider the following set of points: {(0, 0), (1, 0.9), (2, 2.1), (3, 2.8), (4, 3.7), (5, 5.2), (6, 6.3), (7, 6.5), (8, 8.4), (9, 8.9), (10, 10)} A graph of these points will reveal that they are not quite linear but that taken as a whole they have a definite, linear-like overall appearance:

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While each of the line segments connecting the successive points has a slightly different slope than the preceding line segment, it is clear that there is a generally “steady” rise in the y-values as the x-values increase. If we want to be very precise, then the only way to describe the relationship between the y-values and the x-values is to construct a linear equation for each line segment using its two endpoints. In the end we would have ten different equations, each with a different domain; the first several of these are listed below: linear equation for line segment y = 0.9x + 0.0 y = 1.2x – 0.3 y = 0.7x + 0.7 y = 0.9x + 0.1

domain

0 < x ≤1 1 < x ≤2 2 < x ≤3 3 < x ≤4

y = 1.1x – 1.0

9 < x ≤10

While very accurate, this is a rather cumbersome way to describe the relationship and offers no way to predict y-values for x-values that are greater than 10.

Linear Approximation to the Rescue: The “Line of Best Fit” Mathematicians realized that considerable value could be gained if a simpler means of describing relationships such as the one above were devised. The natural way of doing this, of course, was to make a linear approximation to the set of points, i.e., draw a straight line through the set of points and attempt to find the “middle” ground in doing so. A linear approximation is shown below; this line is referred to as the “line of best fit.”

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As you can see, this linear model is a very good representative of the actual set of data points; all of the data points lie fairly close to the straight-line approximation. Since the graph of the linear model contains the points (0,0) and (10, 10), its equation is very simple: linear model: y = x

This very simple linear model may then be used to approximate the y-values for any specified xvalue. For example, if x=6.5 were chosen, then the model tells us that the approximate corresponding y-value is y=6.5. In addition, the model may be used to predict the y-values for xvalues that are larger than those in the originally specified set of points. If we were interested in knowing what the y-value would be for x=17, we could use the model to find out. ******************** Analytic Moment: Some important questions naturally arise in the above approximation process; can you determine what they are?

********************

Linear Regression: Perfecting the Approximation As convenient as the linear model is, there are some important issues to consider. 

The first question is: If each of us drew a straight line through the approximate “middle” of a set of data points and some of our lines were different from each other, which line would be the best representative of the relationship?

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A related question is: How do you construct the “best” linear model?

A third question is: How accurate is the “best” linear approximation? In other words, how close is the “fit” of the actual data points to the straight-line approximation?

Thankfully, mathematicians were able to answer these questions many years ago and we may now make use of their findings. Using the methods of Calculus, mathematicians devised a way to minimize the sum of the distances from the data points to the “line of best fit”; this “line of best fit” became known as the linear regression line and, of course, it had a linear regression equation to describe it. This linear regression equation is also known as a linear regression model for a set of data. All graphing calculators have a built-in statistical feature that allows you to enter a set of data points and obtain the linear regression equation automatically; this form of equation is used quite often in analyzing and reporting data in experiments of various sorts (in medicine, social science, science, etc.) In addition, graphing calculators are programmed to calculate what is known as the correlation coefficient, a numerical measure (as a percentage) of how closely the actual data points are correlated to the linear regression equation.   

a correlation coefficient of 1, or 100%, means that ALL of the data points lie on the graph of the linear regression equation a correlation coefficient of 0.9, or 90%, indicates that there is a rather strong correlation between the positions of the data points and the regression line a correlation coefficient of 0.3, or 30%, indicates that there is a rather weak correlation between the positions of the data points and the regression line

This is similar to the percentages given in weather forecasts: a 30% chance of rain doesn’t mean that it won’t rain, but that rain is not very likely. For our purposes here in this section, we will focus our attention on drawing approximate “best fit” lines by hand with a ruler or straightedge and pencil when we are given a set of data points.

Correlation—A Visual Approach In order to get a better picture of what correlation means, read the Guide to Correlation Coefficients on pp.453–454 in Appendix C. You can easily see the relationships here; the highest correlation values occur when the data points are positioned very close to the line.

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Scattergrams and Regression Equations The concepts of linear regression and correlation may also be applied to mathematical relationships in which there are multiple y-values for each x-value. These types of relationships are common in the social sciences where large amounts of data are collected and analyzed. Graphs of the data are called scattergrams and may be constructed in the usual manner; in these cases, a line of best fit (i.e., a regression line) helps to identify the general trend in the data and provides analysts with a mathematical model by which they may predict future developments. An example of a scattergram and its associated regression line are shown below:

In this case, the relationship does not represent a mathematical function but we can make use of the regression line (a mathematical function) and its equation to help us analyze the data. In the graph above, the regression line passes through the “middle” of the data and provides a visual picture of the general trend in the data. The correlation between the regression line and the actual data points is fairly high as evidenced by the “closeness of fit” between the data points and the line. The regression equation may be determined by choosing two points on the line, finding the slope of the line that passes though them, and then finding the associated y-intercept of the line as described in the previous section. Afterwards, predictions concerning future developments may be made using the regression equation as a model. In the graph above, two points that visually appear to be on the regression line are: (1, 6) and (9, 4.25). The slope of the line through these points is:

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m

4.25  6  1.75  0.21875  9 1 8

Using the linear equation form, y=mx+b, we may substitute values for x, y and m into the equation and solve for b: 6  (0.21875)  (1)  b 6.21875  b Hence, the regression equation is approximately given by: y = –0.21875x + 6.21875 Using this equation, we may predict the corresponding y-values for the x-values for which we currently have no associated data points. For example, if we wanted to know the value for y when x=5.2, we could simply substitute 5.2 for x in the regression equation: y = –0.21875 · 5.2 + 6.21875 = 5.08125 If we wanted to predict the y-value when x=14, we could substitute 14 for x: y = –0.21875 · 14 + 6.21875 = 3.15625 As you can see, the regression equation is a powerful tool for analyzing data. ********************

Exercises and Projects for Fun and Profit Note: To draw an approximate "line of best fit", first plot the data points on a graph and then use a ruler and a pencil to draw a line that passes through the approximate "middle" of the data points. Finding the "middle" is an approximation, so just make your best attempt at drawing this line somewhere near what visually appears to be the "middle" of the data. As examples, please note the graphs shown on p.339 and p.341. Note: See Helps and Hints for Exercises on pp.444–445 in Appendix A.

Exercises in Analysis Scattergrams and linear regression equations: For #1–2, the data below represent the hypothetical heights (in inches) of twenty-seven children between the ages of 2 and 10; three measurements are given for each age. For the given data:

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a) Construct a scattergram of the data (i.e., graph the data on x- and y-axes). b) Draw an approximate “line of best fit” (regression line) through the “middle” of the data by using a ruler or straightedge to draw a line that goes through the “middle” of the data points. c) Determine the coordinates of two points that lie on this “line of best fit”. d) Find the equation of the “line of best fit” (the regression line) using the methods described in the preceding section. e) Use the Guide to Correlation Coefficients on pp.453–454 in Appendix C to estimate the correlation coefficient for your “line of best fit”. f) Use your linear regression equation to predict the average height of a child at age 15. 1. Please complete items a) through e) above for this set of data (see Helps and Hints for Exercises on pp.445–446 in Appendix A): age 2 2 2 3 3 3

height 22.5 25.0 32.5 30.0 32.5 40.0

age 4 4 4 5 5 5

height 32.5 37.5 40.0 30.0 40.0 42.5

age 6 6 6 7 7 7

height 37.5 40.0 42.5 42.5 52.5 57.5

age 8 8 8 9 9 9

height 47.5 50.0 55.0 47.5 55.0 57.5

age 10 10 10

height 57.5 60.0 65.0

2. Please complete items a) through e) above for this set of data (see Helps and Hints for Exercises on pp.445–446 in Appendix A): age 2 2 2 3 3 3

height 27.5 30.0 35.0 27.5 30.0 32.5

age 4 4 4 5 5 5

height 35.0 37.5 42.5 32.5 42.5 47.5

age 6 6 6 7 7 7

height 37.5 40.0 47.5 42.5 52.5 57.5

age 8 8 8 9 9 9

height 45.0 47.5 52.5 52.5 55.0 57.5

age 10 10 10

height 52.5 55.0 65.0

3. Linear regression models: For the data given below on water leakage:

a) Construct a graph of the data points. b) Draw an approximate “line of best fit” (the regression line) through the data points by using a ruler or straightedge to draw a line that goes through the “middle” of the data points. c) Determine the coordinates of two points that lie on this “line of best fit”. d) Find the equation of the “line of best fit” (the regression line) using the methods described in the preceding section. e) Use the Guide to Correlation Coefficients on pp.453–454 in Appendix C to estimate the correlation coefficient for your “line of best fit”.

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f) Use the equation to predict how much water leakage will occur on the 14th day after the leak started. (Is it time to call the plumber yet?) total amount of water leakage = day

1 2 3 4 5 6 7

amount of leakage · number of days 1 day amount of water leakage 2.0 cups 2.3 cups 2.7 cups 2.5 cups 2.6 cups 2.9 cups 3.2 cups

4. Linear regression models: For the data given below on student enrollment: a) Construct a graph of the data points. b) Draw an approximate “line of best fit” (the regression line) through the data points by using a ruler or straightedge to draw a line that goes through the “middle” of the data points. c) Determine the coordinates of two points that lie on this “line of best fit”. d) Find the equation of the “line of best fit” (the regression line) using the methods described in the preceding section. e) Use the Guide to Correlation Coefficients on pp.453–454 in Appendix C to estimate the correlation coefficient for your “line of best fit”. f) Use the equation to predict student enrollment in the 12th semester after the inception of the course. semester

1 2 3 4 5 6 7

student enrollment 230 243 255 280 302 312 335

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Math Is Everywhere! Explore and Discover It! 9.3 Linear Models: Analytic and Predictive

Linear regression models: For Exercises #5–8, construct linear regression models for the given sets of ordered pairs (data points):

a) Construct a graph of the data points. b) Draw an approximate “line of best fit” (the regression line) through the data points by using a ruler or straightedge to draw a line that goes through the “middle” of the data points. c) Determine the coordinates of two points that lie on this “line of best fit”. d) Find the equation of the “line of best fit” (the regression line) using the methods described in the preceding section. e) Use the Guide to Correlation Coefficients on pp.453–454 in Appendix C to estimate the correlation coefficient for your “line of best fit”. 5. {(1, 3), (2, 7.3), (3, 12.0), (4, 16.2), (5, 17.9), (6, 22.9)} 6. {(1, 4), (2, 6.7), (3, 7.3), (4, 9.5), (5, 11.8), (6, 15.4)} 7. {(1, 5), (2, 3.5), (3, 3.1), (4, 1.6), (5, 0.5), (6, 0.2)} 8. {(1, 6), (2, 4.3), (3, 1.6), (4, –0.3), (5, –2.6), (6, –4.5)} ********************

Creative Projects: How Does your Lawn Grow? How Does your Garden Grow?

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Math Is Everywhere! Explore and Discover It! 9.3 Linear Models: Analytic and Predictive

HOW DOES YOUR LAWN GROW? In this project, you will collect a set of measurement data and construct a linear model of the growth of your lawn; afterwards, you will write a report on the nature of your lawn’s growth. Note: This project must be done during the summer when your lawn is in its growth phase rather than in its dormant phase.

Assignment:

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Math Is Everywhere! Explore and Discover It! 9.3 Linear Models: Analytic and Predictive

HOW DOES YOUR GARDEN GROW? In this project, you will collect a set of measurement data and construct a linear model of the growth of bean sprouts; afterwards, you will write a report on the nature of the growth. Note: This project may be undertaken at any time of year. Assignment:

Please take the following steps to grow some sprouts in your home; afterwards, you will write a report on your findings: a) Purchase a small amount (a handful will do, or whatever small amount the shopkeeper will sell you) of whole mung beans (not sprouted). Mung beans may be found at some health food stores and are being used for this project due to their edibility and tastiness; if you can’t find mung beans or if you already have some dried beans or seeds in your home, then you may use dried lentils, lima beans, radish seeds, broccoli seeds, alfalfa seeds or any other dried bean or seed. b) Soak the beans in water overnight (8–12 hours) or until they begin to sprout. After they have just begun to sprout, rinse them well and drain them. You may keep them in a jar or open container such as a dish; a strainer is ideal. You will need to rinse and drain the sprouts at least twice per day to provide enough moisture for them to grow on. Be careful not to let them sit in standing water or they may become waterlogged and begin to rot. (See alternative method for growing sprouts on p.348 at the end of this project.) c) Identify several sprouts for investigation. You might want to put drops of food coloring on them (use a different color for each sprout) so that you can easily identify them during the next several days. d) Using a ruler or tape measure, measure the length of each sprout (minimum of three required) and record the lengths. e) Measure the length of each sprout and record your measurements for each of the next six days. This can be a bit of a challenge since sprouts tend to grow in various directions, but do the best you can to determine the entire length of each sprout. You can try to gently straighten them while you measure but do it gently to avoid breaking them. f) Construct a graph of your data for the sprouts that you measured using the horizontal axis for time (day 1, day 2, day 3, etc.) and the vertical axis for the lengths of the sprouts (you will have at least three data points for each day). g) Draw an approximate “line of best fit” through the “middle” of your data points by using a ruler or straightedge to draw a line that goes through the “middle” of the data points.

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Math Is Everywhere! Explore and Discover It! 9.3 Linear Models: Analytic and Predictive

h) Locate two points on this line, one near the left end and one near the right end, and determine approximate values for their x- and y-coordinates. i) Use these two data points to determine the equation of your “line of best fit.” j) Use the Guide to Correlation Coefficients on pp.453–454 in Appendix C to estimate the correlation coefficient for your “line of best fit”. k) Submit a brief report that includes: 1. a table of your data values for each sprout, i.e., their lengths for each day 2. the equation of your “line of best fit” (i.e., the regression line) 3. an estimate of the correlation coefficient ********************

Alternative method for growing sprouts       

Soak the beans in water overnight (8–12 hours) or until they begin to sprout. In the morning, drain the water out, rinse the sprouts to freshen them, and drain them again. Fold a paper towel until it will fit into a small plastic sandwich bag. Dampen the paper towel so that it is quite wet, i.e., almost dripping wet. Place the folded paper towel inside the sandwich bag and lay the sandwich bag on a flat surface. Place a small number of presoaked beans (approximately a dozen or so) on top of the paper towel inside the sandwich bag. Seal the sandwich bag and place it on a flat surface for the duration of the project. It helps to put them in a warm place or on a window sill where they can receive direct sunlight.

This growing method traps the moisture from the paper towel inside the sandwich bag and effectively creates a miniature greenhouse. The beans or seeds will absorb the moisture from the towel and will begin to sprout and grow without any further intervention. You may measure the sprout lengths while the sprouts are still inside the plastic bag. This method tends to make the beans inedible due to their tendency to begin to rot while in a closed, wet environment but it does make them easier to grow for the purposes of this project. ******************** Have fun! 

********************

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Math Is Everywhere! Explore and Discover It! 10.0 Introduction and Objectives

Chapter Ten Exponential Growth: Drama and Suspense 10.0 Introduction and Objectives The thumbnail sketches on this page represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

10.1 Doubling Power: An Amazing Phenomenon

The Generous Philanthropist

10.2 The Exponential Function: A Deeper Look Population Growth: A Fact of Life y = P(t) = P0 · (1  r) t

10.3 Exponential Models and Applications Chain letters: Illegal for good reasons!

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Math Is Everywhere! Explore and Discover It! 10.0 Introduction and Objectives

Learning Objectives for Chapter 10 1. To enable students to develop their analytical reasoning skills through pattern recognition and problem solving. 2. To enable students to develop their inductive and deductive reasoning skills by applying mathematical relationships to construct exponential models and predict future outcomes. 3. To allow students to investigate occurrences of mathematical relationships in natural phenomena; in particular, students will investigate population growth in various milieu. 4. To allow students to use Web-based mathematical resources and their critical thinking skills in mathematical investigations. 5. To encourage students to reflect on the mathematical relationships that they have established for population growth in terms of the practical ramifications for their later lives. 6. To demonstrate the usefulness of mathematics through exponential modeling. 7. To engage students in active learning through the use of projects and activities. 8. To empower students to become lifelong learners.

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Math Is Everywhere! Explore and Discover It! 10.1 Doubling Power: An Amazing Phenomenon

10.1 Doubling Power: An Amazing Phenomenon “Alice laughed. ‘There's no use trying,’ she said. ‘One can't believe impossible things.’ ‘I daresay you haven't had much practice,’ said the Queen. ‘When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast.’” —Lewis Carroll (Alice’s Adventures in Wonderland)

Introduction This section is designed to introduce you to the nature of exponential growth—another very prevalent type of relationship that we find in the world around us. The exponential relationship is very different from the linear relationships that we explored in the previous chapter and you may be quite surprised at its behavior. To illustrate the nature of this relationship, we’ll begin with a story about a generous philanthropist.

The Generous Philanthropist Once upon a time, there was a generous philanthropist who wished to share some of his wealth with his friends. Being mathematically minded, he decided to make them a very unusual offer: in the coming month, either he would give them: 

one million dollars on the first day of the month

or he would give them:     

one penny on the first day of the month two pennies on the second day of the month four pennies on the third day of the month eight pennies on the fourth day of the month and so on until the end of the month

In other words, he would double the number of pennies that he gave them each day for a month. ******************** Analytic Moment: Which choice would you make if you wanted to receive the most money? ********************

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Math Is Everywhere! Explore and Discover It! 10.1 Doubling Power: An Amazing Phenomenon

To determine whether you made the wiser choice, we’ll investigate the growth of the pennies. The following table shows the amounts that you would receive for the first week of the month if you chose the pennies: day 1 2 3 4 5 6 7 Total received:

amount received \$0.01 \$0.02 \$0.04 \$0.08 \$0.16 \$0.32 \$0.64 \$1.27

This is a rather meager sum of money for the first week, especially considering that there are only four weeks in a month. Let’s continue on to the second week. The amounts that you would receive are as follows: day 8 9 10 11 12 13 14 Total received:

amount received \$1.28 \$2.56 \$5.12 \$10.24 \$20.48 \$40.96 \$81.92 \$162.56

This is once again a rather meager sum compared to the alternative choice (the one that you didn’t take) of \$1,000,000.00. Half of the month has gone by and you have received a grand total of only \$163.83; don’t you wish now that you had chosen the million dollars?  In the third week, we can only hope that matters improve! The amounts that you would receive are shown below: day 15 16 17 18

amount received \$163.84 \$327.68 \$655.36 \$1,310.72

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Math Is Everywhere! Explore and Discover It! 10.1 Doubling Power: An Amazing Phenomenon

\$2,621.44 \$5,242.88 \$10,485.76 \$20,807.68

Well, that’s a bit of improvement, but after three weeks this gives you a grand total of \$20,971.51. Compared to the \$1,000,000.00 that you might have chosen, this is still a paltry sum; in fact, with the end of the month rapidly approaching, it appears that you made the wrong choice. Don’t you wish now that you had chosen the million dollars?  With gloomy thoughts and glum countenances, we’ll continue on to examine the results in the fourth week of the month: day 22 23 24 25 26 27 28 Total received:

amount received \$20,971.52 \$41,943.04 \$83,886.08 \$167,772.16 \$335,544.32 \$671,088.64 \$1,342,177.28 \$2,663,383.04

Much to everyone’s surprise, the amounts that you would have received during this last week have skyrocketed dramatically and, on the 28th day of the month alone, the amount is over one million dollars!! Not only that, but the total for the week is over two and one-half million dollars!!  While it hardly matters at this point, you would also have received the \$20,971.51 from the previous three weeks for a grand total of \$2,684,354.55, all by beginning with one penny twentyeight days earlier. Astounding, isn’t it?? Incidentally, it’s possible that the month in which this takes place has 31 days; in this case, you would have three more days of income: day

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Math Is Everywhere! Explore and Discover It! 10.1 Doubling Power: An Amazing Phenomenon

Combined with your previous earnings of \$2,684,354.55, this would give you a grand total of \$21,474,836.47—over twenty-one times as much money as the alternative choice of one million dollars!!

