Pythagoras' Legacy: Mathematics in Ten Great Ideas 019885224X, 9780198852247

Table of contents :
Cover
Pythagoras’ Legacy: Mathematics in Ten Great Ideas
Copyright
Contents
Preface
Chapter 1: The Pythagorean theorem: The birth of mathematics
Prologue
The Pythagorean theorem
Proof
Discovery of √2
Practical uses
Pattern
Fermat’s Last Theorem
Epilogue
Explorations
1. Bhāskara’s snake and peacock puzzle
2. Abu al-Wafa’s assembly puzzle
3. Apollonius’ problem
4. Measurement problem
5. Gardner’s tricky triangle puzzle
Chapter 2: Prime numbers: The DNA of mathematics
Prologue
The infinity of primes
The Fundamental Theorem of Arithmetic
Searching for the primes
The Riemann Hypothesis
Epilogue
Explorations
1. Dudeney’s prime number magic square
2. Unravel the number
3. Prime number pattern
4. Twin primes
5. A prime number riddle
Chapter 3: Zero: Place-holder and peculiar number
Prologue
Negative numbers
Analytic geometry
Division by zero
The zero exponent
Binary digits
Epilogue
Explorations
1. A contradiction
2. Binary arithmetic
3. Classic snail problem
4. Firefighter puzzle
5. Trick problem
Chapter 4: π (Pi): A ubiquitous and strange number
Prologue
Value
Transcendental numbers
Manifestations
Epilogue
Explorations
1. A way to calculate π
2. A circular walk
3. A reverse puzzle
4. String around a circle
5. Pythagoras meets π
Chapter 5: Exponents: Notation and discovery
Prologue
Exponential notation
Exponential arithmetic
Pascal’s Triangle
Logarithms
Epilogue
Explorations
1. Square numbers
2. Exponential arithmetic
3. Generational logarithm
4. A tough nut
5. Laws of exponents
Chapter 6: e: A very special number
Prologue
Mathematical connectivity
Euler’s identity
Epilogue
Explorations
1. A possible series for e
2. Another series
3. Plotting a function
4. Compound interest
5. The exponential function ex
Chapter 7: i: Imaginary numbers
Prologue
Quadratic equations
Complex numbers
Fundamental Theorem of Algebra
Epilogue
Explorations
1. Imaginary numbers
2. A square root
3. Conjugates
4. Conjugate arithmetic
5. Powers of complex numbers
Chaptet 8: Infinity: A counterintuitive and paradoxical idea
Prologue
Zeno’s paradoxes
The Liar Paradox
Galileo’s and Cantor’s paradoxes
Hilbert’s infinite hotel paradox
Epilogue
Explorations
1. Einstein’s paradox
2. Cantorian method
3. Alexia’s paradox
4. Lateral thinking puzzles
5. Cantor again
Chapter 9: Decidability: The foundations of mathematics
Prologue
Consistency
Axiomatic structure
Undecidability
Epilogue
Explorations
1. Smullyan’s Gödelian puzzle
2. Gardner’s box logic puzzle
3. Dudeney’s logic puzzle
4. A derivative of Gardner’s puzzle
5. Deception logic
Chapter 10: The algorithm: Mathematics and computers
Prologue
Algorithms
Computability
Epilogue
Explorations
1. Measuring algorithm
2. Movement algorithm
3. Alcuin’s river crossing puzzle
4. A complex version
5. Possibility versus impossibility
Answers
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
References and Bibliography
Index

Citation preview

OUP CORRECTED PROOF – FINAL, 20/11/19, SPi

Pythagoras’ Legacy

OUP CORRECTED PROOF – FINAL, 20/11/19, SPi

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PYTHAGORAS’ LEGACY Mathematics in Ten Great Ideas

By

marcel danesi University of Toronto Fields Institute for Research in Mathematical Sciences

1

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1 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Marcel Danesi 2020 The moral rights of the author have been asserted First Edition published in 2020 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2019945039 ISBN 978–0–19–885224–7 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY

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CONTENTS

Preface

ix

1. The Pythagorean theorem: The birth of mathematics

1

Prologue1 The Pythagorean theorem 3 Proof5 Discovery of √211 Practical uses 13 Pattern14 Fermat’s Last Theorem 17 Epilogue18 Explorations20

2. Prime numbers: The DNA of mathematics

22

3. Zero: Place-holder and peculiar number

37

Prologue22 The infinity of primes 24 The Fundamental Theorem of Arithmetic 26 Searching for the primes 28 The Riemann Hypothesis 32 Epilogue33 Explorations34 Prologue37 Negative numbers 40 Analytic geometry 43 Division by zero 46 The zero exponent 50 Binary digits 51 Epilogue53 Explorations54

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vi | Co nte nts 4. π (Pi): A ubiquitous and strange number

56

5. Exponents: Notation and discovery

69

6. e: A very special number

82

7. i: Imaginary numbers

93

Prologue56 Value58 Transcendental numbers 60 Manifestations61 Epilogue64 Explorations66 Prologue69 Exponential notation 71 Exponential arithmetic 73 Pascal’s Triangle 75 Logarithms77 Epilogue79 Explorations80 Prologue82 Mathematical connectivity 84 Euler’s identity 88 Epilogue89 Explorations90 Prologue93 Quadratic equations 95 Complex numbers 97 Fundamental Theorem of Algebra 99 Epilogue100 Explorations102

8. Infinity: A counterintuitive and paradoxical idea

104

9. Decidability: The foundations of mathematics

115

Prologue104 Zeno’s paradoxes 105 The Liar Paradox 107 Galileo’s and Cantor’s paradoxes 108 Hilbert’s infinite hotel paradox 112 Epilogue112 Explorations113 Prologue115 Consistency116

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Co nte nts

| vii

Axiomatic structure 118 Undecidability119 Epilogue122 Explorations123

10. The algorithm: Mathematics and computers

126

Prologue126 Algorithms127 Computability129 Epilogue131 Explorations133

Answers References and Bibliography Index

137 163 169

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PREFACE

The mathematics is not there till we put it there. —Sir Arthur Eddington (1882–1944)

“Number rules the universe.” These words were spoken in the 500s bce by one of history’s most reclusive, yet most significant personages, the Ionian Greek philosopher Pythagoras of Samos. To examine the truth of his own aphorism, he founded a society, known as the Pythagorean Brotherhood, in the Greek colony of Crotone in southern Italy. The ideas the society came up with—all in secret—became the foundation of a new and exciting discipline, math­em­at­ics, devised as an intellectual tool for discovering the “numbers” that are hidden in the make-up of the universe. The term “Pythagorean Brotherhood,” which reflects a previous translation, is likely to be a misnomer, because Pythagoras encouraged women to participate fully in his society. Late in life, he married one of the members, Theano, who was an artist and healer. She took over leadership of the Pythagorean society after her husband, and even though she faced persecution, continued to spread Pythagoreanism throughout Egypt and Greece alongside her daughters. The history of mathematics starts in earnest with one of Pythagoras’ most important proofs—the Pythagorean theorem. This proof was the first in a chain of ground-breaking ideas, all interconnected with each other, that turned mathematics into an “art of the mind.” That chain continues to be extended today. There would be no science, engineering, or philosophy without Pythagoras’ legacy. The purpose of this book is to sketch a history of that legacy by presenting and discussing ten of the greatest ideas in the mathematical chain. It aims to illustrate why mathematics can be designated an intellectual art, a creative enterprise that mirrors any art, from music to painting. Pythagoras actually connected music and mathematics into a system of knowledge that came to be called the Music of the Spheres.

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x | Pre face This book is the result of teaching an undergraduate course on the history of puzzles at the University of Toronto for decades, a course for which mathematical ideas constitute its core. There are many varied and excellent histories of mathematics, but, as I discovered in teaching that course, only a few focus on how mathematical ideas are interconnected imaginatively. Pythagoras saw mathematics as existing outside of the human mind, constituting a “secret code,” as it may be called here, that might bear answers to the very question of existence. Whether true or not, it is something of which, I believe, everyone should be cognizant. The Hungarian writer Arthur Koestler (1905–1983) made the following relevant statement in his book, The Sleepwalkers (1959): “Nobody before the Pythagoreans had thought that mathematical relations held the secret of the universe. Twenty-five centuries later, Europe is still blessed and cursed with their heritage.” Needless to say, there are many other great ideas that I could have treated in this book. I chose these particular ones because they stand out consistently across histories of mathematics. They are also the ones that come up continually in my own course. This book is intended both for those who love mathematics and those who do not. I hope to persuade the latter that mathematics is, well, more than math. For this reason, virtually no technical knowledge is required, since I have taken great care to ensure that everything is explained in plain language. Only basic high school mathematics is required. My goal is to show how mathematics can (and should) be envisioned as a creative art that is both pleasurable and understandable. As Mark Twain once wrote: “Intellectual work is misnamed; it is a pleasure, a dissipation, and is its own highest reward” (quoted in A Connecticut Yankee in King Arthur’s Court, chapter 28, 1889). To make this book as en­joy­ able as possible, each of its ten chapters ends with five exploratory problems that will allow readers to become engaged in the main ideas treated in the chapter directly. Answers and explanations are found at the back of the book. There are fifty such explorations, allowing readers to use this book also as a collection of math problems. I wish to express my gratitude to Daniel Taber and Katharine Ward for supporting this project and for their wonderful advice and suggestions. I am also thankful to Julian Thomas, for his superb work on this book. Any infelicities it may contain are my sole responsibility. Marcel Danesi, 2019 University of Toronto Fields Institute for Research in Mathematical Sciences

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1

The Pythagorean theorem The birth of mathematics

Reason is immortal, all else mortal. —Pythagoras (c. 580–500 bce)

Prologue If we join a stick of length 3 inches, another one of 4 inches, and one more of 5 inches into a triangular shape, the result will be a right triangle—a triangle in which one angle is a right angle (90°). The same type of triangle, bigger in size, would result if we used sticks of lengths 5 inches, 12 inches, and 13 inches (Figure 1.1). Any set of three random whole number lengths will not, however, ne­ces­sar­ ily produce such a triangle. The obvious question this raises is: Why do some three whole-number lengths, when combined, produce a right-angled triangle and others do not? The answer to this question is known as the Pythagorean theorem. It constitutes one of the first great ideas of mathematics. Its signifi­ cance is evident in the fact that it has been taught in schools throughout human history; it has had many scientific applications; it has shown up in numerous other mathematical ideas; it is used commonly in engineering and construction problems; and the list could go on and on. In a phrase, it is one of those ideas that have mattered to human evolution. The goal of this opening chapter is to look at the theorem—named after the ancient Greek mathematician, Pythagoras— and at why it is so profoundly important.

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2 |  TH E PY THAG O R E AN TH E O R E M

5

3 4

5

Figure 1.1  Right triangles

13 12

Sets of three whole number lengths, {a, b, c}, which produce right-angled tri­ angles when joined, were known before Pythagoras, who lived from circa 580 to 500 bce (Neugebauer and Sachs 1945). Around 2000 bce, the Egyptians knew that by stretching a rope around three stakes of 3, 4, and 5 units in length, a right triangle would be formed, and that the angle opposite the largest side of 5 units (the hypotenuse) would measure 90°. In the same era, the Babylonians also knew that some whole number triples, {a, b, c}, stood for the sides of a right triangle. Archeologists discovered a clay tablet, called Plimpton 322, that contains fifteen of them. British mathematician Ian Stewart (2012, 7) describes it as follows: It is a table of numbers, with four columns and 15 rows. The final column just lists the row number, from 1 to 15. In 1945 historians of science Otto Neugebauer and Abraham Sachs noticed that in each row, the square of the number (say c) in the third column, minus the square of the number (say b) in the second column, is itself a square (say a). It follows that a2 + b2 = c2, so the table appears to record Pythagorean triples. At least, this is the case provided four apparent errors are corrected. However, it is not absolutely certain that Plimpton 322 has anything to do with Pythagorean triples, and even if it does, it might just have been a convenient list of triangles whose areas were easy to calculate. These could then be assembled to yield good approximations to other triangles and other shapes, perhaps for land measurement.