******************** This is an example of the amazing and dramatic nature of exponential growth—it is deceptively slow for quite some time and then suddenly skyrockets upward at a shocking pace.

******************** To summarize the results:    

In the first week, there was hardly any growth in the amounts that you received. In the second week, there was only slightly more growth. In the third week, we began to see some signs of improvement but the amounts were still relatively small. In all, it has taken 21 days to accumulate just \$20,971.51. In the fourth week, however, the power of exponential growth was clearly evident; the daily amounts increased from \$20,971.52 to \$1,342,177.28 in just seven days and the accumulated total increased from \$20,971.51 to \$2,684,354.55, more than a hundredfold increase!

Those Prolific E.coli Bacteria A similar situation occurs with the growth of E.coli bacteria. If you have studied biology, you probably learned about cell growth and division. The process of mitosis follows the same doubling pattern that we observed with the philanthropist’s pennies: an E.coli cell will, given the proper medium and nutrients, eventually grow to the point where it will split and divide into two cells. Each of these cells does the same, thus producing a total of four cells a while later. Each of these four cells also splits and divides, and so on.

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Math Is Everywhere! Explore and Discover It! 10.1 Doubling Power: An Amazing Phenomenon

This growth pattern may be described as follows:

generation (independent variable) 1 2 3 4 5

number of E.coli cells (dependent variable) 1 2 4 8 16

As you can see, this is the same exponential growth pattern as that of the pennies. Knowing what happened with the pennies as time went on, we know also that there will be millions and millions of E.coli bacteria in just two dozen or so generations. It would be quite helpful (especially to biologists) if there were a mathematical equation that could describe and quantify this growth so that accurate predictions of the number of cells could be made. Our next objective is to develop such an equation.

An Exponential Model To assist us in our development, let’s put our analytic reasoning skills to work. While there doesn’t appear to be a direct connection between the values of the independent and dependent variables in the table, it is clear that the dependent values double for each successive generation; i.e., each number of cells is a multiple (or power) of 2. It may be helpful, then to add another column to the growth table above that displays the numbers of cells as powers of 2 and a final column that simply indicates the powers of 2 themselves:

generation number

number of E.coli cells

1 2 3 4 5 n

1 2 4 8 16

number of cells as powers of 2 20 21 22 23 24 ?

power of 2

0 1 2 3 4 ?

To establish an equation that describes the number of cells for a given generation, we need to find the nth term of the sequence shown in the table; in particular, we need to determine the values in the third and fourth columns for generation n.

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Math Is Everywhere! Explore and Discover It! 10.1 Doubling Power: An Amazing Phenomenon

******************** Analytic Moment: Using your analytic reasoning skills, can you determine the values in these columns for generation n? ********************

So what is the relationship between the generation number (independent variable) and the number of cells (dependent variable)? If you’ll notice, the power of two associated with a generation is one less than the generation number. If the generation number is a variable such as n, then the power of two associated with it should be one less, i.e., the generation number should be n–1. This is the analytic observation that leads to the mathematical relationship: the number of cells in generation n = 2 n 1

We could formalize this somewhat by writing: Cells in generation n = 2 n 1 C(n) = 2 n 1 This last format is reminiscent of our function notation, f(x), and, indeed, the above relationship is an exponential function. It derives its name exponential from the fact that the independent variable (generation number) appears as an exponent in the relationship. Now that we have constructed this relationship, a biologist may use it to predict how many bacteria will be present in any given generation (assuming unlimited nutrients and space are available); again, this predictive capability is quite valuable. On a related note, the exponential relationship involving the philanthropist’s pennies can be similarly written as: Pennies on day n = 2 n 1 or P(n) = 2 n 1

Conclusion Exponential growth is quite different from the linear growth that we studied in the previous chapter and the story about the generous philanthropist was designed to illustrate the difference in a dramatic way. There is nothing “steady” about exponential growth! We examined the growth of E.coli bacteria in order to facilitate the development of a mathematical relationship to describe this growth. We will further develop this relationship and investigate its applications in the following sections.

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Math Is Everywhere! Explore and Discover It! 10.1 Doubling Power: An Amazing Phenomenon

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Math Is Everywhere! Explore and Discover It! 10.2 The Exponential Function: A Deeper Look

10.2 The Exponential Function: A Deeper Look “Men occasionally stumble on the truth, but most of them pick themselves up and hurry off as if nothing had happened.” —Sir Winston Churchill

Introduction We will now investigate exponential functions—another basic type of relationship model for the world around us. In the process of doing so, we will develop a general equation by which to describe exponential growth. As we indicated in the previous section, exponential relationships are quite dissimilar from the linear relationships that we investigated in the previous chapter— there is nothing “steady” about exponential growth; nonetheless, these relationships are quite prevalent and of much interest and value to scientists, social scientists, and others.

The Exponential Function: A Deeper Look As part of our investigation of E.coli bacteria in the previous section, we constructed the following exponential model (equation) for their growth: Cells in generation n = C(n) = 2 n 1 or y = C(n) = 2 n 1 This equation is an exponential function (i.e., each value of the independent variable, n, produces exactly one corresponding value of the dependent variable, y); it derives its name from the fact that the independent variable is the exponent in the relationship. Just as linear functions have two very important elements, the slope and the y-intercept, exponential functions also have two very important elements: the growth factor and the initial population (or amount).

Populations: The Main Object of Interest in Exponential Functions Since many exponential growth relationships involve populations of living entities as they change over time, mathematicians have conventionally used t (for time) as the independent variable rather than n. In addition, they have used P (for population) as the variable to represent the number of entities in a population rather than C (for cells). Employing these conventions in our exponential function for E.coli growth, we may rewrite the equation as follows: y = P(t) = 2 t 1

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Math Is Everywhere! Explore and Discover It! 10.2 The Exponential Function: A Deeper Look

The Growth Factor As you may recall, the bacteria double in number in each successive generation; i.e., the number of bacteria in each generation is multiplied by 2 in order to obtain the number of bacteria in the succeeding generation. This multiplication by 2 is an inherent part of the growth relationship and we say that the relationship has a growth factor or growth multiplier of 2. If the bacteria were to triple in number, we would say that the relationship had a growth factor of 3, and so on. In essence, then, the growth factor is the number by which each successive population or amount is multiplied in order to obtain the new population or amount. ******************** Analytic Moment: Use your generalization ability to generalize the growth factor of 2 to a growth factor of any value and rewrite the exponential equation to include this generalized factor. ******************** We may generalize the concept of growth factor by replacing the specific value of 2 with a variable that represents the growth factor in general: y = C(n) = ( growth factor ) n 1 or y = P(t) = ( growth factor ) t 1 Traditionally, mathematicians have used the variable a to represent the growth factor, so we may also write: y = P(t)= a t 1 We will investigate this growth factor in greater depth later in this section, but for now we’ll consider the next important exponential feature.

The Initial Population or Amount The other important feature of an exponential growth relationship is the initial population or, in the case of inanimate objects like money, the initial amount. In the case of the E.coli bacteria, the initial amount was one bacterium. There is no obvious evidence of this amount in the E.coli growth equation, but it is there as an understood value of 1:

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Math Is Everywhere! Explore and Discover It! 10.2 The Exponential Function: A Deeper Look y = C(n) = 1 bacterium · 2 n 1 or y = P(t) = 1 bacterium · 2 t 1 y = P(t) = 1 · 2 t 1 y = P(t) = 2 t 1 If the initial number (population) of bacteria had been 8, the equation would have looked like this: y = P(t) = 8 · 2 t 1 ******************** Analytic Moment: Use your generalization ability to generalize the initial value of 1 bacterium so that it represents an initial value of any amount and rewrite the exponential equation to include this generalized value. ******************** We may generalize this to accommodate any initial value by using a variable to represent the initial population: y = P(t) = (initial population) · 2 t 1 or, using a more concise notation for the initial population, y = P(t) = Pinitial · 2 t 1 Note that when t=1, the exponent of the growth factor is 0 so that 2 0  1 ; this means that when t=1, P(1) = Pinitial · 211 P(1) = Pinitial · 2 0 P(1) = Pinitial · 1 = Pinitial Essentially, then, the initial population occurs when t=1.

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Math Is Everywhere! Explore and Discover It! 10.2 The Exponential Function: A Deeper Look

Letting the Initial Population Occur When t=0 In practice, mathematicians have traditionally chosen to have the initial population occur when t=0 rather than when t=1. This makes things a bit simpler in mathematical terms, and it is logical to begin measuring the growth of a population when the “time clock” is set to 0 rather than 1. This requires a minor adjustment to the exponential equation: y = P(t) = Pinitial · 2 t 1 ******************** Analytic Moment: Do you see what minor adjustment is needed in the above equation? ******************** If we want the initial population to occur when t=0, then it is necessary that P(0), i.e., the population when t=0, is equal to the initial population, Pinitial . As it stands at the moment, substituting 0 for t in the above equation produces the following result: P(0) = Pinitial · 2 01  Pinitial  2 1  Pinitial 

1 2

1 The exponent of –1 causes the multiplier to change to , thus reducing the initial population by 2 one-half. Clearly this doesn’t produce the desired result of P(0) = Pinitial . To rectify matters, we will need to modify the expression for the exponent of 2 so that the exponent equals 0 (rather than –1) when t=0. The simplest way to do this is to modify our equation so that the exponent is equal to t itself as follows: P(t) = Pinitial · 2 t

Now, when t=0, P(0) = Pinitial · 2 0  Pinitial  1  Pinitial

And we have accomplished our objective. 

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Math Is Everywhere! Explore and Discover It! 10.2 The Exponential Function: A Deeper Look

Putting It All Together: The Exponential Function Having made generalizations for both the growth factor and the initial population and having made the necessary adjustment in the growth factor exponent, we may now combine these into one generalized exponential equation as follows: y = P(t) = Pinitial · a t

In addition, mathematicians have traditionally used the variable P0 to represent the initial population, Pinitial . The subscript “0” is included as a way of indicating that P0 represents the population at time=0, i.e., at the very beginning of the population growth, before any growth has actually taken place. We may now rewrite the previous equation as: y = P(t) = P0 · a t

This last equation is the traditional exponential function for population growth. If you are dealing with inanimate objects (such as money), mathematicians often use the letter A (for amount) rather than P (for population); hence the exponential function for inanimate growth is often written as follows: y = A(t) = A0 · a t

The Growth Factor and the Growth Rate: An Important Connection Now that we have developed a formula (equation, function, model) for exponential growth, we are almost ready to apply it to situations in the world around us. Before doing so, however, we need to establish an important connection between the growth factor as described above and the growth rate, the rate of change in population or in amount with respect to time. Note: This growth rate is usually represented by the variable r.

In news and reports on issues involving growth, the most commonly reported measurement is the growth rate. This rate of growth is usually reported as a percent, e.g., the population of Anywhere, USA, is growing at a rate of 10% per year. This means that the population of 10 new people , in one year as indicated in Anywhere, USA, will increase by 10%, or every100 current people the table below. We’ll use an initial population of 100 people for the sake of simplicity: year Jan. 1, 2003 Jan. 1, 2004

population 100 110

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Math Is Everywhere! Explore and Discover It! 10.2 The Exponential Function: A Deeper Look

To calculate the population for 2004, we can use either a two-step process or a more efficient one-step process. The two-step process is as follows: Step 1: Multiply the population for 2003 by the growth rate of r = 10%, or 0.10, to obtain the amount of actual growth during 2003: 100 people · 0.10 = 10 people Step 2: Add this amount of growth to the original 2003 population: 100 people + 10 people = 110 people.

A more general description of Step 2 is: original population + 10% of the original population = final population (one year later)

or, including decimal multipliers, 1.0 · original population + 0.10 · original population = final population

We may now factor the common term, original population, from the left-hand side to obtain the more efficient one-step process for calculating the final population: (1.0 + 0.10) · original population = final population or 1.10 · original population = final population

These last two equations provide a method by which we may calculate the final population in one step by using a single multiplier: 1.10. They also reveal the important connection between the commonly reported growth rate and the growth factor in our general exponential equation: the growth factor = 1.0 + the growth rate

In other words, using our conventional variables for the growth factor, a, and the growth rate, r, we may write: a=1+r This relationship between growth factor and growth rate allows us to obtain a deeper understanding of the exponential function and we now have two equally valuable ways in which to express this function (equation, model): y = P(t) = P0 · a t y = P(t) = P0 · (1  r) t

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Math Is Everywhere! Explore and Discover It! 10.2 The Exponential Function: A Deeper Look Note: If a population is growing smaller rather than larger, we consider the growth rate to be a negative value; this, in turn, causes the growth factor to be less than 1. This situation is often referred to as exponential decay.

If a population is declining at an annual rate of 4%, then we say that the growth rate is –4%, or –0.04. Since the growth factor is equal to 1 plus the growth rate, the growth factor is: 1 + (–0.04) = 1 – 0.04 = 0.96 Multiplying the initial population successively by 0.96 will result in successively smaller population values in each of the ensuing years.

A Final Look at the Exponential Function: Its Graph Now that we have developed an algebraic description of the exponential function, it would be quite informative to explore this type of function in a visual manner. A graph of an exponential function has a very predictable shape but one that is very different from the straight line of a linear function. As an example, we’ll explore the graph of the population growth of Anywhere, USA, where the growth rate is 10% per year and the initial population is one person. The exponential model for this scenario is: y = P(t) = P0 · (1  r) t y = P(t) = 1 · (1  0.10) t y = P(t) = 1 · 1.10 t = 1.10 t

For the purposes of this exploration, we’ll let the value for time vary from 0 years to 40 years, i.e., 0 ≤ t ≤ 40. Note: The initial population value of one person is rather unrealistic, but its simplicity will allow us to more easily gain some clarity and insight.

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Math Is Everywhere! Explore and Discover It! 10.2 The Exponential Function: A Deeper Look

As you can see from the above graph, the population exhibits little change for the first twenty years or so; this is reminiscent of the wealthy philanthropist’s penny growth. However, in the last ten years or so, the population rockets upward. ********** Exponential graphs are characterized by very slow, almost imperceptible growth followed eventually by a sudden and dramatic change in the values of the dependent variable (population or amount).

********** As the values of the independent variable, t, increase, the slope of an exponential graph continues to increase as well. This increasing rate of change is what gives exponential functions their power—the values of the dependent variable continually grow faster as time goes on. We have completed our investigation of exponential functions for the moment and we can now begin to apply this knowledge as we explore various relationships in the world around us. As you will see in the next section, exponential growth is quite prevalent.

Conclusion We began this section by investigating the two features of importance in an exponential relationship: the growth factor and the initial population (or amount). After generalizing both of these features to accommodate various values, and after adjusting the equation to allow population to begin when t=0, we constructed the general exponential function: y = P(t) = P0 · a t

We continued on to make a connection between the growth factor, a, and the growth rate, r, so that we could elaborate on our initial function: y = P(t) = P0 · (1  r ) t

This final equation is the complete description of exponential growth. Lastly, we took a brief look at a visual representation of an exponential relationship and noted that the growth was slow at first but eventually reached dramatic proportions as time went on. We’ll examine this further in the next section.

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Math Is Everywhere! Explore and Discover It! 10.2 The Exponential Function: A Deeper Look

Exercises and Projects for Fun and Profit Exercises in Analysis Identify the growth factor and the growth rate in each of the following scenarios: Example (#2): growth rate is 5% = 0.05; growth factor is 1+ 0.05 = 1.05

1. The town of Lakeside, Arizona, is growing at an annual rate of 7%. 2. The city of Crackerville, Florida, is growing at an annual rate of 5%. 3. The village of Fort Dallas, Texas, is losing population at an annual rate of 3%. 4. The town of Wrigley, North Dakota, is losing population at an annual rate of 4%.

Construct an exponential equation for the population in each of the following sets of conditions: Example (#6): P(t) = P0  (1  r ) t ; P(t) = 5000 ·1.07 t Example (#10): 1 + r = 1 + (–0.03) = 0.97; P(t) = 12000 · 0.97 t

5. The city of Orange, Alaska, has a population of 800 and is growing at an annual rate of 2%. 6. The town of Gates, Iowa, has a population of 5,000 and is growing at an annual rate of 7%. 7. The village of Tuck, Ohio, has a population of 18,000 and is growing at an annual rate of 9%. 8. The town of Flint, Texas, has a population of 9,000 and is growing at an annual rate of 4%. 9. The city of Grimes, Utah, has a population of 8,000 and is declining at an annual rate of 5%. 10. The city of Tarn, Iowa, has a population of 12,000 and is declining at an annual rate of 3%. ********************

Creative Project: World Population Growth

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Math Is Everywhere! Explore and Discover It! 10.2 The Exponential Function: A Deeper Look

WORLD POPULATION GROWTH Population growth is most usually an exponential phenomenon. The world population in 2006 was slightly more than six billion and the population is gradually increasing at a rate of slightly more than 1% per year. If this growth rate continues and holds steady for the next sixty years, the world population will approximately double in size. If this trend were to continue indefinitely into the future, the world would at some point be overrun with people and drastic consequences would result. In this project, we will investigate the growth of the world population as well as the population growth of some individual countries around the world. Assignments:

For each assignment, complete the table of values beginning in the year 2005 and calculate the population in each indicated year thereafter assuming that the population doubles as indicated. Sketch a graph of these values to see a dramatic picture of the future population. Note: Do NOT use an exponential function model for these calculations; simply double the population values for each indicated period.

This table of values is based on the assumption that the current annual growth rates will remain stable and that the populations will double as indicated. In all likelihood, the growth rates will decline somewhat for various reasons and the respective populations won’t continue to grow at the current rates; however, the scenarios are still informative. You may use Microsoft Excel to record your population values if you wish and then create an Excel Chart based on those values; or you may construct a graph of your values using pencil and paper in the usual fashion. Assignment #1 Assignment #2 Assignment #3 Assignment #4 World Mexico Pakistan Bangladesh year doubling time: doubling time: year doubling time: doubling time: 60 years 60 years 34 years 34 years 6.4 billion 106.2 million 144.3 million 2005 2005 162.4 million 2065 2039 2125 2073 2185 2107 2245 2141 2305 2175 Source of data (2006): http://www.cia.gov/cia/publications/factbook/index.html

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Math Is Everywhere! Explore and Discover It! 10.3 Exponential Models and Applications

10.3 Exponential Models and Applications “In symbols, one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.” —Gottfried Wilhelm Leibniz

Introduction Now that we have gained an understanding of the exponential relationship, we can begin to construct mathematical models of numerous real-world occurrences. These models will give us the ability to predict the values of populations or amounts in the future and will allow us to reflect constructively on our findings.

Exponential Growth—It’s Everywhere! (well, almost) As noted in the previous section, human population growth is an exponential phenomenon. Using America as an example, the basic process is this: provided that each pair of parents has more than two children as a national average, the number of people in the second generation of these Americans will grow slightly larger. As these children grow and eventually become parents, and if each pair of them continues to have children at the same rate as their parents, then there will be even more children in the third generation than there were in the second. As these third-generation children grow and eventually become parents, and if each pair of them continues to have children at the same rate as their parents, then there will be even more children in the fourth generation than there were in the third. And so on. The key here is that as each generation increases in number, the increase in the number of their children will continue to grow as well. This is an example of multiplication by a growth factor and is a classic example of exponential growth. Other examples include:        

Population growth (or decline) in animal, bird and fish species Spread of communicable diseases (e.g., AIDS, SARS, H5N1, H1N1, etc.) Increase of frequencies of notes in the twelve-tone musical scale (from Chapter 6) Transmission of chain letters Growth in the number of personal computer owners Growth of cell phone usage Growth of spam on the Internet Compound interest (a topic that we shall investigate in detail in Chapter 11)

We will now investigate one of these examples; later, you’ll have opportunity to explore others in the Exercises and Projects for Fun and Profit at the end of this section.

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Math Is Everywhere! Explore and Discover It! 10.3 Exponential Models and Applications

Why Chain Letters are Illegal on the Internet You are most likely familiar with the concept of a chain letter. A person initiates a letter and sends it to a fixed number of people, say five others, with the instructions that each of them is to send it to five additional people and include the same instructions. As you can well imagine, if everyone obeys the instructions, then it won’t be long before there will be thousands and millions of letters being transmitted. Let’s construct an exponential model of this phenomenon and use it to determine just what will take place. The initial amount is one letter; the growth factor is 5, i.e., each “generation” of letters will consist of five times as many letters as the previous generation. This allows us to write the following model (we’ll use n as a variable to represent the number of “generations” and we’ll agree that the 0th generation corresponds to the initial letter in the chain so that the initial amount occurs when n=0): A(n) = A0 · a n number of letters sent after n generations = A(n) = 1 · 5 n = 5 n

If we assume that each recipient of this chain letter would, in turn, send the letter within twentyfour hours, then we can consider a generation to be equivalent to one day and we can replace the variable n with the conventional variable t (for time): number of letters sent after t days = 1 · 5 t = 5 t

The following table of values shows just the very beginnings of this process: generation (n)

day (t)

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

number of letters sent ( 5 n or 5 t ) 1 5 25 125 625 3,125 15,625 78,125 390,625 1,953,125

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Math Is Everywhere! Explore and Discover It! 10.3 Exponential Models and Applications

As you can readily observe, on the tenth day there are almost two million letters being sent and received (theoretically). We can use our exponential model to predict how many letters will be sent on the 15th day, the 20th day or the 30th day (one month after beginning) with ease: Since A(t) = 5 t , A(15) = 515 = 30,517,578,125 Just five days later, there are over 30 billion letters! A(20) = 5 20  95,367,431,640,600

Just five more days later, there are over 95 trillion letters!! This represents approximately 15,894 letters per person for every man, woman and child in the world!!! And now you can understand why sending chain letters via the Internet could have disastrous consequences--the Internet could very rapidly be clogged with appearances of the chain letter. I won’t bother with calculating the number of letters on the 30th day—the total is hopelessly astronomical. ******************** The theoretical chain letter vividly illustrates the dramatic power of exponential growth—it is absolutely astounding!

******************** So when you next receive an email that requests that you send it on to five (or more) other recipients, please politely decline. You can then rest peacefully knowing that you have helped to spare the world from an Internet disaster. 

********************

Exercises and Projects for Fun and Profit Creative Projects: U.S. Population Growth The Fastest Growing State in the U.S.? Population Growth in Florida

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Math Is Everywhere! Explore and Discover It! 10.3 Exponential Models and Applications

U.S. POPULATION GROWTH: WHAT DOES THE FUTURE HOLD? According to U.S. Census Bureau figures (http://www.census.gov/main/www/cen2000.html), the population of the United States grew at a rate of approximately 13.15% during the decade from 1990 to 2000, or at an approximate annual growth rate of 1.243% (0.01243 as a decimal); this corresponds to an annual growth factor of 1.01243. The population in 1990 was 248,709,843 and the population in 2000 was 281,421,906.

Assignments:

Based on the figures above, construct an exponential population growth model, P(t) = P0 · a t , where P0 represents the U.S. population in 2000, a represents the annual growth factor (not the annual growth rate), and t represents the number of elapsed years since 2000. Please answer the following question for each assignment: If this growth rate continues unchanged, what will the population of the U.S. be in the indicated years: # Assignment #1 Assignment #2 Assignment #3 Assignment #4 1. 2050 2040 2025 2033 2. 2100 2080 2050 2066 3. 2150 2120 2075 2099

4. (Applies to all four of the above assignments.) Write a brief report (minimum of one-half page) on how the U.S. population growth might affect your future in practical terms. Note: Be sure to first raise the growth factor to the appropriate power of t and then multiply this result by the initial population afterwards. Note: To raise a number to a power on most scientific calculators (including the one that is accessible via your computer's Start/Programs/Accessories/Calculator/View/Scientific menu), first enter the base number, then press the button labeled y x (sometimes labeled x y ), then enter the exponent value, and press the “=” key. For example, to raise 5 to the 3rd power (i.e., 5 cubed, or 53), enter the base value of 5, press the yx button (or the x y button), enter the exponent value of 3, and press the “=” key.

If you have a graphing calculator, replace the y x above with the “^” key.

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Math Is Everywhere! Explore and Discover It! 10.3 Exponential Models and Applications

THE FASTEST GROWING STATE IN THE U.S.? YOU MAY BE SURPRISED Visit http://www.census.gov/population/www/cen2000/briefs/phc-t2/index.html and determine which state grew at the fastest rate in the 1990s; please use Table 3 for States Ranked by Percent Population Change. See Helps and Hints for Fastest Growing State on p.445 in Appendix A to determine the state’s population in 2000. Afterwards, construct an exponential population growth model, P(t) = P0 · a t , where P0 represents the state’s population in 2000, a represents the annual growth factor of 1.05216, and t represents the number of elapsed years since 2000. Please answer the following question for each assignment: If this growth rate continues unchanged, what will the population of this state be in the indicated years: # Assignment #1 Assignment #2 Assignment #3 Assignment #4 1. 2020 2030 2015 2025 2. 2040 2050 2035 2045 3. 2060 2070 2055 2065

********************

POPULATION GROWTH IN FLORIDA: DRAMATIC TO SAY THE LEAST Collier County is located in southwest Florida and includes the city of Naples. According to U.S. Census Bureau figures (http://www.census.gov/main/www/cen2000.html), the population of Collier County grew dramatically during the decade from 1990 to 2000, from 152,099 in 1990 to 251,377 in 2000. Based on the population in 2000, construct an exponential population growth model, P(t) = P0 · a t , where P0 represents the Collier county population in 2000, a represents the annual growth factor of 1.05152, and t represents the number of elapsed years since 2000. Please answer the following question for each assignment: If this growth rate continues unchanged, what will the population of Collier County be in the indicated years: # Assignment #1 Assignment #2 Assignment #3 Assignment #4 1. 2020 2030 2015 2025 2. 2040 2050 2035 2045 3. 2060 2070 2055 2065

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Math Is Everywhere! Explore and Discover It! 11.0 Introduction and Objectives

Chapter Eleven Financial Planning: Mathematical Secrets to Acquiring Wealth 11.0 Introduction and Objectives The thumbnail sketches on this page represent selected topics in this chapter that you will encounter on your adventure—an adventure that can last a lifetime!

11.1 Compound Interest: Strategy for Wealth

11.2 Savings Plans: Don’t Leave Home Without One!

11.3 Stocks, Bonds and Mutual Funds Investments for the Future

11.4 Home Mortgages: Mathematical Revelations How to Save Many Thousands of Dollars on your Loan

11.5 Credit Card Economics

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Math Is Everywhere! Explore and Discover It! 11.0 Introduction and Objectives

Learning Objectives for Chapter 11 1. To enable students to develop their analytical reasoning skills through pattern recognition and problem solving. 2. To enable students to develop their inductive and deductive reasoning skills through analyzing loans and mortgages. 3. To enable students to calculate and compare simple and compound interest. 4. To enable students to determine an investment strategy for their futures. 5. To allow students to investigate and analyze elements of a financial plan, including investments and annuities. 6. To enable students to analyze the terms of credit card account agreements and to recognize the truth about minimum monthly payments. 7. To allow students to investigate the phenomenon of compound interest and how it relates to successful financial planning for their future retirement. 8. To allow students to investigate the effect of extra principal payments on a home mortgage in order to ascertain the potential financial savings. 9. To encourage the use of a spreadsheet application in order to analyze mathematical data. 10. To engage students in active learning through the use of projects and activities. 11. To empower students to become lifelong learners.

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

11.1 Compound Interest: Strategy for Wealth Introduction Most people will retire from the workforce at some point in the future, either by choice or due to necessity, and will no longer receive their usual professional or occupational income. At that point in life, some other sources of income will be needed to maintain a comfortable standard of living. In essence, then, each of us needs a financial plan for later life. You will need to supplement your federal Social Security retirement income in one or more ways, and investing is the key to a comfortable and perhaps even wealthy retirement. In this section, we will investigate the mathematical secret behind acquiring long-term wealth: compound interest. This natural phenomenon is quite deceptive at first glance, but a deeper look will reveal its depth and power.

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

Simple Interest: Quite Simple Indeed There is a well-known relationship between the interest earned on an investment and the amount of the investment, otherwise known as the principal amount. This relationship is given below: earned interest = amount of investment · rate of interest · length of time of investment or earned interest = principal amount · rate of interest · length of time of investment

By using only the significant words in the above relationship, we may obtain the following wellknown formula for simple interest: interest = principal · rate · time This may be further abbreviated: I = P · R · T or I = PRT In this relationship, the earned interest (dependent variable) depends on the principal, the interest rate and time for its value. While interest may be calculated for various amounts of time, e.g., a week, a month, a quarter of a year (3 months), half a year (6 months) or one year, we will limit our focus to investment times of one year for the sake of simplicity. Principal that is invested in an interest-bearing account for one year produces what is known as annual interest: one year after the initial investment of principal, the account manager (usually the bank’s computer system) calculates the interest earned according to the formula given above. In an account that bears only simple interest, the earned interest is placed in its own account, one that is separate from but related to the account that holds the principal. If the principal remains in the account for a second year of investment with simple interest, then at the end of the second year the account manager again calculates the interest earned according to the formula given above using the original amount of principal from the principal account; the annual interest is hence the same as it was during the first year of investment. This second year’s worth of interest is then added to the separate interest account and the process continues on for as many years as the principal remains invested.

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

As an example, if we invested \$1000 in an account bearing simple interest at a rate of 3% and allowed our principal to remain invested for four years, we would have the following progression:

year principal 1 2 3 4

\$1000 \$1000 \$1000 \$1000

annual interest earned \$30 \$30 \$30 \$30

total interest earned \$30 \$60 \$90 \$120

total value of account at end of year \$1030 \$1060 \$1090 \$1120

Note that the principal amount remains unchanged throughout the duration of the investment.

******************** This is why simple interest is simple—the earned interest is not added to the principal to increase the value of the principal each year but is “simply” treated as interest only. ******************** It would be more advantageous to you as an investor (i.e., lender) if the bank were to add the annual interest to your principal amount each year, thus gradually increasing the amount of your loan to the bank and hence the amount of interest that you would earn each year. This practice was instituted by banks long ago in an effort to attract investors and build their businesses; it became known as compound interest and we shall now investigate its nature and effects.

Compound Interest As noted above, the concept of compound interest involves adding the earned interest to the principal amount at periodic intervals. Banks offer various time intervals for compounding interest, e.g., daily, weekly, monthly, quarterly, semiannually and annually, but for the sake of simplicity we will limit our attention to interest that is compounded annually. At the end of each year in an account that bears interest that is compounded annually, the account manager calculates the earned interest by using the simple interest formula; however, instead of keeping the interest in its own separate account, the manager adds this interest to the current principal, thus increasing the principal that is being invested.

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

As an example, if we invested \$1000 in an account bearing compound interest at a rate of 3% compounded annually and allowed our principal to remain invested for four years, we would have the following progression:

year principal

1 2 3 4

\$1000.00 \$1030.00 \$1060.90 \$1092.73

annual interest earned (compounded annually) \$30.00 \$30.90 \$31.83 \$32.78

total interest earned

total value of account at end of year

\$30.00 \$60.90 \$92.73 \$125.51

\$1030.00 \$1060.90 \$1092.73 \$1125.51

As you can see, the total value of the account at the end of a year, including both principal AND interest combined, is rolled over to the next year as the principal amount. Incidentally, this is reminiscent of “compound subjects” in English grammar: the principal AND the interest.

******************** This is why compound (annual) interest is called “ compound”—the earned interest is added to the principal to increase the value of the principal each year: the previous principal AND the earned annual interest = the new principal ******************** As the principal continues to grow, so do the successive interest amounts which then cause the principal to grow a little more, which then generates larger interest amounts, which then cause the principal to grow even more, which then generates even larger interest amounts, which then cause the … You get the idea!  Looking at the above tables of account values, however, you might wonder whether compound interest will have much of an effect on your investment. After four years, the difference between simple interest earned and compound interest earned is only \$5.51, a gain of a little more than a dollar per year. Based on this observation, your wondering is well founded; at this point I can only say: Trust me—compound interest is both powerful and dramatic and is the mathematical secret to a wealthy retirement. While the increase of \$5.51 may seem relatively small, compound interest is a very powerful process when given enough time. The growth in principal, i.e., the amount in your account, is

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

an example of exponential growth, a topic that we considered in a previous chapter. As such, it has all of the same dramatic and explosive characteristics of exponential growth provided that you leave your investment intact and allow it to continue to grow for a relatively long period of time.

The Exponential Compound Interest Model Our goal here is to determine the effect of compound interest on long-term investments. In order to do so, we’ll apply the mathematical skills that you learned in the previous chapter and we’ll construct a mathematical model of the exponential growth involved with compound interest. In order to see the effects of compound interest in a visual way, you may then construct graphs of this model using various initial values for principal, interest rate and time; you’ll have opportunities to carry this out in the Exercises and Projects for Fun and Profit at the end of this section. To begin our model-building process, we’ll consider the formula for simple interest: I = PRT. To simplify matters, we’ll assume that the interest on an account is compounded annually, i.e., once per year, so that the value for time in the interest formula is T=1. The simple interest formula now reads: I = P·R·1 or simply I = PR

A Sales Tax Diversion: The One-step Method To help us develop the formula for compound interest, we will revisit the Florida sales tax model that we examined in a previous section on Linear Functions: sales tax on item = 0.07 · price of item To calculate the total cost of an item, the sales tax is added to the price of the item: total cost = price of item + sales tax on item Determining the total cost of an item, then, is a two-step process: 1) calculate the sales tax on the item 2) add the sales tax to the price of the item Since we are most often interested in the total cost of an item rather than the sales tax itself, it would be convenient to develop a simpler formula for the total cost. To do so, we may substitute the expression for sales tax into the total cost formula: total cost = price of item + 0.07 · price of item 379 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

This may be simplified further by recognizing that the first instance of the variable price of item has an implicit and understood multiplier of 1.0: price of item = 1.0 · price of item Consequently, we may write: total cost = 1.0 · price of item + 0.07 · price of item By factoring the price of item variable on the right-hand side, we can then write: total cost = (1.0 + 0.07) · price of item total cost = 1.07 · price of item

******************** This final equation calculates the total cost of the item, including the sales tax, in ONE STEP rather than the two steps that we used previously. ********************

Applying the One-step Method to Compound Interest When investing money in an interest-bearing account, we are generally most interested in the total amount of money that is currently in our account rather than in the amount of interest that we may have earned; this is very similar to being interested in the total cost of an item rather than in its sales tax. Refocusing our attention in this manner, we may write the following relationship concerning an investment made for one year: amount of money in account after 1 year = original principal + interest earned during the year We’ll abbreviate this with the use of symbols: amt in account after 1 year = A(1) = P + I where:  

P represents the original principal and I represents the annual interest earned on the principal

Using the formula for simple interest for one year, I = PR, and using our substitution method once again, we can replace the Interest, I, with its equivalent expression, PR: 380 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

amt in account after 1 year = A(1) = P + PR

Factoring out the common element P yields: amt in account after 1 year = A(1) = P · (1 + R) or A(1) = P · (1 + R)

******************** This formula provides a one-step method for determining the total amount of money in an account after an investment for one year with interest that is compounded annually. ******************** Note: This is very similar to the one-step method for determining the total cost of an item: total cost = 1.07 · price of item or total cost = price of item · 1.07

Incidentally, in this equation, the term 1.07 could be replaced with a more general (1 + R), i.e., 1.0 plus the sales tax rate, to accommodate any value for sales tax rate: total cost = price of item · (1 + R)

In conclusion, the one-step method for determining the total amount of money in an account after an investment for one year with interest that is compounded annually is a very effective mathematical procedure. We will now apply this to long-term investments involving compound interest and develop a formula for the amounts in these types of investment accounts.