Whatever they meant, the triples were known before Pythagoras, but, as Stewart suggests, without any likely awareness of the abstract structure they concealed, namely that c2 = a2 + b2. This was established by Pythagoras by the method of proof—the founding methodology of mathematics. Note that this structure manifests itself concretely in the above triples: {a, b, c)



{3, 4, 5}

® c 2 = a 2 + b2 2

2

® 5 =3 +4



2

® 25 = 9 + 16

{5, 12, 13} ® 13 = 5 + 12 ® 169 = 25 + 144 2

2

2

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The Pythagorean theorem  |  3



The Pythagorean theorem Pythagoras’s life and accomplishments are shrouded in mystery, leading to many myths and legends about him. Historians believe that he was born on the island of Samos in the eastern Aegean Sea. According to one legend, Pythagoras was the son of Apollo, an emissary from Zeus, and a miracle worker. What is known for certain is that he founded a secret society among aristocrats around 530 bce in the Greek colony of Crotone (in southern Italy). Those in the inner circle were called mathēmatikoi, which meant “those who pursued and loved knowledge.” Suspicious of the society, the people eventually killed most of its members in a political uprising. The surviving Pythagoreans continued on for a while, disappearing completely in the 400s bce. Proof of the theorem that c2 = a2 + b2 was only one of their seminal achievements in mathematics—they also discovered even and odd numbers, prime numbers, relations among num­ bers and geometrical figures, and several more. The Pythagoreans did not leave any written evidence of a proof, so we can only surmise how they carried it out. Concrete proofs of the theorem appeared in several important Chinese mathematical works—The Arithmetic Classic of the Gnomon and the Circular Paths of Heaven, dated by some historians to around the third century bce (Li Yan and Du Shiran 1987) and the Nine Chapters on the Mathematical Art, written by several generations of math­emat­ icians, with its final edition being traced back to the second century ce (Swetz and Kao 1977). All these proofs were based in geometrical reasoning—that is, on the demonstration that the sum of the areas of the squares on the two sides of a right-angled triangle {a, b} will always equal the area of the square on the hypotenuse {c}—the longest side, opposite the right angle (Figure 1.2). No one knows the actual proof that the Pythagoreans devised. Some his­tor­ ians believe it was a dissection proof, such as the following one. We first draw a right-angled triangle with sides {a, b} and hypotenuse {c}. Then, we construct a

C

A B

Figure 1.2  Geometrical structure of the Pythagorean theorem

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4 |  TH E PY THAG O R E AN TH E O R E M b a

b

Figure 1.3  Dissection proof of the Pythagorean theorem

a

c

c

b

c

c

a

a

b

square with length (a + b), the sum of the lengths of the two sides of the tri­angle. This is equivalent to joining four copies of the triangle together in the way shown in Figure 1.3. The diagram suggests how to carry out a proof as follows: 1 . The area of the internal square is c2, since c is the length of a side. 2. The area of the large square is (a + b)2. 3. This can be expanded (using high school algebra) to (a2 + 2ab + b2). 4. The area of a triangle, as you might also recall from high school, is half the base times the height. For each of the triangles joined together in the diagram it is, therefore, ½ab, where a is the base and b the height. 5. There are four of them; so the overall area covered by the four triangles is: 4(½ab) = 2ab. 6. If we subtract this from the area of the large square, (a2 + 2ab + b2) –2ab, we get (a2 + b2). 7. This now corresponds to the area of the internal square, which is equal to c2. 8. So, by the axiom of equality (things equal to the same thing are equal to each other), we can conclude that c2 = a2 + b2. 9. The proof is now complete. Of course, the original proof, whatever it was, would not have used algebraic symbols like those above, which were used here for the sake of convenience. The reasoning would have been similar, though. There are around 400 different proofs of the theorem, indicating how intri­ guing and significant it has been perceived to be across time (Loomis 1968; Posamentier 2017, 147–168). Among those who devised proofs for it was the Alexandrian mathematician Euclid (c. 300 bce), who also developed a proof for the converse of the theorem, namely that if the square on the longest side of any triangle is equal to the sum of the squares on the other two sides, then the triangle is a right triangle. Other famous proofs include the one by the twelfthcentury Indian mathematician Bhāskara (c. 1114–1185) in his treatise on

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Proof  |  5

­ athematics called the Līlāvatī (1150), by Renaissance painter and scientist m Leonardo da Vinci, and by the American statesman James Garfield, who was sworn in as the twentieth president of the United States on March 4, 1881. Garfield published his proof in an 1876 issue of the New England Journal of Education while still a member of the House of Representatives.

Proof Historians trace the notion of proof to the philosopher Thales of Miletus (c. 624–545 bce) (Maor 2007), defined as a demonstration that something is ne­ces­sar­ily true (or logically consistent) by connecting facts and axioms in a step-by-step logical fashion until an inescapable conclusion is reached. There is no one way to devise a proof, as the many proofs for the Pythagorean theorem saliently emphasize. However, there are several general types that were estab­ lished in Greek mathematics. One method involves applying previous knowledge in order to prove some­ thing as true or consistent. This is called proof by deduction. For the sake of illustration, consider the following typical problem of school geometry: Prove that the vertically-opposite angles formed when two straight lines intersect are equal. We start by drawing two straight lines AD and CB. Two of the four verticallyopposite angles formed by their intersection are labeled x and y. One of the angles formed between the two is labeled z (as shown in Figure 1.4). We are asked to prove that x and y (being vertically-opposite angles) are equal. There are, of course, two other vertically-opposite angles formed by the intersection of the two lines, but they need not concern us because the method of proof and the end result are the same. The proof hinges on previous know­ ledge—specifically that a straight line is an angle of 180°. Consider CB first. As a straight line, it is an angle of 180°. Now, notice that CB is composed of two smaller angles on the diagram, x and z. So, logically, these two must add up to 180°—a deduction that can be represented with the equation: x + z = 180°. A

B x

C

z

y D

Figure 1.4  A deductive proof

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6 |  TH E PY THAG O R E AN TH E O R E M Next, consider AD. Notice that it, too, is composed of two smaller angles on the diagram, y and z. Being a straight line, the two angles also add up to 180°—a fact that can be similarly represented with an equation: y + z = 180°. The two equations are listed below: (1) x + z = 180° (2) y + z = 180° These can be rewritten as follows: (3) x = (180° - z ) (4) y = (180° - z ) Since equation (3) shows that x is equal to (180° − z), and equation (4) that y is equal to the same expression, (180° − z), we deduce, by the axiom of equality (things equal to the same thing are equal to each other), that x = y. We now con­ clude that any two vertically-opposite angles produced by the intersection of two straight lines are equal, because we did not assign a specific value to either angle. The most remarkable thing about a proof such as this one is the way in which the various steps follow from each other, like the parts of a story—a math story in this case. Mathematician Ian Stewart (2008, 34) puts it aptly as follows: What is a proof? It is a kind of mathematical story, in which each step is a lo­gic­al consequence of the previous steps. Every statement has to be justified by referring it back to previous statements and showing that it is a logical consequence of them.

Another main type of proof is called proof by induction. This involves reason­ ing from particular facts in order to extract from them a general conclusion. Here is another high-school example: Develop a formula for the number of degrees in any polygon—a plane figure with at least three straight sides and three angles.