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

Developing a Mathematical Model for Compound Interest First Year: As noted above, the amount in an investment account after one year with interest that is compounded annually is: amt in account after 1 year = A(1) = P · (1 + R) By the definition of compound interest, the entire amount in the account after one year, A(1), is considered as the principal amount for the second year of the investment; consequently, during the second year of the investment, the principal is: principal during the second year = A(1) = P · (1 + R) Second Year: To determine the amount in the account after two years, we follow the same one-step procedure that we used at the end of the first year: amt in account after 2 years = A(2) = P during the second year · (1 + R) or A(2) = P during the second year · (1 + R)

Substituting A(1)= P · (1 + R) for P during the second year, we may write: A(2) = (P · (1+R)) · (1 + R) As before, the amount in the account after two years is considered as the principal amount for the third year of the investment: principal during the third year = A(2) = (P · (1+R)) · (1+R) Third Year: To determine the amount in the account after three years, we follow the same one-step procedure that we used at the end of both the first and second years: amt in account after 3 years = A(3) = P during the third year · (1 + R) or A(3) = P during the third year · (1 + R)

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

Substituting A(2)= (P · (1+R)) · (1+R) for P during the third year, we may write: A(3) = ((P · (1+R)) · (1+R)) · (1 + R)

To review our developing sequence of these amounts, they are as follows: A(1) = P (1+R) A(2) = (P· (1+R)) · (1+R) A(3) = ((P· (1+R)) · (1+R)) · (1+R) Since (1+R) · (1+R) = (1  R) 2 and so on, we can simplify the above sequence and, by doing so, help ourselves recognize the mathematical pattern here: A(1)  P  (1  R) A(2)  P  (1  R) 2 A(3)  P  (1  R ) 3 We can also include an exponent for the expression for A(1) so that all of the equations will have the same basic exponent structure: A(1)  P  (1  R)1 A(2)  P  (1  R) 2 A(3)  P  (1  R ) 3

******************** Analytic Moment: Here is another opportunity to use your inductive reasoning skills and determine the next value in this developing sequence, the value for A(4). You may then extend this reasoning to determine values for A(5), A(6), etc. ********************

Ultimately, we want to construct an equation (model) for the amount in an account after an arbitrary number of years, i.e., after n years.

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

******************** Analytic Moment: Here is another opportunity to use your inductive reasoning skills and exercise your ability to generalize by determining the value for A(n), the nth term in the sequence. ********************

Hopefully, this presented no difficulty and you arrived at the conclusion: amount in an account after n years = A(n)  P  (1  R) n This simple formula has the same basic structure as the exponential models that we investigated in a previous chapter: amount after t years = initial amount · a t or

A(t )  A0  (1  R) t And, in fact, the relationship for compound interest is an exponential model. This means that compound interest has the same powerful and dramatic characteristics as all other exponential models: given enough time, a small investment will eventually experience dramatic growth. It is this long-term growth that will allow you to retire as a wealthy person if you invest early in life.

Additional Models for Compound Interest The model developed above involves interest that is compounded annually; however, there are also models for interest compounded for other time periods. In the following examples, A0 represents the initial investment amount, R represents the annual interest rate, and t represents the number of years for which the initial amount is invested. Compounded quarterly: A(t )  A0  (1  Compounded monthly: A(t )  A0  (1 

R 4t ) 4

R 12t ) 12

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

Compounded daily: A(t )  A0  (1 

R 365t ) 365

Compounded n times per year: A(t )  A0  (1 

R nt ) n

For a given initial investment, interest rate and length of time, the amount in the account becomes successively larger as the frequency of compounding increases, i.e., the more often, the better. So what if a bank compounded your interest more often than once a day? What if a bank compounded your interest 500 times per year, or 1000 times per year, or even 10,000 times per year? The results are most interesting, and I’ll leave that matter for you to investigate!  Ultimately, increasing the frequency without limit results in the following model for compound interest in which the interest is compounded continuously: Compounded continuously: A(t )  A0  e Rt , where e is Euler’s constant (  2.718)

This model is sometimes written as: A(t )  P0  e Rt or, more simply, A  P e Rt

Conclusion After presenting the basic concept of investing money in an interest-bearing account and our role as lender rather than borrower, we examined the concept of simple interest. We then went on to examine the related concept of compound interest. While the development of the compound interest model was a bit involved and tedious, the result is well worth our efforts. We now have a remarkably simple mathematical formula (model) with which we may predict the outcomes of various initial conditions. Mathematical models are highly desirable for this predictive ability as you shall see in the Investigative Project that is to follow; in fact, models allow us to generate answers to various “What if?” questions that are of great practical interest.

********************

Exercises and Projects for Fun and Profit Creative Project: The Long-term Results of Compound Interest

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

THE LONG-TERM RESULTS OF COMPOUND INTEREST Introduction

In order to see the effects of long-term investments involving compound interest, we’ll now investigate some possible scenarios. The three factors that will affect the outcomes are:   

P: principal amount—the original investment amount R: the interest rate, or rate of return, on your investment T: the number of years that the investment is maintained

Of these three, the interest rate is the most sensitive, i.e., small changes in this rate will have enormous impact on the long-term outcome. Equally important is the length of time of the investment. As you may recall from the section on Exponential Models, exponential growth is relatively slow for quite some time; however, after a considerable amount of time a critical point occurs in the growth process and a rapid and dramatic change takes place. It is this critical point that we wish to achieve or exceed in our financial investments.

The Secret to Financial Wealth and Independence: Invest early! A Very Feasible Scenario: When you graduate from college and begin your career (hopefully before you begin to raise a family), determine that you will save \$2000 each year for five years. This will produce a total of \$10,000 that you may then invest in an index fund or in some other investment vehicle with a good rate of return. You may then, if you wish, invest nothing more—just sit back and wait while your investment earns its annual compound interest; of course, if you are willing and able, you may continue to add to your investment amounts in the ensuing years for even greater savings and wealth. However, for the purposes of this project, we will assume that you will invest only the original \$10,000 at age 25 and that you will leave the investment intact at least until retirement or beyond, i.e., until you reach the current retirement age of 65 or more.

Is \$10,000 a reasonable amount to set as an investment goal after five years? This \$2000 per year is equivalent to \$166.67 per month or \$38.46 per week. Assuming that you are hired at a beginning salary of \$25,000 per year (or hopefully even more), this represents only 8% of your total income, an amount that you may feasibly set aside for savings and investments. After all, it is quite easy to spend \$40 per week on items that are not necessities of life, so foregoing some of these items is all that is required.

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

Compound Interest Calculations

In the following assignment, use the compound interest formula below and either a calculator or a spreadsheet application such as Microsoft Excel to generate outcomes, i.e., amounts in your account, for the indicated values of principal (P), interest rate (R) and time (T). The amount in your account after n years is given by: A(n)  P  (1  R) n or, if we use the letter T to represent the number of years: A(T )  P  (1  R) T Note: In each of the first three exercises, two of the variables are held constant while the remaining variable (either P, R or T) is allowed to vary.

Assignment: Please answer the five questions below: Please submit a table of values for each of the first three exercises that indicates the amount in your account for each of the variable values. You may construct these tables with:

  

Microsoft Word by inserting a table Microsoft Excel by creating a worksheet paper and pencil

1. The Effect of Time: A(T )  10000  (1.10) T For P=\$10,000 and R=10%=0.10, calculate ten different amounts in your account, one for each of the ten time values shown below. Your table should contain one column for the time values and one column for the corresponding amounts in your account: T = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 years

2. The Effect of Interest Rate: A(40)  10000  (1  R) 40 For P = \$10,000 and T = 40 years, calculate six different amounts in your account, one for each of the six interest rate values shown below. Your table should contain one column for the interest rate values and one column for the corresponding amounts in your account: R = 6%, 8%, 10%, 12%, 14%, 16%

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Math Is Everywhere! Explore and Discover It! 11.1 Compound Interest: Strategy for Wealth

3. The Effect of Principal: A(40)  P  (1.10) 40 For R = 10%=0.10 and T = 40 years, calculate three different amounts in your account, one for each of the three principal values shown below. Your table should contain one column for the principal values and one column for the corresponding amounts in your account: P = \$5,000, \$10,000, \$20,000 Analysis and Reflection

5. The Penalty for Procrastination: If you invested \$10,000 at 10% interest as in Question #1 above but if you didn’t begin investing until age 45 (after perhaps raising a family and earning a much higher salary), then: a) What would the amount in your investment account be at age 65 (i.e., 20 years later)? You may refer to your answer for Question #1 to determine this amount. b) Since this time period of 20 years is exactly half of 40 years, you might expect that your account would have approximately half as much value as \$10,000 invested at 10% interest for a period of 40 years. However, this is not the case; please explain why it isn’t so (you may refer to your graph in Question #4 above).

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Math Is Everywhere! Explore and Discover It! 11.2 Savings Plans: Don’t Leave Home Without One!

11.2 Savings Plans: Don’t Leave Home Without One! Introduction The process of putting aside financial savings is a very important issue for all of us, both individually and collectively as a nation. These savings are essential for immediate needs and medical emergencies such as hospitalization but they are equally as essential and even more important with regard to our future needs after retirement when previous employment income has ceased.

Social Security Our federal Social Security system is designed to ensure that each wage earner sets aside a percentage of his or her earnings so that these saved funds (and their accumulated interest) will be available for living expenses after retirement. For most taxpayers, the Internal Revenue Service currently retains 6.2% of each person’s salary; employers are required to contribute an equal and additional amount for each employee. In some years, Congress has voted to reduce the Social Security tax rate; in 2011, it was reduced to 4.2% for employees while employers still paid 6.2%. (Medicare tax is also collected from each wage earner; the current rate is 1.45%.) All of these funds are deposited with the federal government where the Social Security Administration (SSA) oversees the investing of these funds and the maintaining of records for each individual. The SSA periodically sends reports to every U.S. wage earner that details how much money has been contributed to the system for that individual and gives a projection of monies that will be available upon retirement. The SSA maintains an extensive website at http://www.ssa.gov/ that contains a wealth of information including financial calculators to help you determine your retirement benefits based on your employment history and projected future earnings before retirement. They also provide a rough-estimate calculator for college students who are at least 21 years of age; this calculator estimates your retirement benefits based only on your current age and current annual salary and uses statistical data to predict the outcomes (http://www.ssa.gov/planners/calculators.htm). There are a number of resources for planning your retirement as well. Upon retirement, a person’s Social Security contributions are then doled out in monthly amounts in order to help the retired person pay for his or her living expenses. Unfortunately, these monthly amounts for most people will not be sufficient to allow them to continue living the same lifestyle to which they were accustomed; consequently, other sources of income must be planned well ahead of time if they want to live comfortably. The recent congressional debates concerning the future financial viability of the Social Security system underscore the need to plan for other sources of retirement income.

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Math Is Everywhere! Explore and Discover It! 11.2 Savings Plans: Don’t Leave Home Without One!

Financial Planning The anticipated shortfalls in the Social Security system mean that each of us must develop a long-term financial planning strategy. Spending all of our income each year without putting any aside will surely result in an impoverished state upon retirement. Given that we choose to save some money each week, month and year, it won’t do to simply stash the money under our mattresses, either; money under the mattress draws no monetary interest and is the least effective (and somewhat dangerous!) means of saving your money. Savings accounts at banks, government bonds, certificates of deposit and the stock market are types of investments with potential for generating more income and increasing one’s personal wealth. The first three of these are very stable and secure investments but provide only modest returns; the stock market, while less stable in the short term, has a proven track record. Over the last eighty years, the stock market has averaged over 10% annual return on investments. While I don’t recommend that everyone should buy and sell stocks as an investment practice, I highly recommend that you learn about the market and eventually, perhaps, invest some of your money in it so that you can maximize your returns and build your financial savings. In the Exercises and Projects for Fun and Profit at the end of this section, you will have the opportunity to learn about some basic concepts concerning financial investments. In the next section, you will gain more in-depth knowledge about stocks, bonds and mutual funds.

Retirement Plans In addition, there are some long-term savings plans directly related to retirement that are regulated to some extent by the federal government. These retirement plans come in various forms and allow you to put some of your money in tax-deferred savings accounts under certain restrictions, i.e., you pay no taxes on your earned and invested money until you begin to withdraw it. Some examples are: Individual Retirement Accounts (IRA’s), Roth IRA’s and 401(k) plans. Some employers make contributions to their employee retirement plans as part of the benefit package that comes with the employee’s position; employees are then allowed to invest extra monies in their plan up to specified limits. Annuities are another type of investment vehicle that guarantees a form of income during retirement years; however, this type of investment has to be carefully considered and is not suitable for everyone. If you choose to purchase an annuity from an insurance company, your investment accrues interest and is tax-deferred until you begin receiving payments from the insurer after age 59½. The insurer then pays you a fixed amount each month until your death. The amount in your annuity after making regular payments into the fund over a period of time is determined by the following formula in which P represents the regular payment amount, R is the annual interest rate, n is the number of payment periods per year as well as the number of times per year that interest is compounded, and t is the number of years:

390 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.2 Savings Plans: Don’t Leave Home Without One!   1   A(t )  P      

nt  R  1  n  R   n 

Conclusion Financial planning is a most important yet often neglected aspect of life. In the Creative Project at the end of this section, you will have the opportunity to explore some Web-based financial resources and learn more about the financial planning topics mentioned earlier in this section. My hope is that you will realize the importance of beginning your savings plan early in life and that you will successfully carry it through to a happy and financially secure retirement in later life!  ********************

Exercises and Projects for Fun and Profit Creative Project: Intro to Savings and Investment ********************

INTRO TO SAVINGS AND INVESTMENT This project is designed to provide you with an overview of savings and investments along with information on common retirement plans. In the next section, we will engage in a long-term investment project (with play money, of course,  ) so that you can experience what it’s like to be an investor and track your investment’s financial progress. ******************** Assignment: In order to obtain some general knowledge about investments and annuities, use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on:  investing in stocks  annuities Afterwards, please write a brief report (minimum of one-half page) on what you found to be the most important aspects of investing for a beginner.

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Math Is Everywhere! Explore and Discover It! 11.3 Stocks, Bonds and Mutual Funds

11.3 Stocks, Bonds and Mutual Funds Introduction Financial investments are necessary to help ensure a comfortable future and retirement. While there are many types of potentially profitable investments such as purchasing high-quality art, collecting antiques and buying real estate, the most common types of long-term investment vehicles are stocks, bonds and mutual funds. In this section, we will explore these vehicles through the use of some Web-based resources, learn of an amazing story involving a stock market investment and undertake a simulated investment in the stock market. In the process, you will learn a lot about the realm of investing—a topic that will play an important role in your future.

Risk and Diversification Many financial planners highly recommend that investors diversify their investments, i.e., purchase different types of investments and different types of stocks as opposed to purchasing only one type. This is reminiscent of the old adage, “Don’t put all of your eggs in one basket.” As you will probably agree, this seems like wise advice. A collection of investments is commonly referred to as an investment portfolio. How you build your portfolio has a lot to do with how much risk you are willing to take. Higher risk generally corresponds to potentially higher returns while lower risk generally corresponds to lower returns. Each investor needs to determine the level of risk that he or she is willing and able to undertake with regard to their investments. This determination depends on how much money the investor has available and on how much must be set aside for immediate needs. Money that is not needed in the foreseeable future may be more readily invested in higher risk investments such as the stock market; these stocks may then be held in anticipation of the historical long-term gains of the stock market even if stocks suffer losses in value in the short term. Balancing risk by diversifying your investments is a sound practice and involves purchasing some low-risk investments such as bonds as well as some higher-risk investments such as stocks. A related practice is the purchase of some low-risk stocks (commonly referred to as blue chip stocks) as well as some high-risk stocks (the start-up companies whose futures are uncertain). Again, the percentages of funds allotted to each of these categories will depend on the amount of risk that the investor is willing to take.

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Math Is Everywhere! Explore and Discover It! 11.3 Stocks, Bonds and Mutual Funds

Conclusion In the Exercises and Projects for Fun and Profit at the end of this section, you will have the opportunity to explore these concepts in more depth and will also take part in a stock investment simulation in which you will track the short-term gains and losses of your investment. The stock market, in particular, is a most interesting and engaging investment vehicle! I trust that you will enjoy these projects and hope that you will gain a sense of some of the issues involved in making good investments and of the value and importance of financial planning early in life. ******************** Best wishes for a successful financial future!

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Exercises and Projects for Fun and Profit Creative Projects: Intro to Stocks, Bonds and Mutual Funds Investment Strategies A True and Amazing Story Stock Market Investment Report

393 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.3 Stocks, Bonds and Mutual Funds

INTRO TO STOCKS, BONDS AND MUTUAL FUNDS The stock market can be a large and scary place for those who are unfamiliar with its ins and outs (but familiar with its ups and downs). However, over the course of the last century, the stock market has proven to be the most profitable investment vehicle and has outperformed bonds, certificates of deposit, real estate investments and others. Consequently, it would behoove us as potential investors to learn about the market and how it works. This project is designed to provide you with some general knowledge about the investment vehicles known as stocks, bonds and mutual funds and to provide a knowledge base for the Creative Project entitled Stock Market Investment Report that follows in this section. There is a wealth of information available and I trust that it will be useful both now and in your future. ******************** Assignment: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on stocks, bonds and mutual funds. Afterwards, please write a short report (minimum of one page) that answers the following questions: 1. 2. 3. 4. 5.

Which investment vehicle (stocks, bonds or mutual funds) involves the least risk? Does purchasing a share of a company’s stock give you part ownership in that company? Describe the nature of a balanced fund. Describe the nature of an index fund. Define the par value for a bond.

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Math Is Everywhere! Explore and Discover It! 11.3 Stocks, Bonds and Mutual Funds

INVESTMENT STRATEGIES Financial planning is best done when you are young, and the sooner the better. Parents who begin savings plans for their children will help them realize the maximum returns on their investments in later years. All college students should begin formulating a savings plan for their futures and begin to implement it as soon as they begin an occupation or profession that provides them with a regular income. In this project, you will gain some basic knowledge about investment vehicles and the keys to creating a successful investment portfolio. This foundation, together with the other knowledge learned in this chapter, will enable you to build a secure financial future for you and your family. ******************** Assignment: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on:   

investing in stocks investing in bonds investing in mutual funds

Afterwards, please write a short report (minimum of one page) that answers the following questions: 1. What did you find to be the most important aspects of investing for beginners? 2. Describe your investment strategy for the purpose of generating some retirement savings, i.e., how you would invest \$10,000, what types of investments (stocks, bonds or mutual funds) you would purchase, and in what amounts. 3. Describe your self-assessed risk level in doing so and how this relates to your overall investment strategy. ********************

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Math Is Everywhere! Explore and Discover It! 11.3 Stocks, Bonds and Mutual Funds

Math Is Everywhere! Explore and Discover It! 11.3 Stocks, Bonds and Mutual Funds

STOCK MARKET INVESTMENT REPORT Investing your money, whether in a savings account, a government bond, or a company’s stock, is an important part of financial planning. In this project, we are going to simulate an investment in the stock market and record the progress of the investment over the course of the second half of the semester (approximately eight weeks in Fall and Spring; five weeks in Summer). In the Business & Finance section of your local newspaper you will find a daily listing of the companies included in the New York Stock Exchange (NYSE) and a number of values associated with their stocks. You can also find this information on the Internet; a number of financial sites provide an option to “Get Quotes”. Use the search engine found at www.google.com to conduct some Internet research on stock quotes. The most important items in a stock quote are: name of the company (abbreviated, e.g., WMT stands for WalMart) opening value of the stock (per share) for the day (Open) closing value (per share) for the day (Close) Note: Some sites list only the Open value, others list only the Close value, while some refer to the current value of the stock as Value. The most important value for this project is the opening (Open) or closing (Close) value for the day; this is sometimes referred to as Value. You may choose either of these values to track and record for the duration of this project. The opening value (Open) represents the price of one share of a company’s stock at the opening of the stock market on Wall Street in New York City on a given day at 9:30 a.m.; the closing value (Close) corresponds to the value at 4:30 p.m. when the stock market closes. The opening or closing value for the day is the value that you will be recording once a week and tracking for the duration of the project. If the value of your stock at some time after you begin this project is higher than it was when you started, then your stock has increased in value and your investment is worth more than when you started.

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Math Is Everywhere! Explore and Discover It! 11.3 Stocks, Bonds and Mutual Funds

Assignment: You will have \$1000 of “pretend” money to invest (or as reasonably close to \$1000 as possible) in this project. The objectives are: 1. 2. 3. 4.

choose three stocks in which to invest, either at random or by personal choice “purchase” an appropriate number of shares of each stock track their progress weekly (once a week on approximately the same day each week) report the results and give a brief analysis Note: You may use the search engine at google.com and enter the search expression get quotes for stocks to find stock share prices.

Your final report should include: 1. 2. 3. 4.

the names of your selected stocks the initial value of each stock (Open or Close value when you began the project) the number of shares purchased for each stock a table of the opening (Open) or closing (Close) values for each stock on a weekly basis (record them once a week on approximately the same day each week) 5. a line graph for each of the stocks (based on the table of values above) showing a visual picture of its growth 6. a final evaluation of the gains or losses of each of your stocks including possible reasons for the gains or losses based on items and events in the news, both national and international. As an example, the recent upswing in the outsourcing of certain manufacturing jobs may cause a rise in some stock values and a decline in others. Note: To calculate the gain or loss for a particular stock, find the difference in value between the initial cost of a share of the stock (i.e., the Open value in the first week of the project) and the final cost of a share of the stock (i.e., the Open value in the last week of the project) and then multiply this difference by the number of shares that you purchased. 7. a conclusion stating how much total net gain or loss you experienced for all of your stocks combined

******************** Happy investing! Since this is just pretend money, you don’t have to worry about whether you might lose money on your investments—just enjoy the process and learn something about the stock market at the same time.  ********************

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Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

11.4 Home Mortgages: Mathematical Revelations Introduction In this section, we will examine the details involved in a home mortgage—the largest loan you may ever need. We will provide the basic facts on how the bank evaluates your application for a mortgage loan and then investigate the mathematics behind mortgage payments. In the process, you will learn the mathematical secret to saving many thousands of dollars on your mortgage payments.

Mortgage Factors While renting a house or apartment is convenient and sometimes necessary and/or advisable, owning your own home is desirable because it is actually an investment from which you may gain monetary value. Rent payments, in contrast, are a total loss and produce no financial return—only outgoing expense. The difficulty most home buyers face is the lack of cash to pay for the entire cost of the home; consequently, they must borrow much of this needed money from a bank or other financial institution whose business it is to make such loans. These types of loans are known as mortgages and are often the single largest loan that you will ever need in your life. There are a number of factors involved in a lender’s decision concerning whether to grant you a loan (mortgage); these factors include:           

your gross annual (and monthly) income your current loans of significant proportions (e.g., automobile loans, student loans and, in general, any loan that requires a fixed monthly payment) your credit card debt your credit history (also known as your credit record) the cost of the home the amount of down payment and the resulting amount of the loan (mortgage) the length of the loan (the most common term is for 30 years at a fixed rate) the lender’s annual interest rate on your loan (this is also known as the annual percentage rate, or APR) the amount of real estate taxes on the property (property taxes) the cost of homeowner’s insurance the monthly loan repayment amount (including principal and interest)

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Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

Basically, the lending institution wants to ascertain whether, in their educated and experienced opinion, you are likely to be able to make your mortgage payments in a regular and timely manner. If your financial status is less than what the lender feels is desirable, or if the lender feels that you may be at risk of not being able to meet your prospective mortgage obligations, the lender will most likely deny your application for a loan. We’ll now take a closer look at what is involved in this decision.

Mortgage Mechanics In order to illustrate the process by which a lender makes a decision on your loan application, we’ll create a hypothetical scenario with the following facts:             

your gross annual income: \$40,000 your gross monthly income: \$3,333 (\$40,000/12) your current monthly loan payments: \$200 car payment, \$200 student loan payment your credit card debt: negligible your credit history: excellent (i.e., you always pay your bills on time) cost of the home that you would like to purchase: \$88,000 amount of down payment: \$8,000 mortgage amount: \$80,000 (i.e., the actual amount that you are borrowing) type of mortgage: a fixed rate for 30 years the lender’s annual mortgage percentage rate (APR): 8% property taxes on the home: \$1,200 per year, or \$100 per month cost of homeowner’s insurance: \$800 per year, or \$67 per month monthly loan payments (principal and interest): \$587.01 (This amount is determined by looking it up in a mortgage amortization schedule; these are available on the Internet, from bookstores and from lenders themselves. We’ll investigate this in more detail later in this section.)

Step 1: Subtract your current monthly loan payments from your gross monthly income to obtain your monthly available income:

\$3,333 \$400 \$2,933

Step 2: Calculate 25% of this amount to obtain an estimate of the amount that the lender feels that you will be able to put toward your mortgage payments: \$2,933 · 0.25 = \$733.25

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Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

Note: Some lenders use slightly higher percentages in their estimate but generally these values are between 25% and 30%.

Mortgage Options There are two basic features of a mortgage that have options: the type of interest rate offered and the length of the mortgage. Before purchasing a home, you need to consider which of these options best suits your needs and circumstances. Many financial institutions offer a choice between fixed rate and adjustable rate mortgages. A fixed rate mortgage will maintain the same interest rate throughout the life of the loan; an adjustable rate mortgage (ARM) has an interest rate that may vary within certain limits. ARMs are usually based on the prime lending rate which is established by the Federal Reserve Board and are usually a specified number of percentage points above the prime rate. If the prime rate

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Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

should rise during the life of your mortgage, your monthly payments also will rise; on the other hand, if the prime rate falls, your monthly payments also will drop. Note: When the prime rate is at a relatively or historically low value, an ARM is quite risky even though it offers an initially lower interest rate (and lower monthly payment) than a fixed rate mortgage. In all likelihood, the prime rate will increase at some point in the not-too-distant future and the associated monthly payments will also rise. The other common mortgage option involves the length of the loan. The common terms are for either 15 years or 30 years; the 30-year mortgage is much more prevalent. As you can readily infer, the monthly payment on a 15-year mortgage is significantly higher than that on a 30-year mortgage; however, the amount of interest paid over the life of the loan is considerably less since the loan is repaid twice as fast as the 30-year mortgage.

The U.S. Banking Rule on Loan Repayments Many years ago, the U.S. Government established a banking rule concerning the repayment of loans. Basically, the rule states that when a loan payment is made, the payment is first applied to the interest on the loan; afterwards, whatever amount of the payment is left will then be applied to the principal. While this seems like a reasonable way of doing business, the effects are quite dramatic when the loan amounts are large (as they are with mortgages).

Amortization: A Closer Look at Your Loan Repayment Amortization is the term used to describe the process by which you pay off your mortgage. Banks and other financial organizations publish amortization schedules that list how much principal and interest you will pay on a monthly basis depending on your loan amount, the interest rate and the length of the loan. We’ll now take a closer look at your loan repayment details and, in particular, examine your first few payments. Interest on a mortgage is calculated like any other interest: I=PRT In the case of your hypothetical mortgage, the principal is \$80,000, the interest rate is 8% and the length of time is one month, i.e., one-twelfth of a year: I = \$80,000 · 0.08 ·

1 12

I = \$533.33

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Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

The portion of your monthly payment that represented the sum of the principal and the interest due was: \$587.01 The U.S. Banking Rule requires that \$533.33 of your first monthly payment be allocated to pay the current interest; the remainder is then applied to reduce the amount of the principal (i.e., the amount of the loan principal that is still unpaid): \$587.01 principal and interest due – \$533.33 interest paid \$ 53.68 remainder

This means that a whopping sum of \$53.68 will be applied toward reducing the principal for the next month: \$80,000.00 loan amount (principal owed) – \$53.68 principal paid \$79,946.32 principal owed

And you will owe almost as much as you did in the first month! The interest in the second month is: 1 I = \$79,946.32 · 0.08 · 12 I = \$532.98 This is only 35 cents less than the interest in the first month! And it means that there will again be relatively little of the payment left to put toward reducing the principal: \$587.01 principal and interest due – \$532.98 interest paid \$ 54.03 remainder

This reduces the principal as shown below: \$79,946.32 principal owed – \$54.03 principal paid \$79,892.29 principal owed

As you can already see, this is going to be a long, slow process!!  For approximately the first twenty years of your mortgage, most of your monthly payment is going to be applied to paying off the interest on the loan.

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Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

In the final analysis, you will make monthly payments for 30 years on your principal and interest, i.e., 360 monthly payments, at \$587.01 each: \$587.01 x 360 \$211,323.60 total payment This is a grand total of \$211,323.60 that you will have paid to the lending institution by the time your mortgage is paid off. In view of the fact that your original loan amount was only \$80,000, this represents an enormous amount of interest on the loan: \$211,323.60 – \$80,000.00 \$131,323.60 interest In other words, you will have paid more than one and one-half times the value of the home in interest payments! Shocking, isn’t it? However, all is not lost—there is a mathematical and lender-approved solution to this predicament. Read on!

The Secret to Saving Many Thousands of Dollars on Your Mortgage We will now examine the amortization schedule for the first two years of your mortgage: First year: monthly payment

interest paid

1 2 3 4 5 6 7 8 9 10 11 12

533.33 532.98 532.62 532.25 531.89 531.52 531.15 530.78 530.40 530.03 529.65 529.26

principal cumulative cumulative principal paid principal interest balance paid paid remaining 53.68 53.68 533.33 79946.32 54.03 107.71 1066.31 79892.29 54.39 162.10 1598.93 79837.90 54.76 216.86 2131.18 79783.14 55.12 271.98 2663.07 79728.02 55.49 327.47 3194.59 79672.53 55.86 383.33 3725.74 79616.67 56.23 439.56 4256.52 79560.44 56.61 496.17 4786.92 79503.83 56.98 553.15 5316.95 79446.85 57.36 610.51 5846.60 79389.49 57.75 6375.86 668.26 79331.74

404 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

Second year: monthly payment

interest paid

13 14 15 16 17 18 19 20 21 22 23 24

528.88 528.49 528.10 527.71 527.31 526.91 526.51 526.11 525.70 525.30 524.88 524.47

principal cumulative cumulative principal paid principal interest balance paid paid remaining 58.13 726.39 6904.74 79273.61 58.52 784.91 7433.23 79215.09 58.91 843.82 7961.33 79156.18 59.30 903.12 8489.04 79096.88 59.70 962.82 9016.35 79037.18 60.10 1022.92 9543.26 78977.08 60.50 1083.42 10069.77 78916.58 60.90 1144.32 10595.88 78855.68 61.31 1205.63 11121.58 78794.37 61.71 1267.34 11646.88 78732.66 62.13 1329.47 12171.76 78670.53 62.54 12696.23 78607.99 1392.01

Please note the amounts for cumulative principal paid at the end of the first year, \$668.26, and at the end of the second year, \$1392.01. If we subtract these two values, we can determine the amount of principal that you repaid during the second year: \$1392.01 – \$668.26 = \$723.75

The point here is that each of these amounts of principal paid per year is less than the value of one monthly mortgage payment of \$754.01 (including taxes and insurance).

Mortgage Prepayments Most mortgage lenders today allow you to make extra payments in addition to your regular monthly payments at no financial penalty to you; in fact, you may (and should!) designate that all of any extra payment be applied to reduce the principal amount of your mortgage. ******************** The secret to saving many thousands of dollars on the cost of your mortgage lies in making one extra payment per year, either an amount equal to the value of your total monthly payment or a round value such as \$1000.

********************

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Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

Although it doesn’t at first seem that these extra payments would make much difference, you’ll be amazed at their effect. Let’s assume that you choose to put aside an extra \$1000 per year toward repaying your mortgage principal. As you saw above in the tables for the first two years of payments, \$1000 is more than the entire regularly paid principal per year. By designating that this \$1000 be applied to the principal, you are effectively reducing the length of your mortgage by more than one year! As soon as you make this extra payment, the amount of your principal is dramatically reduced and the amortization process jumps immediately to the month in the schedule that reflects the reduced principal amount. For example, if you were to make a \$1000 payment toward your principal along with your first monthly payment, the reduced principal balance would change from \$79,946.32 to \$78,946.32. This lower balance is equivalent to a balance that normally occurs between the end of the 18th month and the end of the 19th month. In essence, then you have advanced, i.e., shortened, your payment schedule by approximately a year and a half! If you continue to make this extra payment each year, you will shorten the length of your mortgage considerably. The dramatic shortening effect will diminish with time as higher percentages of the monthly payments are applied toward the principal, but nonetheless there is a definite reduction in required mortgage payments. ******************** Analytic Moment: If you make an extra principal payment of \$1000 each year, make a conjecture concerning the number of years that will be needed to pay off your 30-year mortgage?

******************** The actual time required to pay off your mortgage is reduced to approximately 20 years. But at this point, then, you no longer need to make any payments and you effectively eliminated the remaining 10 years of payments. Let’s illustrate this savings with some specific numbers: 10 years x 12 months per year 120 months 120 months x \$587.01 monthly payment \$70,441.20 loan savings

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Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

What was the cost of obtaining these savings? The combined total of your extra payments: \$1000 extra payment per year x 20 years \$20,000 total extra payments

We may subtract these payments from your total savings to determine your net savings: \$70,441.20 loan savings – \$20,000.00 extra payments \$50,441.20 net savings

Even after subtracting this “cost” from the resulting benefit, this still represents a savings of over \$50,000! This is an enormous savings!! This is greater than the amount that you would receive as your hypothetical salary of \$40,000 (before taxes) after working for an entire year!! ********** Once again, a concerted effort to make some relatively small and temporary financial sacrifices has resulted in monetary savings that far outweigh the sacrifices. This is a timeless lesson but, unfortunately, one that many people do not apply to their own financial situations. **********

Incidentally, some lending institutions have begun offering early mortgage prepayment plans whereby they will electronically transfer your monthly mortgage payment in biweekly installments from your bank account to theirs, i.e., they will withdraw half of your monthly payment every two weeks. The hidden advantage here is that they then make twenty-six payment transfers each year which represents thirteen monthly payments rather than the usual twelve. This process guarantees that you will make an extra mortgage payment each year; however, it is most important that you verify with the institution that the extra payment will be put entirely toward reducing the principal and not simply be treated as an ordinary payment that covers both interest and principal. The advantage to the lender is that they receive your payment monies sooner than they otherwise would have received them and they can immediately reinvest those monies to earn even more money. In essence, it's a win-win situation. Most lenders allow early prepayment of a mortgage without penalty but be sure to verify this with a lender before applying for a mortgage.

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Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

Exercises and Projects for Fun and Profit Exercises in Analysis Monthly Payments: Visit the search engine google.com and enter the search expression amortization calculator. Using one of the listed calculators, determine the combined monthly principal and interest payments for the following conditions:

question loan amount interest rate term of mortgage 1. \$70,000.00 7.0% 30 years 2. \$80,000.00 8.0% 30 years 3. \$100,000.00 8.0% 30 years 4. \$90,000.00 7.0% 30 years 5. \$110,000.00 7.0% 30 years 6. \$120,000.00 6.0% 30 years 7. \$95,000.00 7.5% 30 years 8. \$85,000.00 6.5% 30 years 9. \$75,000.00 8.5% 30 years 10. \$95,000.00 8.5% 30 years

********************

Creative Project: The All-Powerful Interest Rate

408 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

THE ALL-POWERFUL INTEREST RATE This is a three-part project designed to allow you to investigate the effect of interest rates on your home mortgage, to illustrate the process of prepayment on your principal and to help you know when it may be advisable to refinance a mortgage. Assignments:

For each assignment, the objectives are to perform the indicated calculations and write a report that includes: 1. your table of calculated values in Part 1 2. the number of years that were required to repay the mortgage and how many thousands of dollars were saved through early prepayment in Part 2 3. whether or not you should accept the bank's offer to refinance in Part 3 ********** Part 1: Interest Rates: Just How Much Effect Do They Have? It is interesting to observe how much effect small changes in the interest rate have on the total cost of a mortgage. Visit the search engine google.com and enter the search text amortization calculator.

For each assignment, use one of the listed calculators to calculate the monthly mortgage payment and the total cost of the loan (total amount repaid) for 30 years at the indicated fixed interest rate. Use the indicated interest rates for each assignment. Please see Helps and Hints for The All-Powerful Interest Rate on p.445–446 in Appendix A: Assignment Assignment Assignment Assignment #1 #2 #3 #4 mortgage mortgage mortgage mortgage amount amount amount amount \$110,000 \$100,000 \$90,000 \$80,000 \$110,000 \$100,000 \$90,000 \$80,000 \$110,000 \$100,000 \$90,000 \$80,000 \$110,000 \$100,000 \$90,000 \$80,000 \$110,000 \$100,000 \$90,000 \$80,000 \$110,000 \$100,000 \$90,000 \$80,000 \$110,000 \$100,000 \$90,000 \$80,000 \$110,000 \$100,000 \$90,000 \$80,000 \$110,000 \$100,000 \$90,000 \$80,000

interest rate

monthly mortgage payment

total cost of loan

6.00% 6.25% 6.50% 6.75% 7.00% 7.25% 7.50% 7.75% 8.00%

409 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

Part 2: Saving Many Thousands of Dollars on Your Mortgage For each assignment, use an amortization calculator and enter whichever of the indicated values below are pertinent (see Helps and Hints for The All-Powerful Interest Rate on p.446 in Appendix A). Then calculate the amortization schedule.

item Assignment #1 Assignment #2 Assignment #3 Assignment #4 \$110,000 \$100,000 \$90,000 \$80,000 Principal 7.0 7.0 7.0 7.0 Annual interest rate 12 12 12 12 Payments per year 30 30 30 30 # of years 360 360 360 360 # of payments

******************** For the purposes of demonstrating the process of early prepayment of your mortgage, we will use Assignment #1 as an example; please use this as a guide for the other assignments.