We start with a triangle first—the polygon with the least number of sides and angles—three. The sum of the angles in a triangle is, as established from a previ­ ous theorem, 1800. Next, we consider a quadrilateral—a four-sided polygon. ABCD in Figure 1.5 is one such figure, which has been divided into two tri­ angles (triangle ABD and triangle ACD) with a diagonal line. The diagonal line could have been drawn from B to C, of course. But the result would be the same. With this simple insight, we have discovered that the sum of the angles in the quadrilateral is equivalent to the sum of the angles in two triangles, namely 180° + 180° = 360°. Next, let’s consider the case of a penta­gon—a five-sided polygon. ABCDE in Figure 1.6 is one such figure.

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Proof  |  7 A

B

C

Figure 1.5  A quadrilateral divided into two triangles

D

A

B

C

D

E

Figure 1.6  A pentagon divided into three triangles

Since the pentagon can be divided into three triangles as shown (triangle ABC, triangle BCD, and triangle CDE), we have again uncovered a hidden fact—namely that the sum of its angles is equivalent to the sum of the angles in three triangles: 180° + 180° + 180° = 540°. Continuing on in this way, we will find that the number of triangles in a hexa­ gon (a six-sided polygon) is four, in a heptagon (a seven-sided polygon) five, in an octagon (an eight-sided polygon) six, and so on. We are now in a position to generalize as follows: the number of triangles that can be drawn in any polygon is “two less” than the number of sides that make up the polygon. For example, in a quadrilateral we can draw two triangles, which is “two less” than the number of its sides, or (4 – 2); in a pentagon, we can draw three tri­angles, which is, again, “two less” than the number of its sides, or (5 – 2); and so on. In the case of a tri­ angle, this rule also applies, since we can draw in it one triangle (itself), which is “two less” as well, or (3 – 2). In an n-gon—a polygon of n sides, where n is un­speci­ fied—we can thus draw (n – 2) triangles. Let’s summarize all this in chart form: Sides in a Polygon

Internal Triangles

3 (= triangle)

(3 – 2) = 1 triangle

4 (= quadrilateral)

(4 – 2) = 2 triangles

5 (= pentagon)

(5 – 2) = 3 triangles

6 (= hexagon)

(6 – 2) = 4 triangles

7 (= heptagon)

(7 – 2) = 5 triangles

…

…

n (= n-gon)

(n – 2) triangles

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8 |  TH E PY THAG O R E AN TH E O R E M Since we know that there are 180° in a triangle, then there will be (4 – 2) × 180° degrees in a quadrilateral, (5 – 2) × 180° degrees in a pentagon, and so on, up to (n – 2) × 180° in an n-gon (see Table 1.1). We have our required formula for the number of degrees in any polygon: 180° × (n – 2) or 180° (n – 2). Now, how can we be sure that this truly holds? Well, if the pattern applies to the nth-sided polygon and then can be shown to apply to the (n + 1)th case, we will have our definitive proof, because the “n” can be any number. In our polygon proof above, we showed that the nth case held, as expressed by 180° (n – 2). Now, let’s consider the (n + 1)th case. A polygon of (n + 1) sides will contain [(n + 1) – 2] triangles. If we let m = (n + 1), then we can rewrite this as follows: [(n + 1) – 2] = [(m – 2)] or just (m – 2). So, in this case the sum of the degrees of the polygon is 180° (m – 2), which has the exact same “symbol form” of a polygon of n sides, namely 180° (n – 2). We have sim­ ply replaced n with m. This shows that the formula applies to the (n + 1)th case. We can now rest assured that the proof always holds. Why? Because like a dom­ ino effect, we can use the same reasoning for the (n + 2)th case, the (n + 3)th case, and so on ad infinitum. A third type of proof is called reductio ad absurdum (“reduction to the absurd”), also known as proof by contradiction. This involves assuming the opposite of something and then showing that this assumption leads to an absurdity or a contradiction. The Pythagorean theorem, c2 = a2 + b2, implies that the sum of (a + b) is larger than c, that is, the sum of the lengths of the two sides is greater than the length of the hypotenuse, or in algebraic symbols, (a + b) > c. How do we prove this? We start by assuming the opposite, namely that (a + b) ≤ c, which is read as “(a + b) is less than or equal to c:” Table 1.1  Number of triangles in a polygon Sides

Number of triangles

Sum of degrees

3

(3 – 2) = 1

180° × (3 – 2) = 180°

4

(4 – 2) = 2

180° × (4 – 2) = 360°

5

(5 – 2) = 3

180° × (5 – 2) = 540°

6

(6 – 2) = 4

180° × (6 – 2) = 720°

7

(7 – 2) = 5

180° × (7 – 2) = 900°

…

…

…

n

(n – 2)

180° × (n – 2)

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Proof  |  9

1. Assume that (a + b) ≤ c. 2. Square both sides: (a + b)2 ≤ c2. 3. Expand (a + b)2: a2 + 2ab + b2 ≤ c2. 4. So, (a2 + b2) is much less than c2, as can be seen by simply considering the term 2ab from (3). The reason is that a2 + 2ab + b2 is a larger num­ ber than just (a2 + b2). In sum: a2 + b2 ≤ c2. 5. This is not possible logically, since we know already, from the Pythagorean theorem, that a2 + b2 = c2. 6. Thus, the original assumption that (a + b) ≤ c leads to a contradiction, and must therefore be false. 7. This implies that its opposite is true, namely that (a + b) > c. As David Berlinski (2013, 83) wryly quips, proof by contradiction “assigns to one half [of the mind] the position it wishes to rebut, and to the other half, the ensuing right of ridicule.” As an aside, note that the relation (a + b) > c holds for all triangles, with c being the longest side of any triangle. This does not change the above reasoning, of course. The first to lay the foundation for mathematics on methods of proof was Euclid (mentioned above). Euclid taught at the Museum, an institute in Alexandria, Egypt. He probably studied in Athens, moving to Alexandria at the invitation of the Egyptian ruler Ptolemy I. It is said that when Ptolemy asked him if there was a shorter way to learn geometry than by plowing laboriously through his text, the Elements of Geometry, Euclid replied ironically: “There is no royal road to geom­ etry.” Euclid ended each proof with the phrase, “Which proves what we wanted to demonstrate,” a phrase abbreviated in Roman times to QED, for Quod erat demonstrandum (“Which was to be demonstrated”). This abbreviation became the stamp of truth in mathematics—remaining so to this very day. The ancient mathematicians were aware that not everything in mathematics can be proved. When something is resistant to known methods of proof it is  called a conjecture. As an example, consider the conjecture by Prussian ­math­em­at­ician Christian Goldbach (1690–1764) in the eighteenth century, who discovered that he could write every even integer greater than 2 as the sum of two primes (numbers that have no divisors other than themselves and 1):