As you will note in Assignment #1, the calculator result for monthly payment for principal and interest is \$731.83. Assuming that the annual property taxes are \$1800 (\$150 per month) and that homeowner’s insurance costs \$1200 per year (\$100 per month), the total mortgage payment is \$981.83 per month. Assuming that you make an extra payment of \$1500 at the end of each year (for any given assignment) to reduce the principal amount of your mortgage, determine how many years will be required to pay off the mortgage. In order to accomplish this, you’ll need to employ the “leap-frog” method in which you subtract \$1500 from the principal balance at the end of each year and then locate the month farther down the schedule that shows an approximately equal principal balance. Example: The principal balance in Assignment #1 at the end of the first year (12 months) is \$108,882.64. After making an extra \$1500 payment toward the principal, the new principal balance is then \$107,382.64. Searching the last column for a principal balance that is approximately equal, you will find two values, one higher and one lower, in months 26 and 27:  

end of month 26: \$107,476.56 end of month 27: \$107,371.68

Please choose the balance that is closest to the target balance. In this case, you would choose the balance in month 27. In essence, your extra payment has effectively reduced your principal remaining balance and your next regular monthly payment begins at this

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Math Is Everywhere! Explore and Discover It! 11.4 Home Mortgages: Mathematical Revelations

point in the amortization schedule. In other words, the extra principal payment has accelerated the loan repayment by approximately 15 months. You would then make twelve regular monthly payments from month 27 onwards (in the amortization schedule), i.e., you would make regular payments in each of the following months: 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38 and 39. At the end of the following year, i.e., at the end of the second year of your mortgage (or at the end of month 39 in the amortization schedule), your remaining principal balance is \$106,064.32. You then make another extra payment of \$1500, and “leap-frog” down the schedule to find the approximate month in which the balance is \$1500 less than \$106,064.32, i.e., the month in which the principal balance is approximately \$104,564.32. This occurs at the end of month 52, where the principal balance is \$104,541.15. You’ll need to continue this “leap-frog” process until you reach a principal balance of \$0, i.e., until you pay off the mortgage. Then count and record the number of years that were required to accomplish the repayment. Subtract the required number of years to repay the loan from the original loan term of 30 years, and then multiply the resulting number of years by 12 to obtain the number of monthly payments that are effectively saved. Finally, multiply the number of months saved by the regular monthly payment amount to determine how many thousands of dollars you would save by making extra annual payments. Astounding, isn’t it?  ******************** Part 3: Should You Refinance a Mortgage When Interest Rates Drop? As you can see in the results from Part 1, there is a significant difference in total mortgage costs for each interest rate. When interest rates drop, as they sometimes do, the rule of thumb is that it is advantageous to refinance your mortgage if you can reduce your mortgage rate by at least 1% and if you can recover the refinancing costs within two years from the date of refinance.

In other words, you will be hypothetically saving money each month due to lower monthly principal and interest payments; multiplying this savings by 24 months (2 years) will give you an idea of how much refinancing cost you can afford. If your bank or lending institution offers to refinance your mortgage for a fee that is less than this, you should consider refinancing. For this last investigation, you may use the values that you calculated for a given assignment in Part 1. Let’s assume that your original mortgage carried an interest rate of 7% and that interest rates went down in the years immediately afterwards. Let’s also assume that your bank has offered to refinance your mortgage at an interest rate of 6% for a fee of \$1500 a few years after you had initiated the mortgage. For each assignment mortgage amount from Part 1, determine if this is an offer that you should accept.

411 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.5 Credit Card Economics

11.5 Credit Card Economics Introduction In this day and age, credit cards are an obvious and definite convenience and in some respects have almost become a necessity. Their convenience, however, is counterbalanced by the debt that they allow people to accrue. Generally, the interest rates on credit cards are rather high and can create a financially burdensome situation for those who do not pay off their credit card balances on a monthly basis. In this section, we will examine the mathematics of credit card economics and investigate the realities of credit card debts. In the Exercises and Projects for Fun and Profit at the end of this section, we will point out the importance of avoiding the credit card “minimum payment” trap and of eliminating credit card debt as quickly as possible. We will also look at the mechanics of credit reports and credit ratings and inform you on how to maintain an excellent credit rating.

Credit Card Debt: Easy to Acquire—Harder to Eliminate It all seems so easy: just present your credit card to the cashier and the object of your desire is yours for the taking. Whether it is a new stereo, some new clothes or dinner at a restaurant, these luxuries are immediately accessible via credit cards. No thought needs to be given to the aftermath of the purchase (please pardon the play on words  )—just sign on the dotted line. In earlier generations, when money was more highly regarded and when saving for the future or for times of lack was more prevalent, purchasing something on credit was unheard of. Layaway plans were a related phenomenon but those plans required that you saved and deposited the money in advance of the receipt of the item involved. Only when the item was fully paid for did you gain possession of it. In today’s generation, the situation has changed. Now, you may purchase an item and take possession of it immediately even though you may not have saved the necessary monies for its purchase. Credit card companies have extended a previously unavailable purchasing power to their constituents; in doing so, they have also required payment of interest on any unpaid balance at the end of each monthly billing cycle. Due in part to societal changes in attitude toward money and savings, many people today think little of charging a purchase to their credit card. They also seem to give little thought to the consequences of not paying off their bills at the end of each billing cycle and have incurred increasingly larger credit card debts in recent years.

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Math Is Everywhere! Explore and Discover It! 11.5 Credit Card Economics

******************** Statistics show that credit card debt for the average credit card user in the United States has grown to more than \$8000 in 2007. This is a radical change from just fifty years ago when personal credit debt was almost nonexistent. ********************

The High Cost of Credit Card Debt What isn’t so obvious at first glance is the fact that unpaid credit card balances can end up costing the user far more than the original purchase due to the recurring monthly interest payments involved. It is a natural tendency on the part of someone who has incurred some credit card debt to simply pay the minimum monthly payment requested by the card issuer rather than to pay off the entire balance. This tendency leads to even further debt and actually compounds the problem. In the Exercises and Projects for Fun and Profit at the end of this section, you will explore the nature of this process and discover an important mathematical truth about credit card debt. The solution, of course, is to pay off any credit card balance at the end of each billing cycle. This eliminates any finance charges (interest charges) and is, in effect, like paying cash for each item purchased rather than taking out a loan to do so. When you pay off your balances in this manner, credit cards actually provide a financial advantage in that you can keep your money in the bank up to a month longer than if you had paid cash and your money earns interest during that time.

Credit Reports and Your Credit Rating All of your credit card balances and your payment activities are recorded and later collected by consumer reporting agencies. These agencies are allowed to provide this information upon request to legitimate entities that have a legal right to access it. These entities include your creditors (including credit card companies), government agencies, employers, insurers and others with a compelling interest in your financial history. If you pay your bills on time each month, then you will build up a good credit rating; conversely, if you are late with your payments or default on any of your loans, then you will earn a poor credit rating. This credit rating is important in terms of obtaining loans, home mortgages, credit cards, increases in your credit limits, etc. It is a good idea to check your credit report before applying for any loans to ensure that the report is correct and accurate. In Florida, the state legislature recently passed a law that requires reporting agencies to provide a free copy of their report to each resident on an annual basis so that any errors or discrepancies may be detected and corrected.

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Math Is Everywhere! Explore and Discover It! 11.5 Credit Card Economics

Currently, you may obtain a copy of your credit report by contacting any of the three major credit bureaus: Equifax, Experian, and Trans Union. Their contact information is available on the Internet. In the Exercises and Projects for Fun and Profit at the end of this section, you will have an opportunity to learn more about credit reports and ratings.

Conclusion Credit cards can be financially useful and advantageous; however, there are clear and present dangers associated with their misuse. Despite the growing trend in America for borrowers to take on increasingly larger credit debt loads, it is most important that you keep your credit debt to a minimum or eliminate it entirely. Meanwhile, paying your bills on time will help you earn a good credit rating on your credit report. Best wishes for your financial success!  ********************

Exercises and Projects for Fun and Profit Creative Project: Credit Reports and Ratings: An Inside Look The Truth about Credit Card Debt ********************

CREDIT REPORTS & RATINGS: AN INSIDE LOOK Credit reports and credit scores are often unseen but important aspects of a person’s financial status. Gaining some knowledge of how these scores are determined will be valuable with regard to your financial future. Assignment: Use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on credit scores. Afterwards, write a short report (minimum of one page) on the features that were most informative or noteworthy.

414 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.5 Credit Card Economics

THE TRUTH ABOUT CREDIT CARD DEBT In this project, we will investigate the reality of credit card debt and the financial burden that results from making only minimum monthly payments on your outstanding balances. Most people are quite unaware of how quickly this burden grows and how much money that they will end up paying in interest before they make their final payment on the purchased item. After completing this project, you should have a good understanding of why it is so important to avoid the “minimum payment” trap, and I would encourage you to make every effort to do so. For the purposes of this project, we’ll assume that you have made some credit card purchases and that your average daily balance at the end of the current month is as indicated in each assignment. We will also assume that you choose to make minimum monthly payments for the life of this balance instead of paying off the entire balance at the end of the month and that you make no further purchases with this credit card until this balance is completely paid off. With regard to credit card terms, we will assume that the:  annual percentage rate (APR) for calculating interest (finance charge) is as indicated in each assignment  monthly billing period ends on the last day of each month  monthly interest (i.e., monthly finance charge) is calculated on the average daily balance at the end of each monthly billing period  minimum monthly payment is 2% of the current balance or a minimum of \$10, whichever is greater In actuality, credit card companies use somewhat differing methods to calculate finance charges and minimum payments. In recent years, the federal government has imposed new rules on credit card companies to address the issue of the minimum payment trap. In the tables below, there are six columns containing the financial data that is pertinent to your loan from the credit card company. We will use the loan amount and interest rate from Assignment #1 (\$1000, APR=16%) to illustrate the loan repayment process. The tables represent a loan repayment schedule and detail the following information: 

Month: the number of months since the debt was incurred

Balance: current balance at the beginning of the indicated month

Interest: the monthly interest (finance charge). This is calculated by multiplying the balance by the APR to obtain the annual interest and then dividing by 12 to obtain the monthly interest. In our example first month, the interest is: \$1000 · 0.16 = \$160.00 annual interest \$160.00 = \$13.33 monthly interest (finance charge) 12 415 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.5 Credit Card Economics 

Min paymt: the minimum monthly payment. This is calculated by multiplying the balance by 2% (0.02) and comparing the result to \$10.00. If the result is greater than \$10.00, then it becomes the minimum monthly payment; otherwise, \$10.00 is used:

\$1000.00 · 0.02 = \$20.00 current balance · 0.02 = minimum monthly payment 

New bal: the new balance that results from adding the monthly interest (finance charge) to the current balance and subtracting the minimum monthly payment:

\$1000.00 + \$13.33 – \$20.00 = \$993.33 current balance + interest – minimum monthly payment = new balance 

Cum interest: the cumulative interest at the end of the indicated month. This is calculated by adding each month’s interest to the sum of the interests from the previous months. In the first month, there is no previous sum involved so the cumulative interest is simply \$13.33. At the end of the second month, the interest from that month is added to the interest from the first month to obtain a cumulative interest of \$13.33 + \$13.24= \$26.57. The cumulative interest represents the total amount of finance charges that have accrued to the loan at the end of the indicated month. month 1 2

balance 1,000.00 993.33

interest 13.33 13.24

min paymt 20.00 19.87

new bal 993.33 986.71

cum interest 13.33 26.57

Assignments:

For each of the four assignments, please follow the example calculations shown above and complete the tables using the specified values given for selected months. Begin the calculations for each new row by copying the existing New Bal value into the Balance column of the new row and then proceed to calculate the values in each successive column to the right. More specifically:   

Complete rows for months 10 through 12 for Question #1 Complete three rows for Question #2 using the specified starting months Complete three rows for Question #3 using the specified starting months

Afterwards, answer Questions #4 and #5.

416 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! 11.5 Credit Card Economics

Assignment #1: loan=\$1000, APR=15%

1. Following the sample calculations on pp.415–416, please complete rows 10 through 12 in the table to get an idea of how the loan repayment will progress through the first year. month 9 10 11 12

balance 941.55

interest 11.77

min paymt 18.83

new bal 934.49

cum interest 109.18

2. Complete the three rows of the table below to see the repayment data when the balance reaches \$500.00, approximately half-way through the process: month 104 105 106 107

balance 507.86

interest 6.35

min paymt 10.16

new bal 504.05

cum interest 826.58

3. Complete the last three rows of the table to see the repayment data when the balance finally reaches \$0.00. At this point, the minimum payment is \$10.00 for each month. month 184 185 186 187

balance 29.53

interest 0.37

min paymt 10.00 10.00 10.00 10.00

new bal 19.90

cum interest 1,122.51

4. Calculate the following: a) total cost of your loan including the cumulative finance charges (cumulative interest), i.e., the total amount of money that you repaid to the credit card company. This includes the original loan amount plus the cumulative interest for the entire life of the loan. b) cumulative interest as a percent of the original loan. To find this percent, divide the cumulative interest by the initial loan amount. (Hint: The answer is greater than 1.00, i.e., greater than 100%.) c) length of time to repay. Convert the months required to complete the repayment into years to see how long the process took in terms of years. Shocking, isn’t it?

5. Write a brief report on your observations concerning the “minimum payment” trap. ********************

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Math Is Everywhere! Explore and Discover It! 11.5 Credit Card Economics

Assignment #2: loan=\$2000, APR=17%

1. Following the sample calculations on pp.415–416, please complete rows 10 through 12 in the table to get an idea of how the loan repayment will progress through the first year. month 9 10 11 12

balance 1,908.55

interest 27.04

min paymt 38.17

new bal 1,897.41

cum interest 249.13

2. Complete the three rows of the table below to see the repayment data when the balance reaches \$1000.00, approximately half-way through the process: month 139 140 141 142

balance 1,026.54

interest 14.54

min paymt 20.53

new bal 1,020.56

cum interest 2,378.64

3. Complete the last three rows of the table to see the repayment data when the balance finally reaches \$0.00. At this point, the minimum payment is \$10.00 for each month. month 367 368 369 370

balance 34.35

interest 0.49

min paymt 10.00 10.00 10.00 10.00

new bal 24.84

cum interest 4,018.12

4. Calculate the following: a) total cost of your loan including the cumulative finance charges (cumulative interest), i.e., the total amount of money that you repaid to the credit card company. This includes the original loan amount plus the cumulative interest for the entire life of the loan. b) cumulative interest as a percent of the original loan To find this percent, divide the cumulative interest by the initial loan amount. (Hint: The answer is greater than 1.00, i.e., greater than 100%.) c) length of time to repay. Convert the months required to complete the repayment into years to see how long the process took in terms of years. Shocking, isn’t it?

5. Write a brief report on your observations concerning the “minimum payment” trap. ********************

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Math Is Everywhere! Explore and Discover It! 11.5 Credit Card Economics

Assignment #3: loan=\$3000, APR=19%

1. Following the sample calculations on pp.415–416, please complete rows 10 through 12 in the table to get an idea of how the loan repayment will progress through the first year. month 9 10 11 12

balance 2,901.45

interest 45.94

min paymt 58.03

new bal 2,889.36

cum interest 420.44

2. Complete the three rows of the table below to see the repayment data when the balance reaches \$1500.00, approximately half-way through the process: month 208 209 210 211

balance 1,525.32

interest 24.15

min paymt 30.51

new bal 1,518.96

cum interest 5,627.95

3. Complete the last three rows of the table to see the repayment data when the balance finally reaches \$0.00. At this point, the minimum payment is \$10.00 for each month. month 643 644 645 646

balance 38.25

interest 0.61

min paymt 10.00 10.00 10.00 10.00

new bal 28.86

cum interest 9,997.61

4. Calculate the following: a) total cost of your loan including the cumulative finance charges (cumulative interest), i.e., the total amount of money that you repaid to the credit card company. This includes the original loan amount plus the cumulative interest for the entire life of the loan. b) cumulative interest as a percent of the original loan To find this percent, divide the cumulative interest by the initial loan amount. (Hint: The answer is greater than 1.00, i.e., greater than 100%.) c) length of time to repay. Convert the months required to complete the repayment into years to see how long the process took in terms of years. Shocking, isn’t it?

5. Write a brief report on your observations concerning the “minimum payment” trap. ********************

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Math Is Everywhere! Explore and Discover It! 11.5 Credit Card Economics

Assignment #4: loan=\$4000, APR=21%

1. Following the sample calculations on pp.415–416, please complete rows 10 through 12 in the table to get an idea of how the loan repayment will progress through the first year. month 9 10 11 12

balance 3,920.70

interest 68.61

min paymt 78.41

new bal 3,910.89

cum interest 623.74

2. Complete the three rows of the table below to see the repayment data when the balance reaches \$2000.00, approximately half-way through the process: month 416 417 418 419

balance 2,019,68

interest 35.34

min paymt 40.39

new bal 2,014.63

cum interest 13,897.57

3. Complete the last three rows of the table to see the repayment data when the balance finally reaches \$0.00. At this point, the minimum payment is \$10.00 for each month. month 1381 1382 1383 1384

balance 34.56

interest 0.60

min paymt 10.00 10.00 10.00 10.00

new bal 25.17

cum interest 25,197.81

4. Calculate the following: a) total cost of your loan including the cumulative finance charges (cumulative interest), i.e., the total amount of money that you repaid to the credit card company. This includes the original loan amount plus the cumulative interest for the entire life of the loan. b) cumulative interest as a percent of the original loan To find this percent, divide the cumulative interest by the initial loan amount. (Hint: The answer is greater than 1.00, i.e., greater than 100%.) c) length of time to repay. Convert the months required to complete the repayment into years to see how long the process took in terms of years. Shocking, isn’t it?

5. Write a brief report on your observations concerning the “minimum payment” trap. ********************

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

APPENDIX A HELPS and HINTS for EXERCISES and PROJECTS

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

Notes:

422 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

HELPS AND HINTS FOR EXERCISES & PROJECTS The Helps and Hints for Exercises and Projects are listed in the order in which the respective exercise or project appears in the textbook; chapter and section numbers as well as page numbers are included as guides. ********************

1.2 Helps and Hints for Exercises

pp.16–18

1. St. Ives: 1 is an acceptable answer (i.e., the narrator may be the only one going to St. Ives if the man that he meets is going in the other direction); however, there are a few other acceptable answers as well. To find the other answers, I would highly recommend that you draw a diagram and then use your calculator to perform the necessary multiplications. Try to find all possible logical solutions. 3. The Missing Dollar: This problem is fairly simple if you use the strategy of drawing a diagram to describe who has the various amounts of money at each point in the story. Please give it a try! Expanded version of Exercise #5: 5. Exact measures: Pouring liquids into containers of various sizes and attempting to get an exact amount of liquid into a container has long been a part of recreational mathematics. The following problem dates back several centuries at least and, as might be expected, requires some persistence and analytical reasoning in order to obtain a solution. This newer version uses a water supply in the form of a faucet with a shut-off handle. You are provided with a water supply in the form of a faucet with a shut-off handle and two empty jars, one of which holds exactly four cups of liquid while the other holds exactly seven cups of liquid. Neither of the jars has any measurement markings so there is no way to partially fill them, with accuracy, directly from the faucet. Your objective is to obtain EXACTLY five cups of water in the seven-cup jar by completely filling a jar from the faucet when necessary and pouring water from one jar to the other until you have achieved your objective. You may empty a jar of water by pouring it on the ground when necessary. Please identify the problem-solving strategies that you use to solve this classic and record your solution. (The strategies are enumerated in the previous section of the text.) ********************

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Solving this classic problem is a matter of trial and error and some persistence—it may seem impossible at first but there really is a way to do it. An intermediate and necessary goal is to get exactly one cup of water in the four-cup container. Specifically, you need to begin with the following steps:  

Step 1: Fill the four-cup container with water. Step 2: Pour the water from the four-cup container into the seven-cup container so that the four-cup container is again empty and the seven-cup container contains exactly four cups of water. Note: When you fill either the four-cup or the seven-cup container from the faucet, you must completely fill them. Partial filling is not allowed since there are no measurement marks on the containers; in particular, you cannot fill a container until it is half full. Note: You may pour water from the small container into the large container if and when necessary and vice versa.