4=2+2 6=3+3 8=3+5 10 = 5 + 5 or 7 + 3 ¼

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0 |  TH E PY THAG O R E AN TH E O R E M We can always find two primes with which to write an even number. But all we can do is assume this to be true. So far, no one has been able to prove Goldbach’s conjecture, at least to everyone’s satisfaction. Goldbach also discovered that any number greater than 5 could be written as the sum of three primes:

6=2+2+2 7=2+2+3 8=2+3+3 9=3+3+3 10 = 2 + 3 + 5 11 = 3 + 3 + 5 ¼

Actually, these conjectures have spawned much research, leading to a number of interesting findings. Computer-based searches have established the validity of Goldbach’s two conjectures for very large numbers. But for mathematicians this is not good enough—only a proof will satisfy them. The proof, if it will ever come, involves prime numbers which, as will be discussed, present many intractable problems. There are many conjectures that tantalize math­emat­ icians, yet shut them out from finding a proof. But this certainly does not stop them from trying. The history of proofs constitutes the history of mathematics. The more elu­ sive a proof is, the more it is hunted down, even if it may seem to have no implications above and beyond the proof itself. French mathematician and sci­ entist Henri Poincaré (1854–1912) formulated a conjecture in 1904 that can be reduced to a simple assertion for our purposes—any object without holes in it has to be a sphere. As obvious as this might seem, there might exist some object (real or hypothetical) that contradicts this assertion. Poinacaré’s conjecture was finally proved by Russian mathematician Grigori Perelman in 2002, who posted his 400-plus-page proof on the Internet (O’Shea 2007; Gessen 2009). Since antiquity, the search to convert conjectures into proofs has been the fuel behind a large part of mathematical progress, revealing how the human imagination is relentless in exploring problems. As mathematician David Wells (2012, 140) has aptly pointed out: Proofs do far more than logically certify that what you suspect, or conjecture, is actually the case. Proofs need ideas, ideas depend on imagination and im­agin­ation needs intuition, so proofs beyond the trivial and routine force

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D i s cov ery of √ 2  |  



you to explore the mathematical world more deeply—and it is what you dis­ cover on your exploration that gives proof a far greater value than merely con­ firming a fact.

Discovery of √2 When the Pythagorean theorem is applied to an isosceles right-angled triangle with each of its two sides equal to 1, a very strange number for the length of the hypotenuse, √2, emerges (Figure 1.7). As it turns out the length of this line is indefinite, 1.4142136…, despite the evidence of our eyes that it has a particular length on the diagram. The Pythagoreans were so distressed by this discovery that they decided to keep it secret, because it did not fit in with their view of a harmonious world mirrored in the properties of whole numbers. Yet, there it was, appearing as the hypot­en­use of an isosceles triangle. It was Euclid who eventually accepted √2 as le­git­im­ate, labeling it as an irrational (“not rational”) number. A rational number can be expressed as a ratio (fraction) of two integers. Expressed as a decimal, a rational number always terminates after a finite number of digits (after the deci­mal point) or else repeats the same sequence of digits over and over: Ratio

Decimal

3/1

3.0

3/4

0.75

6/5

1.2

15/21

0.714285. . . (repeated ad infinitum)

…

…

The expression √2 does not fit into this pattern. So, where or how does it fit in with the set of numbers? The first thing to do is to show that it is not a rational

45° 1

√2

45° 1

Figure 1.7  The appearance of √2

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2 |  TH E PY THAG O R E AN TH E O R E M number. Euclid started by noting that the general form of a rational number is p/q. The exception is that q cannot equal 0, as will be discussed subsequently. In the case of the integers, the denominator q is always 1—for example, 4 is really 4/1, 5 is 5/1, and so on. Euclid proved that √2 could not be written in the form p/q. He did this by assuming the contrary—that it could in fact be written in that form—thus using reductio ad absurdum: 1. Assume that √2 = p/q. 2. Square both sides: (√2)2 = p2/q2. 3. Therefore: 2 = p2/q2 (the square of √2 is 2). 4. Multiply both sides by q2, thus eliminating the cumbersome ­denominator on the right side of the equation: 2q2 = p2. 5. Now, p2 is an even number because it equals 2q2. (Any number, n, multiplied by 2 will produce an even number, which can be expressed with the general formula 2n, or two times any number.) 6. It follows that p itself is even, since p2, as just demonstrated, is equal to an even number, namely 2q2. The square of an even number is always even. You can try this out yourself with even numbers at random. 7. We can now use the general formula for an even number: p = 2n. 8. We plug this into equation (4) above: 2q2 = p2 → 2q2 = (2n)2 = 4n2. 9. To recapitulate: 2q2 = 4n2. 10. This equation can be simplified by dividing both sides by 2: q2 = 2n2. 11. This shows that q2 is an even number, and thus that q itself is an even number and can be written as 2m (to distinguish it from 2n). Therefore, q = 2m. 12. Now, Euclid went right back to his original assumption—namely that √2 was a rational number, or that √2 = p/q. 13. In this equation, he substituted what he had just proved, namely, that p = 2n and q = 2m. Therefore: √2 = 2n/2m. 14. By simplifying the equation, we get: √2 = n/m. Now, the problem is that we find ourselves right back to where we started. We have simply ended up replacing p/q with n/m. We could continue on in this way, always coming up with a ratio with different numerators and de­nom­in­ ators (n/m, x/y, . . .) ad infinitum. We have thus reached an absurdity. What caused it? The fact that √2 was assumed to have the form p/q. It obviously does not. Thus, Euclid proved that √2 is not a rational number. Interestingly, the ­so-called Yale Tablet (YBC 7289), which is traced back to around 1700 bce,

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Practi ca l u s e s  |  3



shows that the ancient Babylonians may also have been aware of irrational numbers. It contains a sexagesimal approximation of √2 (Neugebauer and Sachs 1945). Sexagesimal (in this case) refers to a fraction with a denominator of 60. The unanticipated appearance of √2 illustrates that discovery in mathematics is often serendipitous. Without the irrationals, the calculus would not have come into existence since, as Richard Dedekind so eloquently argued, it deals with continuous magnitudes (like √2), not discrete ones.

Practical uses The Pythagorean theorem has had so many practical applications to science and engineering that it would take a large tome simply to list them. A con­ crete example will suffice. Suppose that a tunnel must be dug right through the middle of a mountain. Since the length of the tunnel cannot be measured phys­ic­al­ly, the Pythagorean theorem suggests a plan for doing so without direct measurement. Point A on one side of the mountain and point B on the other are chosen such that both points remain visible from C to the right. C is chosen so that angle ACB is a right angle (90°). Then, by aligning A with A´ (the entrance to the mountain on one side) and B with B´ (the entrance to the mountain on the other side) the required length can be seen to be A´B´ (Figure 1.8). The problem is now solved easily with the Pythagorean theorem. 1. Measure AC and BC. 2. Plug the values into the Pythagorean formula AB2 = AC2 + BC2. A

A´ C

90° [AB—(AA´ + BB´)] = length of tunnel

B´ B

Figure 1.8  An engineering problem

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4 |  TH E PY THAG O R E AN TH E O R E M 3. This yields a measure for AB. 4. Measure the distances AA´ and BB´. 5. Subtract these two distances (added together) from AB to get the length of A´B´; that is, AB — (AA´ + BB´) = A´B´. 6. That is the length required to dig a tunnel through the mountain. As this shows, knowledge of how to represent a real-life physical situation in a mathematical way is a truly remarkable achievement of human evolution. The Pythagorean theorem allows us to solve engineering problems mathematically before we tackle them in real terms. With the emergence of geometry as a the­ or­et­ic­al discipline, the ancient engineers were able to represent a situation mathematically before carrying out a physical intervention. As Ian Stewart (2008, 46) so aptly observes: Using geometry as a tool, the Greeks understood the size and shape of our planet, its relation to the Sun and the Moon, even the complex motions of the remainder of the solar system. They used geometry to dig long tunnels from both ends, meeting in the middle, which cut construction time in half. They built gigantic and powerful machines, based on simple principles like the law of the lever.

Pattern The Pythagoreans studied patterns between numerical and geometrical forms, coming up with remarkable discoveries. One of these was that of figurate numbers—numbers that can be displayed geometrically. For example, square inte­ gers, such as 12 (= 1), 22 (= 4), 32 (= 9), and 42 (= 16) can be shown to have the form of squares (Figure 1.9). This simple correlation suggests a hidden pattern—every square number is the sum of consecutive odd integers.

1 =1 4 =1+ 3 9 =1+ 3 + 5 16 = 1 + 3 + 5 + 7 25 = 1 + 3 + 5 + 7 + 9 ¼

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Pattern  |  5

16 9

4 1

1×1

2×2

3×3

4×4

Figure 1.9  Square numbers (from Wikimedia Commons)

1

3

6

10

Figure 1.10  Triangular numbers (from Wikimedia Commons)

Intrigued by this kind of discovery, the Pythagoreans studied other ­geom­etry– numerical correlations. One of these was triangular numbers (see Figure 1.10). The number of dots in the figure for the first triangular number is 1; the number that makes up the figure for the second triangular number is 3 = 1 + 2; the number that makes up the third triangular number is 6 = 1 + 2 + 3; and so on. From this, it becomes obvious that each successive triangular number is obtained by adding a row of dots containing one more dot than the number of dots in the previous triangular number. This leads to the discovery that the nth triangular number is the sum of the first n counting numbers:

1st triangular number :

=

1

=

1

nd

=

3

=

1+ 2

rd

=

6

=

1+ 2 + 3

th

=

10 =

1+ 2 + 3+ 4

th

=

15 =

1+ 2 + 3+ 4 + 5

th

=

21 =

1+2 +3+ 4 + 5+ 6

2 triangular number : 3 triangular number : 4 triangular number : 5 triangular number : 6 triangular number : ¼ nth triangular number :

=



1+ 2 + 3 +¼+ n

The Pythagoreans came up with many other fascinating discoveries. Two were called amicable and perfect numbers. The numbers 284 and 220, for instance, are called amicable because the proper divisors of one of them, when added

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6 |  TH E PY THAG O R E AN TH E O R E M together, produce the other. A proper divisor of a positive integer, n, is any ­div­isor other than n itself. The proper divisors of 284 are 1, 2, 4, 71, and 142, and the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110 (Posamentier 2017, 46): 220 = 284 =

1 + 2 + 4 + 71 + 142 (the proper divisorsof 284) 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 (the proper divisors of 220)

A perfect number is one that equals the sum of its proper divisors. For example, the proper divisors of 6 are 1, 2, and 3, since all three numbers will divide evenly (without a remainder) into 6. Now, if we add these together we get 6 = 1 + 2 + 3. The next perfect number is 28. Its proper divisors are 1, 2, 4, 7, 14, and if we add these together we get 28 = 1 + 2 + 4 + 7 + 14 . Very few perfect numbers have been discovered. Numbers that are not perfect are called either excessive or defective. An excessive number is one whose proper divisors, when added together, produce a result that exceeds its value. The number 12, for example, is excessive because the sum of its proper divisors, 1, 2, 3, 4, and 6 (1 + 2 + 3 + 4 + 6 = 16) exceeds its value. A defective number is one whose proper divisors, when added together, produce a result that is smaller than its value. One ex­ample is 8, since the sum of its proper divisors 1, 2, and 4 (1 + 2 + 4 = 7) is less than its value. Now, one can ask if this is nothing more than “number play.” As it turns out, these numbers have been used in various domains of mathematics, leading to subsequent discoveries. A perfect number did not exist until the Pythagoreans discovered it by playing with patterns. From this, other discoveries were made possible. Euclid proved in his Elements that even perfect numbers can be repre­ sented with the formula, 2n−1(2n–1), provided that (2n–1) is a prime number. Table 1.2 shows the first four perfect numbers.

Table 1.2  Euclid’s formula for perfect numbers n Euclid’s formula

Perfect number

2

2n−1(2n−1) = 21 (22 − 1)

2×3=6

3

2n−1(2n−1) = 22 (23 − 1)

4 × 7 = 28

5

2 (2 −1) = 2 (2 − 1)

16 × 31 = 496

7

2 (2 −1) = 2 (2 − 1)

64 × 127 = 8128

n−1 n−1

n n

4 6

5 7

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F ermat ’ s La st Theorem  |  7

No odd perfect number has ever been discovered, but no one has ever proved why this is so. This is now one of the oldest unsolved problems in mathematics. Incidentally, Euclid’s formula became part of prime number theory, as will be discussed in the next chapter. The Pythagoreans believed that perfect numbers had mystical or divine meanings. Even the early medieval religious thinker, St. Augustine of Hippo (354–430 ce) made the following observation about them in his theological treatise, City of God, written around 420 ce: Six is a perfect number, not because God created the world in six days, rather than the other way around. God created the world in six days because six is perfect.