7. Prize Money: For this problem, please follow the example on pp.10–12 in the textbook. Using the algebraic approach as indicated will lead you directly to the solution or you may use the table method to construct tables for each of the winner's scenarios and search the table entries for matching values in the two rightmost columns. The first two equations are: a + 200 = b, and b + 200 = 5a. Subtract 200 from both sides of the first equation to get "a" by itself and then substitute this value of "a" into the second equation and solve it for "b". These classic problems are a bit challenging, but be persistent—that counts for a lot! ********************

1.2 Helps and Hints for Guards and Prisoners

p.19

This presents a real challenge, especially if you restrict yourself to paper and pencil. It may be quite helpful to use some physical manipulatives such as three quarters and three pennies to represent the guards and the prisoners and then try moving them across a boundary line that represents the river. It will then be easier to visualize the process and to arrive at a solution. You just have to be sure that, at each step along the way, the number of prisoners is never greater than the number of guards on the same side of the river (including those in the boat after it arrives with its occupant(s)).

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

The notes below apply to all four assignments that are included in this project; helps and hints for each individual assignment are listed afterwards. Note 1: It is acceptable to have prisoners without guards on either side of the river; the prisoners won't try to escape... Note 2: Each time that the boat crosses the river counts as one river crossing. One round trip (across and back) equals two river crossings. Note 3: A prisoner or a guard in the boat counts as one of the people on whichever side of the river the boat is on. For example, if there were one guard and one prisoner on one side of the river and if the boat were to arrive carrying one prisoner, at that point the prisoners would outnumber the guard. Note 4: Someone (either a guard or a prisoner) needs to row the boat across the river during each crossing. Swimming and towing the boat with ropes are not permitted. Note 5: Only one or two people can cross the river in the boat during each crossing. Note 6: There is only one boat available. To get started, you will need to use the strategies of trial and error and eliminate the impossible so that only the possible remains. ********************

Assignment #1:

(3 guards and 3 prisoners)

There are three possibilities for the first river crossing: a) 2 guards cross over b) 1 guard and 1 prisoner cross over c) 2 prisoners cross over Of these three possibilities, the first one is impossible because then there would be 3 prisoners and only 1 guard on the original side of the river and this violates the conditions of the problem. If 1 guard and 1 prisoner cross over, then only the guard may return in the boat. If the prisoner were to return in the boat to the original side, then there would be 3 prisoners and only 2 guards there after the boat arrived (see Note 3 above). If 2 prisoners cross over, then, of course, 1 prisoner would have to return in the boat.

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Let’s assume that 2 prisoners cross the river during the first river crossing and that 1 prisoner rows the boat back across the river during the second river crossing. Afterwards, there are 3 guards and 2 prisoners on the original side and 1 prisoner on the destination side. For the third river crossing, there is only one possibility: 2 prisoners cross over  

If 2 guards cross over, then there would be 2 prisoners but only 1 guard on the original side. If 1 guard and 1 prisoner cross over, then there would be 2 prisoners and only 1 guard on the destination side when the boat arrived.

To summarize the first few river crossings: 1. 2 prisoners cross over 2. 1 prisoner returns 3. 2 prisoners cross over Your tasks are to continue with these methods of trial and error and eliminating the impossible and then to determine the total number of river crossings required in order to transport everyone safely across the river. ********************

Assignment #2:

(4 guards and 2 prisoners)

There are three possibilities for the first river crossing: a) 2 guards cross over b) 1 guard and 1 prisoner cross over c) 2 prisoners cross over Any of these are valid moves. Let’s assume that 2 prisoners cross the river during the first river crossing and that 1 prisoner rows the boat back across the river during the second river crossing. Afterwards, there are 4 guards and 1 prisoner on the original side and 1 prisoner on the destination side. For the third river crossing, there are only two possibilities: a) 2 guards cross over b) 1 guard and 1 prisoner cross over

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

Of these two possibilities, the second one is impossible because then there would be 2 prisoners and only 1 guard on the destination side of the river when the boat arrives on the shore and this violates the conditions of the problem. Therefore, 2 guards must cross the river during the third river crossing. To summarize the first few river crossings: 1. 2 prisoners cross over 2. 1 prisoner returns 3. 2 guards cross over Your tasks are to continue with these methods of trial and error and of eliminating the impossible and then to determine the total number of river crossings required in order to transport everyone safely across the river. ********************

Assignment #3:

(3 guards and 2 prisoners)

There are three possibilities for the first river crossing: a) 2 guards cross over b) 1 guard and 1 prisoner cross over c) 2 prisoners cross over Of these three possibilities, the first one is impossible because then there would be 2 prisoners and only 1 guard on the original side of the river and this violates the conditions of the problem. If 1 guard and 1 prisoner cross over, then either the guard or the prisoner may return in the boat. If 2 prisoners cross over, then, of course, 1 prisoner would have to return in the boat. Let’s assume that 2 prisoners cross the river during the first river crossing and that 1 prisoner rows the boat back across the river during the second river crossing. Afterwards, there are 3 guards and 1 prisoner on the original side and 1 prisoner on the destination side. For the third river crossing, there are only two possibilities: a) 2 guards cross over b) 1 guard and 1 prisoner cross over

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

Of these two possibilities, the second one is impossible because then there would be 2 prisoners and only 1 guard on the destination side of the river when the boat arrives on the shore and this violates the conditions of the problem. Therefore, 2 guards must cross the river during the third river crossing. To summarize the first few river crossings: 1. 2 prisoners cross over 2. 1 prisoner returns 3. 2 guards cross over Your tasks are to continue with these methods of trial and error and of eliminating the impossible and then to determine the total number of river crossings required in order to transport everyone safely across the river. ********************

Assignment #4:

(4 guards and 3 prisoners)

There are three possibilities for the first river crossing: a) 2 guards cross over b) 1 guard and 1 prisoner cross over c) 2 prisoners cross over Of these three possibilities, the first one is impossible because then there would be 3 prisoners and only 2 guards on the original side of the river and this violates the conditions of the problem. Let’s assume that 2 prisoners cross the river during the first river crossing and that 1 prisoner rows the boat back across the river during the second river crossing. Afterwards, there are 4 guards and 2 prisoners on the original side and 1 prisoner on the destination side. For the third river crossing, there are three possibilities: a) 2 guards cross over b) 1 guard and 1 prisoner cross over c) 2 prisoners cross over Of these, the second one is impossible because then there would be 2 prisoners and only 1 guard on the destination side of the river when the boat arrives on the shore and this violates the conditions of the problem.

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

Therefore, 2 guards or 2 prisoners must cross the river during the third river crossing. Let’s assume that 2 guards cross the river during the third river crossing and that 1 guard rows the boat back across the river during the fourth river crossing. To summarize the first few river crossings: 1. 2. 3. 4.

2 prisoners cross over 1 prisoner returns 2 guards cross over 1 guard returns

Your tasks are to continue with these methods of trial and error and of eliminating the impossible and then to determine the total number of river crossings required in order to transport everyone safely across the river. ********************

1.2 Helps and Hints for Cross-number Puzzle

pp.20–21

Assignment #1: (Please read Cross-number Puzzle Terminology on p.449 in Appendix B.) To solve this puzzle, it will be most helpful to make lists (with the help of a calculator) of all of the possible answers for each clue based on the number of digits needed according to the puzzle's structure. For example, for #1 Across and #5 and #7 Down, the only cubes with three digits are: 125, 216, 343, 512, and 729. With a few exceptions (viz., #6 Across, #1 Down and #9 Down which you may omit from this process), the clues have only a very limited number of possible solutions and, with the help of a calculator, this process will not take very long to complete. This is a very important part of the puzzle-solving process! Making these lists will also reveal the key clue to unlocking the puzzle—the only clue with exactly one solution (#10 Down)! It then is a matter of using that first solution to help determine solutions for clues whose entries are connected to the entries in the first solution. You may have to try to solve two or three clues together in order to determine what works for all of them at the same time. This is the nature of a cross-number or crossword puzzle. In particular, use a calculator to find the fourth powers of 6, 7, 8 and 9; you'll find that these are the only possibilities (that contain four digits) for #8 Across.

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You then have to consider these four possibilities for #8 Across along with the possible answers for clues #7 Down, #9 Down and #11 Across all at the same time since they all are interdependent. As an additional hint, the solution for #8 Across is 1296. Have fun! And be persistent!  ********************

Assignment #2: (Please read Cross-number Puzzle Terminology on p.449 in Appendix B.) To solve this puzzle, it will be most helpful to make lists (with the help of a calculator) of all of the possible answers for each clue based on the number of digits needed according to the puzzle's structure. For example, for #1 Across and #11 Across, the only cubes with three digits are: 125, 216, 343, 512 and 729. With a few exceptions (viz., #1 Down, #5 Down and #10 Down which you may omit from this process), the clues have only a very limited number of possible solutions and, with the help of a calculator, this process will not take very long to complete. This is a very important part of the puzzle-solving process! Making these lists will help to reveal the key clues to unlocking the puzzle: #8 Across and #7 Down are the best clues with which to begin. #6 Across will also be helpful in conjunction with these first two... In particular, the solution for #8 Across is 4913. You may have to try to solve two or three clues together in order to determine what works for all of them at the same time. This is the nature of a cross-number or crossword puzzle. Have fun! And be persistent!  ********************

Assignment #3: (Please read Cross-number Puzzle Terminology on p.449 in Appendix B.) To solve this puzzle, it will be most helpful to make lists (with the help of a calculator) of all of the possible answers for each clue based on the number of digits needed according to the puzzle's structure. For example, for #1 Across, #4 Across and #11 Across, the only squares with three digits are: 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961 With a few exceptions (viz., #1 Down, #5 Down and #10 Down which you may omit from this process), the clues have only a very limited number of possible solutions and, with the help of a

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

calculator, this process will not take very long to complete. This is a very important part of the puzzle-solving process! Making these lists will help to reveal the key clues to unlocking the puzzle: #2 Down, #9 Down and #8 Across are the best clues with which to begin. #7 Down will also be helpful in conjunction with these first three... In particular, the solution for #8 Across is 2197. You may have to try to solve two or three clues together in order to determine what works for all of them at the same time. This is the nature of a cross-number or crossword puzzle. Have fun! And be persistent!  ********************

Assignment #4: (Please read Cross-number Puzzle Terminology on p.449 in Appendix B.) To solve this puzzle, it will be most helpful to make lists (with the help of a calculator) of all of the possible answers for each clue based on the number of digits needed according to the puzzle's structure. For example, for #4 Across and #1 Down, the only squares with three digits are: 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961 With a few exceptions (viz., #9 Down and #10 Down which you may omit from this process), the clues have only a very limited number of possible solutions and, with the help of a calculator, this process will not take very long to complete. This is a very important part of the puzzlesolving process! Making these lists will help to reveal the key clues to unlocking the puzzle. #11 Across and #7 Down are the best clues with which to begin since they each have only one solution. #8 Across can then be solved in conjunction with #7 Down In particular, the solution for #8 Across is 6561. You may have to try to solve two or three clues together in order to determine what works for all of them at the same time. This is the nature of a cross-number or crossword puzzle. Have fun! And be persistent!  ******************** 431 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

2.1 Helps and Hints for Exercises

pp.44–45

Some riddle clues: 1. 2. 3. 4. 5.

It is a type of feline... You use one of these when driving to an unfamiliar location. An animal that has been out in the sun too long. Think of the basic elements of life. This riddle is much like Bilbo's Challenge #5 on p.34 (answer on p.47). Hope these are helpful!  ********************

2.2 Helps and Hints for Exercises

pp.57–59

Some helps and hints are listed below: 11. Work from right to left and follow the pattern backwards by dividing each time instead of multiplying each time. 17–20: As indicated in the introduction to #17–20, each sequence involves three interwoven sequences. 17. Observe every third term and find the next term that follows the pattern, e.g., 5, 11, 17, ___ In this case the common difference is 6 and the next term is 23. Then look at the second term (8) and every third term thereafter, e.g., 8, 14, 20, ___ and do the same... 25. The first sequence of successive differences is found by subtracting each term in the original set of fourth powers from the one on its right: 16 – 1 = 15, 81 – 16 = 65, 256 – 81 = 175, etc. 15, 65, 175, ... then forms a new sequence of successive differences. The second row of successive differences is found in the same way: 65 – 15 = 50, 175 – 65 = 110, etc. 50, 110, ... then forms a new sequence of second successive differences.

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

Then, keep finding new sequences of successive differences until you arrive at a sequence in which all terms have the same value. This value will be the answer to the question. Be sure to use a calculator when performing the subtractions—one small error can throw the whole process off! Hope this helps!  ********************

2.3 Helps and Hints for Exercises

pp.77–80

Please see the Example Exercises provided on pp.83–84. Some additional example solutions are listed below: For #29, the idea is to create a simple equation that describes the number of common differences needed to get from the first given term, 9, to the ending term, 69, like this: 9 + d + d + d + d = 69 9 + 4d = 69 4d = 60 d = 15 Then successively add d=15 to the first term to find the missing terms: 9 + 15 = 24 (first missing term) 24 + 15 = 39 (second missing term) 39 + 15 = 54 (third missing term) 54 + 15 = 69 (proof that the common difference produces the correct last term) #31 and #33 are similar. For #35–39, please review the section entitled, Working in Reverse: Constructing the nth term, a(n), from its Sequence Values, on p.72. Then read Example Exercise #36 on pp.83–84. Hopefully, you will then be able to determine how to construct an nth term. For the record, the answer to #35 is: a(n) = 6 + (n – 1) · 5 You could also multiply (n – 1) by 5 and collect like terms to get: a(n) = 6 + 5n – 5 or a(n) = 5n + 1 Either of these answers are acceptable as the final answer: a(n) = 6 + (n – 1) · 5 or a(n) = 5n + 1

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

2.3 Helps and Hints for Generalization—A Powerful Tool p.82 The "a" and the "n" are important parts of the generalization process. The generalization in this project (p.82) moves beyond numbers to use letters (variables) instead. In essence, the project asks you to fill in the "?" in the bottom right entry in the table. You can do this by carefully observing the examples of sequences and their corresponding formulas given in the first three rows. In each example sequence, the entry in the rightmost column begins with "a(n)=" and is followed by five distinct items: 1. The first term from column 1 2. The "+" symbol 3. The expression "(n – 1)" 4. The multiplication symbol (represented by a small dot) 5. The common difference from the middle column Your objective in this project is to fill in these same five items in the bottom row, the one in which variables are used instead of numbers. Hope this helps!  ********************

2.4 Helps and Hints for Exercises

pp.97–98

Each letter represents a single digit between 0 and 9, inclusive. Different letters represent different digits—no duplicates allowed; if A=2, then B cannot equal 2 and so on... Please see the Example Exercises provided on pp.100–101. Some additional example solutions are listed below; please find as many solutions as possible for each exercise: 1. A+B=B: 0+1=1 is one of several solutions. 3. A+B=CC: 2+9=11 is one of several solutions. 5. A+A=BB: There are no solutions since no single digit added to itself results in 11 (or 22 or 33...). 7. A+B=CD: CD indicates that the sum is a two-digit number, each digit of which is different from each other and different from the digits used in the addends (A and B).

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

Some solutions are: 2+8=10, 3+7=10, 3+9=12, ... 3+8=11 is not a valid solution since C and D must be different digits. 5+5=10 is not a valid solution since A and B must be different digits. 9. GA+A=FY: Some solutions are: 15+5=20, 17+7=24, 18+8=26, ... 16+6=22 is not a valid solution since F and Y must represent different digits. 11. DA+A=VEE: Only one solution—I'll let you see if you can find it by using analytical reasoning. 13. BL+A=STS: 93+8=101 is one of several solutions... 15. BB+CC=ADE: 44+88=132 is one of several solutions... 17. ONE+ONE+ONE=TWO: There is no “carry-over” in the third column, so O=1, O=2 or O=3. Since O also represents the sum in the first column, E+E+E (or the ending digit in the sum of E+E+E) must have the same value as O. 107+107+107=321 is one of several solutions… Hope this helps!  ********************

2.5 Helps and Hints for Exercises

pp.112–113

Each letter represents a single digit between 0 and 9, inclusive. Different letters represent different digits—no duplicates allowed; if a=2, then b cannot equal 2 and so on... Some example solutions are listed below; please find as many other solutions as possible: 1. a+b=11: 5+6=11 is one of several solutions. 3. a+b=c: 1+4=5 is one of several solutions. 5. a+b=c: Only one solution (or two solutions when you reverse the order of the addends)—I'll let you see if you can find it (or them) by using analytical reasoning. 7. a b + c d = e f: No solution. This can only be discovered by trial and error and a good bit of persistence.

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

9. a b + c d = e f: No solution. This can only be discovered by trial and error and a good bit of persistence. 11. a b c + d e = f g h: 289+57=346 is one of several solutions. 13. a · b = c d: 2·7=14 is one of several solutions. Hope this is helpful!  ********************

2.5 Helps and Hints for Magic Squares

p.116

Constructing a 3x3 magic square can be quite a challenge, especially if you simply use the trialand-error approach. This project affords an excellent opportunity to use your analytical reasoning skills to effectively narrow the possibilities. A very good analytical line of reasoning would begin by recognizing that a 3x3 magic square has the same structure as the childhood game of tic-tac-toe. A good analytical question is: What is the most important square in the 3x3 magic square? (To answer this, you might recall the most important square in the game of tic-tac-toe.) A second and related question is: What is the most important number in the list of entry numbers for the 3x3 magic square? I'll let you put the answers to these two questions together as you carry out your investigation and I'll trust that these will lead you in a fruitful direction. The answers to these questions are: the middle square in the middle row and the number in the middle of the list of entries when the list is written in numerical order. Hopefully, the answers to these two questions will lead you to the solution.  As an additional hint, the desired sum will be the sum of the nine entry values divided by 3. ********************

3.1 Helps and Hints for Polygonal Numbers: A Geometric View pp.126–127 For #1 and #2, simply continue the patterns shown on p.126 to construct larger triangular and square numbers. For #3: Please note that the first pentagonal number contains only one dot just as the first triangular

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

and square numbers did. To construct the second and third pentagonal numbers, please take the following steps: a. Construct a regular pentagon that has five dots spaced evenly around its perimeter. Each side of this pentagon will contain two dots, one at either end. This is the second pentagonal number; it contains a total of five dots. b. Extend the bottom side of this pentagon to the right by a distance equal to the spacing of the original dots and place a dot there. This will create an extended bottom side that contains three dots. c. Extend the left side adjacent to the bottom side by a distance equal to the spacing of the original dots and place a dot there. This will create an extended adjacent side that contains three dots. d. Then form the other three sides of this larger pentagon by placing dots at evenly spaced intervals so that there are three dots on each side of the larger pentagon. You have now constructed a larger pentagon that contains a smaller pentagon within. Determine how many dots the third pentagonal number has by counting the total number of dots in both the larger and the smaller pentagons (each dot is to be counted only once); together, these dots comprise the third pentagonal number. ******************** For #4: Please note that the first hexagonal number contains only one dot just as the first triangular and square numbers did. To construct the second and third hexagonal numbers, please take the following steps: a. Construct a regular hexagon that has six dots spaced evenly around its perimeter. Each side of this hexagon will contain two dots. This is the second hexagonal number; it contains a total of six dots. b. Extend the bottom side of this hexagon to the right by a distance equal to the spacing of the original dots and place a dot there. This will create an extended bottom side that contains three dots. c. Extend the left side adjacent to the bottom by a distance equal to the spacing of the original dots and place a dot there. This will create an extended adjacent side that contains three dots. d. Then form the other four sides of this larger hexagon by placing dots at evenly spaced intervals so that there are three dots on each side of the larger hexagon.