Fermat’s Last Theorem French mathematician Pierre de Fermat (1601–1665) was reading Diophantus’ Arithmetica, centuries later, when he wrote, in the margin of that book, the fol­ lowing enigmatic words: To divide a cube into two cubes, a fourth power, or in general any power what­ ever above the second, into two powers of the same denomination, is im­pos­ sible, and I have assuredly found an admirable proof of this, but the margin is too narrow to hold it.

Fermat claimed (or believed) that his proof would show that only for the value n = 2 do solutions of cn = an + bn exist: namely, the Pythagorean equation c2 = a2 + b2. Fermat’s enigmatic statement was revealed by his son, Samuel de Fermat, who was cataloging his father’s unpublished papers after his death in 1665. Samuel tried doggedly to find his father’s promised proof, but he could not. It came to be known as Fermat’s Last Theorem. For centuries afterward, mathematicians across the world were intrigued by Fermat’s claim, trying to come up with a proof, but always to no avail, although a number of special cases were settled. German mathematician Karl Friedrich Gauss (1777–1855) proved that c3 = a3 + c3 had no positive integral solutions; Fermat himself proved the untenability of c4 = a4 + b4, in 1659; French math­em­ at­ician Adrien-Marie Legendre (1752–1833) showed that c5 = a5 + c5 had no solutions; and Johann Dirichlet (1805–1859) demonstrated that c14 = a14 + b14 had no solutions. Ernst E. Kummer (1810–1893) proved the theorem for all but three numbers less than 100. But no general proof had come forth until June

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8 |  TH E PY THAG O R E AN TH E O R E M 1993, when British mathematician Andrew Wiles declared that he had finally proved it. In December of that year, however, mathematicians found a gap in his argument. In October of 1994 Wiles, together with Richard L. Taylor, filled that gap to virtually everyone’s satisfaction. The Wiles–Taylor proof was pub­ lished in May 1995 in the Annals of Mathematics (see Wiles 1995, Taylor and Wiles 1995). Wiles’ proof is the result of connecting and modifying previous ideas and formulas used to explore the theorem. Two ideas, in fact, were crucial to his proof, the elliptic curve and the modular form. The history of such forms and the actual details leading to the Wiles–Taylor proof are too complex for the present purposes. Fermat’s Last Theorem still haunts some math­emat­ icians, however, for the simple reason that the Wiles–Taylor proof was cer­ tainly not what Fermat could have possibly envisioned. The proof depended on math­em­at­ic­al work subsequent to Fermat. In a pure sense, therefore, the Wiles–Taylor proof is not a historical resolution to Fermat’s Last Theorem. Fermat left behind a true mathematical mystery. What possible “simple proof ” could he have been thinking of as he read Diophantus’ Arithmetica? As Ian Stewart (1987, 48) aptly puts it, “Either Fermat was mistaken, or his idea was different.”

Epilogue The Pythagoreans were dedicated to the cause of discovering truth with math­ ematics. Pythagoras encouraged women to participate fully in his secret soci­ ety, at a time when this was not socially acceptable. The indirect evidence for this is that, according to a widely held legend, he married one of his students, Theano. An accomplished cosmologist and healer, she headed the Pythagorean society after her husband’s death, and, even though she faced persecution, con­ tinued to spread the Pythagorean philosophy throughout Egypt and Greece, alongside her daughters. Do we discover mathematics or do we invent it and then discover that it works? Was √2 “somewhere” in the world ready to be discovered, or did the Pythagorean theorem produce it inadvertently? Plato believed that math­em­at­ ic­al ideas pre-existed in the world—the mathematician simply gives them form. Like the sculptor takes a clump of clay and gives it the form of a human body, so too mathematicians take a clump of reality and give it symbolic (math­em­at­ic­al) form. The truth is already in the clump. Some find this

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E pi log u e  |  9

­ erspective difficult to accept, leaning towards constructivism, or the idea p that mathematical ideas are constructed to tell us what we want to know about the world. But, as Berlinski (2013, 13) suggests, the Platonic view is not so easily shrugged off: If the Platonic forms are difficult to accept, they are impossible to avoid. There is no escaping them. Mathematicians often draw a distinction between con­ crete and abstract models of Euclidean geometry. In the abstract models of Euclidean geometry, shapes enjoy a pure Platonic existence. The concrete ­models are in the physical world.

There might even be a neurological basis to the Platonic view. As Pierre Changeux (2013, 13) muses, Plato’s trinity of the Good (the aspects of reality that serve human needs), the True (what reality is), and the Beautiful (the aspects of reality that we see as pleasing) is actually consistent with modern-day neuroscience: So, we shall take a neurobiological approach to our discussion of the three universal questions of the natural world, as defined by Plato and by Socrates through him in his Dialogues. He saw the Good, the True, and the Beautiful as independent, celestial essences of Ideas, but so intertwined as to be in­sep­ ar­able . . . within the characteristic features of the human brain’s neuronal organization.

As the historian of science Jacob Bronowski (1973, 168) has insightfully writ­ ten, we hardly recognize today how important the Pythagorean theorem was to human progress. It was a discovery that reached out into the world, brought it into human consciousness, and then changed the world: The theorem of Pythagoras remains the most important single theorem in the whole of mathematics. That seems a bold and extraordinary thing to say, yet it is not extravagant; because what Pythagoras established is a fundamental characterization of the space in which we move, and it is the first time that it is translated into numbers. And the exact fit of the numbers describes the exact laws that bind the universe. If space had a different symmetry the theorem would not be true.

We could conceivably live without the Pythagorean theorem. After all, it simply tells us formally what we know intuitively—that a diagonal distance is shorter than taking an L-shaped path to a given point. But its formulation projected intuition into the light of understanding. And we are much more enlightened as a result.

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20 |  TH E PY THAG O R E AN TH E O R E M

Explorations As mentioned in the preface, the final section of a chapter presents five “­ex­plor­ations” based on the subject matter of the chapter. Answers are found at the back.

1. Bhāskara’s snake and peacock puzzle This is a puzzle devised by the Indian mathematician Bhāskara (mentioned above) in his famous work, the Līlāvatī, which involves use of the Pythagorean theorem: A snake’s hole is at the foot of a pillar which is 15 cubits high and a peacock is perched on its summit. Seeing the snake, at a distance of thrice the pillar’s height, gliding toward his hole, the peacock stoops obliquely upon him. Say quickly at how many cubits from the snake’s hole do they meet, both proceed­ ing an equal distance?

2.  Abu al-Wafa’s assembly puzzle The medieval Persian scholar Abu al-Wafa’ Buzjani (940–998 ce) devised the following tricky puzzle about triangles, which challenges us to think outside the box: Draw three identical triangles, and one smaller triangle, similar to them in shape, so that all four can be made into one large triangle. Here’s a hint. The smaller triangle does not have to be drawn separately. It will emerge as a shape in the middle, after joining the three larger triangles in a particular way.