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

You have now constructed a larger hexagon that contains a smaller hexagon within. Determine how many dots the third hexagonal number has by counting the total number of dots in both the larger and the smaller hexagons (each dot is to be counted only once); together, these dots comprise the third hexagonal number. ********************

3.1 Additional Help for Polygonal Numbers: A Geometric View pp.126–127 Since this project can be quite challenging, use the search engines found at www.merlot.org and/or www.google.com to conduct some Internet research on each of the various polygonal numbers to guide to you the correct constructions:    

Triangular Numbers Square Numbers Pentagonal Numbers Hexagonal Numbers Hope these visual explanations are helpful... 

********************

4.1 Helps and Hints for Exercises

p.150

Some example solutions are listed below: In #1, you need to use expanded notation in base 7 (see p.144) and simply multiply and add: 456 in base 7 = (4 · 7 2 ) + ( 5 · 7 1 ) + (6 · 7 0 ) = (4 · 49) + (5 · 7) + (6 · 1 ) = 196 + 35 + 6 = 237 in base 10. #2–5 follow the same pattern. In #6, you need to work in reverse by using either the long method (shown on pp.145–148 in Examples 5 and 6) or the elegant but mysterious short method (shown on pp.148–149 in Examples 7 and 8).

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

To use the short method, divide 345 by 4 and continue dividing as follows: 345/4 = quotient of 86 with a remainder of 1 86/4 = quotient of 21 with a remainder of 2 21/4 = quotient of 5 with a remainder of 1 5/4 = quotient of 1 with a remainder of 1 1/4 = quotient of 0 with a remainder of 1 Afterwards, read the remainders in reverse order (from bottom to top): 11121 So 345 in base 10 is equivalent to 11121 in base 4. #7–10 may be done in a similar fashion. ********************

4.1 Helps and Hints for The Banker’s Dilemma

p.152

See Hints and Additional Hints provided on pp.153–154 ********************

5.3 Helps and Hints for Exercises

pp.203–205

Some example solutions are listed below: Caesar ciphers 1. This involves a Caesar shift cipher as described on top of p.196. 3. If you write two very evenly spaced alphabets on two separate sheets of paper and then slide one alphabet horizontally across another, moving the alphabet one character over each time, you can test each shift to see if the corresponding letters make the encoded text make sense. When you reach the correct number of letter shifts, the code will then be intelligible. In this case, when you have the ciphertext alphabet's A is above the plaintext alphabet's D, the code will be easily solved. Cryptograms 5. The first word is a contraction, like can't, won't, isn't, etc. This will help you determine what M and C are....

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

7. The third word is NEED. Cryptoquotes 9. Please read the Helps and Hints for Cryptoquotes that follow in order to get a feel for what is involved. The first word in the cryptoquote is TO.

********************

5.3 Helps and Hints for Cryptoquotes: Another Favorite

p.207

Please read the cryptanalysis techniques listed at the bottom of p.197 and illustrated on pp.198–201. These techniques will be needed in order to decode the cryptoquotes. (A cryptoquote does not involve a lettershift as some of the Exercises did.) ********************

Assignment #1: Examining word length will be of assistance. The two one-letter words, K and P, will decode to A and I but not necessarily in that order; for example, either K or P might decode to the letter A. Then notice that the "words" PXT and WZR each occur twice in the quote. Thinking of common three-letter words in English will help you guess the correct letter identities here. Two of the most common are THE and AND. Determining which letter represents E and substituting the correct letters for PXT and WZR into the cryptoquote will lead you to the solution. This should get you on the right track, and then hopefully you can complete the rest of the puzzle. Hope this helps!  ********************

Assignment #2: Examining word length will be of assistance. The two one-letter words, X and R, will decode to A and I but not necessarily in that order; for example, either X or R might decode to the letter A. Then notice that the "word" BSM occurs twice in the quote. Thinking of common three-letter words in English will help you guess the correct letter identities here. Two of the most common are THE and AND.

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

Determining which letter represents E and substituting the correct letters for BSM and X and R into the cryptoquote will lead you to the solution. This should get you on the right track, and then hopefully you can complete the rest of the puzzle. Hope this helps!  ********************

Assignment #3: Examining word length will be of assistance. Notice that the "word" FWX occurs twice in the quote. Thinking of common three-letter words in English will help you guess the correct letter identities here. Two of the most common are THE and AND. The "word" FWQ is an old-fashioned word that is no longer in use. Two additional clues are: R=I and A=R. Determining which letter represents E and substituting the correct letters for FWX, R and A into the cryptoquote will lead you to the solution. Notice also that the first "word" is repeated and that it is followed in each case by an exclamation point. This should get you on the right track, and then hopefully you can complete the rest of the puzzle. Hope this helps!  ********************

Assignment #4: Examining word length will also be of assistance. The only one-letter word, G, will decode to either A or I. Notice that each of the "words" GAF and EPR occurs twice in the quote. Thinking of common three-letter words in English will help you guess the correct letter identities here. Two of the most common are THE and AND. Substituting the correct letters for GAF and EPR into the cryptoquote will lead you to the solution.

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

Notice also that:  

the word HRQRAEB-LCQR is hyphenated. the first "word" ZCHERA is followed by a comma and that the third "word" KPCZFORA is also followed by a comma.

This should get you on the right track, and then hopefully you can complete the rest of the puzzle. Hope this helps!  ********************

6.1 Helps and Hints for Exercises

pp.221–222

Some example solutions are listed below: Note: In answering some of the exercises, you will need to know how to find the root of a number on your calculator. For example, to find the twelfth root of 2: On a scientific calculator, first type in the number 2, then press the “y^x” key (or the “x^y” key), type a left parenthesis, then type “1/12” (i.e., 1 divided by 12), type a right parenthesis, and press the “=” key. If you have a graphing calculator, please use the caret key, ^, in place of the yx or the xy key To find numerical roots other than the twelfth root, simply replace the number 12 in the above example with the number of the root that you would like to find, e.g., use 4 for the fourth root, 7 for the seventh root, etc. 1. In the equation shown on p.222, 440 · r 4 = 880, both sides can be divided by 440 to produce the equivalent equation of: r raised to the fourth power = 2, or r 4 = 2 Solving this equation for r, i.e., finding the fourth root of 2 on your calculator, will give you the solution for r, the common ratio. (See p.217 for an example of solving a related equation.) 2. In the equation shown on p.222, 440 · r 7 = 880, both sides can be divided by 440 to produce the equivalent equation of: r raised to the seventh power = 2, or r 7 = 2

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

Solving this equation for r, i.e., finding the seventh root of 2 on your calculator, will give you the solution for r, the common ratio. (See p.217 for an example of solving a related equation.) 3. The answer for part a) is given; the answers for parts b) and c) are based on the description of the octave and the fact that the high note has a frequency that is three times that of the low note in a given octave. 4. The solution may be found in a similar manner as in #1 and #2. 5. Please read pp.219–220 for a description of how the formula was developed for a note that is n half-steps above A4; the formula for a note that is n half-steps below A4 is very similar. To go up the scale, you need to multiply; to go down the scale, you need to divide. Hope this helps!  ********************

9.1 Helps and Hints for Exercises

pp.313–316

Some example solutions are listed below: 2. Average outdoor temperature depends on geographical latitude 6. a) miles traveled = 30 mpg · number of gallons used m = 30 · g m = 30g b) gal miles 5 150 10 300 15 450 10. y = 0.5x – 12 is a function since each x-value has one and only one corresponding y-value. x y 2 –11 4 –10 6 –9

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

9.2 Helps and Hints for Exercises

pp.331–332

Please see the Example Exercises provided on pp.334–335.

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9.3 Helps and Hints for Exercises

pp.342–345

An example solution is listed below for Exercise #1: a. Graph all of the data points. b. Draw an approximate line of best fit. See Note on p.342 just preceding the Exercises for directions on how to draw the line. c. Locate two points on this line and label their x- and y-coordinates as accurately as possible. Two estimated points are: (3, 31) and (9, 55). d. Find the equation of this line. First find the slope (see p.319): m

y 2  y1 55  31 24  4  93 6 x 2  x1

Then use one of the points on the line and this slope to find the value for b in y=mx+b. Substitute the values for m, x and y into the equation y=mx+b and solve for b: y = mx + b 31 = (4)(3) + b 31 = 12 + b 19 = b Write the complete regression line equation using the values for m and b that you found: y = 4x + 19 e. Use the Guide to Correlation Coefficients on pp.453–454 in Appendix C to determine an approximate correlation coefficient for the data points and the regression line.

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

f. Use the equation that you found in part d) above and use x=15 to predict the y-value (height of a child at age 15). y = 4(15) + 19 y = 60 + 19 y = 79 Hope this helps!  ********************

10.3 Helps and Hints for Fastest Growing State

p.372

After you have determined the fastest growing state, you will need to find the state’s population in the year 2000. To do so, please visit http://www.census.gov/main/www/cen2000.html and select the pertinent state in the Select a State dropdown list in the section entitled Data Highlights. The population will be displayed at the top of the state’s data page. ********************

11.4 Helps and Hints for The All-Powerful Interest Rate

p.409

In Part 1, you need to use an amortization calculator and enter the specified mortgage amount and each of the percents to determine each monthly payment amount (including principal and interest) and total cost of the loan, respectively. You have to reset the calculator after each percent. In Part 2, you first count 12 months of payments for the first year in the amortization schedule and find the principal balance (or balance) at the end of this first year. Then, as in the example in the textbook, pretend that you make an extra payment at that time and subtract this \$1500 payment from the principal balance at the end of the first year to find the new balance. Then look farther down in the schedule to find the balance that most closely approximates this new balance. Then you "leap" down to the corresponding month in the schedule and begin counting 12 more months for the second year of mortgage payments to find the principal balance at the end of this second year. Repeat the process of subtracting \$1500 and finding the new balance. Then look

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Math Is Everywhere! Explore and Discover It! Appendix A: Helps and Hints for Exercises and Projects

farther down in the schedule to find the balance that most closely approximates this new balance, "leap" down to the corresponding month in the schedule and begin counting 12 more months for the third year of mortgage payments, and so on…. Each 12-month period plus the ensuing "leaped" number of months counts as one year in the repayment process. Hope this helps. 

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Math Is Everywhere! Explore and Discover It! Appendix B: Cross-number Puzzle Terminology

APPENDIX B CROSS-NUMBER PUZZLE TERMINOLOGY

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Math Is Everywhere! Explore and Discover It! Appendix B: Cross-number Puzzle Terminology

Notes:

448 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Appendix B: Cross-number Puzzle Terminology

CROSS-NUMBER PUZZLE TERMINOLOGY This terminology refresher pertains to the puzzle clues for the Cross-number Puzzles in Section 1.2, pp.20–21. ********************

Prime Numbers Prime numbers are positive whole numbers that are evenly divisible only by themselves and by the number 1. Examples: The first several primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, … 1 is not technically considered to be a prime. 4 is not a prime because 2·2=4 and 4/2=2; 6 is not a prime because 2·3=6 and 6/2=3, etc. Numbers greater than 1 that are not primes are also called composite numbers. ********************

Factors and Prime Factors When one number is multiplied by another, both of the numbers are called factors and the resulting number is called a product. If either of the numbers being multiplied together happens to be a prime, then that number is also called a prime factor. Example: 2·6=12 2 and 6 are factors and 12 is the product of those factors. Since 2 is a prime, it is also a prime factor. 6 is not a prime, so it is simply a factor of 12 but not a prime factor of 12. In the reverse direction, if we start with a given number such as 12, we can then find all of the possible factors for 12 and identify which of these are prime factors. Example: Possible factorizations of 12 (using only two factors in each factorization): 1·12=12, 2·6=12, 3·4=12 Observing these factors, we can see there are only two that are primes: 2 and 3. So 12 has only two prime factors. ********************

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Math Is Everywhere! Explore and Discover It! Appendix B: Cross-number Puzzle Terminology

Squares A perfect square (also referred to more simply as a square) is a product resulting from a whole number that has been squared, i.e., used twice as a factor. Examples: 1, 4, 9 and 16 are called perfect squares: 1 squared = 12 = 1·1 = 1 2 squared = 22 = 2·2 = 4 3 squared = 32 = 3·3 = 9 4 squared = 42 = 4·4 = 16

This is sometimes written as: This is sometimes written as: This is sometimes written as: This is sometimes written as:

1^2 = 1 2^2 = 4 3^2 = 9 4^2 = 16

When we say the “square of 6”, we mean the number 6 multiplied by itself, i.e., “6 squared”, or 62 =6·6=36. When we say the “square of a prime”, we mean a prime number multiplied by itself; see note above on prime numbers. ********************

Cubes A perfect cube (also referred to more simply as a cube) is a product resulting from a whole number that has been cubed, i.e., used as a factor three times. Examples: 1, 8, 27 and 64 are called perfect cubes: 1 cubed = 13 = 1·1·1 = 1 2 cubed = 23 = 2·2·2 = 8 3 cubed = 33 = 3·3·3 = 27 4 cubed = 43 = 4·4·4 = 64

This is sometimes written as: This is sometimes written as: This is sometimes written as: This is sometimes written as:

1^3 = 1 2^3 = 8 3^3 = 27 4^3 = 64

********************

Fourth Powers The fourth power of a number is a product resulting from a number that has been raised to the fourth power, i.e., used as a factor four times. Example: 81 is called the fourth power of 3: 3 raised to the fourth power = 34 = 3·3·3·3 =81

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Math Is Everywhere! Explore and Discover It! Appendix C: Guide to Correlation Coefficients

APPENDIX C GUIDE TO CORRELATION COEFFICIENTS

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Math Is Everywhere! Explore and Discover It! Appendix C: Guide to Correlation Coefficients

Notes:

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Math Is Everywhere! Explore and Discover It! Appendix C: Guide to Correlation Coefficients

GUIDE TO CORRELATION COEFFICIENTS A correlation coefficient is a measure of how strong the correlation is between a “line of best fit”, or regression line, and the associated data points on a graph. When the data points are tightly clustered about the line, the correlation is high (or strong); when the data points are scattered broadly away from the line, the correlation is low (or weak). Correlation coefficients range in value from 0 to 1.00 (or 100%) for regression lines having positive slope and from 0 to –1.00 (–100%) for regression lines having negative slope. This coefficient is often referred to as the variable r. The sample graphs and correlation coefficients below illustrate various strengths of correlation for regression lines having positive slope; the strengths are similar for regression lines having negative slope except that their correlation coefficients have negative values rather than positive. You may use these sample graphs as a guide when determining approximate correlation coefficients for pertinent exercises and projects in this textbook. To determine an approximate correlation coefficient, first plot a given set of data points and draw a “line of best fit” through the approximate “middle” of the data points. Afterwards, compare your completed graph with the sample graphs below and select the one below that most closely resembles your graph with regard to the closeness of the data points to the regression line. The correlation coefficient of the selected graph will then be a good approximation of the correlation coefficient for your graph. For the purposes of the exercises and projects in our textbook, you may choose whichever of the four graphs below that best approximates your completed graph and then record the associated correlation coefficient (0.99, 0.90, 0.50 or 0.30) as the answer to the pertinent question. ******************** Sample graph 1: Correlation coefficient = 0.99 = 99% Extremely high correlation between the regression line and the actual data points, i.e., all of the data points are very close to the line. 25 20 15 10 5 0 0

1

2

3

4

5

6

7

453 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Appendix C: Guide to Correlation Coefficients Sample graph 2: Correlation coefficient = 0.90 = 90% Strong correlation between the regression line and the actual data points, i.e., the data points are clustered near the line. 70 60 50 40 30 20 10 0 0

2

4

6

8

10

12

Sample graph 3: Correlation coefficient = 0.50 = 50% Moderate correlation between the regression line and the actual data points, i.e., the data points are moderately spread away from the line. 70 60 50 40 30 20 10 0 0

2

4

6

8

10

12

Sample graph 4: Correlation coefficient = 0.30 = 30% Weak correlation between the regression line and the actual data points, i.e., the data points are broadly spread away from the line. 70 60 50 40 30 20 10 0 0

2

4

6

8

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12

454 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.

Math Is Everywhere! Explore and Discover It! Appendix D: Fractal Structures: Italian Parsley

APPENDIX D FRACTAL STRUCTURES: ITALIAN PARSLEY

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Math Is Everywhere! Explore and Discover It! Appendix D: Fractal Structures: Italian Parsley

Notes:

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Math Is Everywhere! Explore and Discover It! Appendix D: Fractal Structures: Italian Parsley

FRACTAL STRUCTURES: ITALIAN PARSLEY Many plants have fractal or fractal-like structures although they go largely unnoticed. As my wife and I were having dinner one evening, she suddenly exclaimed, “This looks like a fractal!” The object of her attention was a stalk of parsley on her dinner plate and, of course, I had to investigate. Much to my amazement, I found that it was true—the stalk of parsley most definitely exhibited a fractal-like structure. I was not so much amazed at the parsley as at the fact that I had never noticed this fact even though I had eaten parsley for many years. It was a “Eureka!” moment— one that we will never forget. The photo below shows a stalk of Italian parsley that displays a distinctive fractal-like structure (the same structure is also found in curly-leaf parsley). The next time that you purchase some parsley, take time to observe it carefully and you will find even more fractal detail. Happy investigating!

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Math Is Everywhere! Explore and Discover It! Appendix D: Fractal Structures: Italian Parsley

Notes:

458 Math is Everywhere, Fifth Custom Edition, by Jim Rutledge. Published by Pearson Learning Solutions. Copyright ' 2012 by Pearson Education, Inc.