3.  Apollonius’ problem The following ingenious problem in geometry was posed in a lost work by Apollonius of Perga (c. 262–190 bce), but reported by Pappus of Alexandria (c. 290–350 ce) in his Collection (c. 340 ce). This type of problem again involves a form of geometric thinking that is also highly imaginative. Can you construct a circle that is tangent to three others in the plane—that is, that touches three circles at once? (See Figure 1.11.)

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 E xplorati on s  |  2

Figure 1.11  Apollonius’ problem

4.  Measurement problem Here is a measurement problem that requires the Pythagorean theorem for its solution: The length of a rectangular floor is twice its width. The area of the floor is 32 square feet. In the lower left corner there is a bug that wants to get to the op­pos­ ite corner. What is the shortest path for the bug?

5.  Gardner’s tricky triangle puzzle The following puzzle was invented by the late Martin Gardner, one of the great puzzle-makers of the twentieth century. Given an obtuse triangle (a triangle with an angle greater than 90°; see Figure 1.12), is it possible to cut the triangle into seven smaller triangles, all of them acute (triangles with all of their angles less than 90°)?

Figure 1.12  Gardner’s problem

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2

Prime numbers The DNA of mathematics

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate. —Leonhard Euler (1707–1783)

Prologue As we saw in the previous chapter, the Pythagoreans were intrigued with math­ em­at­ic­al patterns. They must have noticed, at some point, that some whole numbers could be decomposed into factors, and others could not. For example, the number 12 has the factors 3 × 4 = 12 and 2 × 6 = 12; 4 can itself be expressed as the product of 2 × 2 = 4, and 6 as the product of 3 × 2 = 6. So, the number 12 can be decomposed into its irreducible skeletal factors as follows:

12 = 3 ´ 4 = 3 ´ (2 ´ 2) = 3 ´ 2 ´ 2 = 3 ´ 22 12 = 2 ´ 6 = 2 ´ (3 ´ 2) = 2 ´ 3 ´ 2 = 3 ´ 22

So, 3 and 2 are the skeletal factors of 12, since these numbers cannot be decom­ posed further. This yielded a fundamental insight into the structure of the ­integers that was a truly paradigmatic event in the history of mathematics. The  Pythagoreans had discovered, in effect, that some numbers cannot be

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 Pro lo g u e  |  23

decomposed further and are the building blocks of all other numbers. Thymaridas of Paros (c. 400–350 bce), a member of the Pythagoreans, called prime numbers rectilinear, because they can only be represented in one dimen­ sion on a number line, whereas composite numbers, such as 6, can be repre­ sented in two dimensions by a rectangle of sides 2 and 3. A prime number, more specifically, is any positive integer that has no factors other than itself and the number 1 (which is a factor in every single number). For instance, the number 7 is prime because 7 and 1 are its only factors: 7 × 1 = 7. The discovery of the prime numbers was monumental for early mathematics. They are its DNA, leading to fundamental questions about the nature of num­ bers. Below are a few of these: 1. Since the primes become scarcer as the numbers grow larger—there are twenty-five primes in the first 100 integers, between 101 and 200, there are twenty-one primes, between 201 and 300, there are sixteen primes, and so on in decreasing fashion. So, do they eventually come to an end? 2. Two primes that differ by 2, such as 5 and 7, are called twin primes. How many twin primes are there? 3. As we saw in the previous chapter, Christian Goldbach found that he could write every even integer greater than 2 as the sum of two primes: 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, etc. Can this ever be proved? 4. Is there a formula that generates primes and only primes? French scholar Marin Mersenne (1588–1648) thought that he had come up with one such formula: (2n−1). The formula does indeed generate a high number of primes, known as Mersenne numbers. But many exceptions have been found. Euclid answered the first question above; the others remain largely resistant to proof, for some reason. Some proofs, nonetheless, have come about ser­en­ dip­it­ous­ly. Mathematicians have proved that there is at least one prime ­number  between any number greater than 1 and its double. For example, between 2 and its double 4 there is one prime, 3; between 11 and its double 22 there are three primes, 13, 17, and 19; between 50 and 100 there are ten primes, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. So, we are always guaranteed to find primes between n and 2n if we make n greater than 1 (n > 1). This started as a conjecture by French mathematician Joseph Louis François Bertrand (1822–1900), and first proved by Russian mathematician Pafnuty

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24 |  PR I M E N U M B ER S : TH E D NA O F MATH E MATI C S Lvovich Chebyshev (1821–1894). Subsequent proofs are those by Indian mathematician Srinivasa Ramanujan (1887–1920), and the Hungarian Paul Erdös (1913–1996). This chapter deals with the prime numbers. There are many books and web­ sites that discuss the theoretical issues that prime numbers have raised over the centuries. The goal here is to look at the primes in a generic and non-technical fashion, because they matter deeply to mathematics, not to mention the pro­ gress of human civilization. Pythagoras even thought that prime numbers were part of a secret code which, if deciphered, would allow us to unlock the secrets and mysteries of the cosmos itself.

The infinity of primes Shortly after the Pythagoreans discovered the primes, a fundamental question arose: Do the primes end? They do not; they go on ad infinitum. A proof was devised by Euclid in Book IX of his Elements. He used proof by contradiction, also known as reductio ad absurdum, as discussed previously (Hardy and Woodgold 2009), which is summarized here. We start with the contrary hypothesis—namely that there is a finite set of primes (P):

P = {p1 , p2 , p3 , ¼ pn }

The symbol pn stands for the last (and largest) prime; each of the other symbols stands for a specific prime in their order of occurrence, starting with the small­ est one: p1 = 2, p2 = 3, p3 = 5, p4 = 7, p5 = 11, and so on. Next, we multiply all the primes in the set, in order to produce a composite number, C, that is divisible by each of the primes in P; that is, by p1, p2, p3, . . . pn, since they all divide evenly into C.  The decomposition of C into its prime factors (factors which, when multiplied, produce C), can thus be shown as follows:

C = {p1 ´ p2 ´ p3 ´ ¼ ´ pn }

At this point Euclid came up with a clever thought: What would happen if we add 1 to C? In terms of our formula above this means adding 1 to both sides:

C + 1 = {p1 ´ p2 ´ p3 ´ ¼´ pn } + 1

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Th e infinit y o f pr im e s  |  25



Let’s call the number produced in this way M, rather than C + 1, for the sake of convenience: M = {p1 ´ p2 ´ p3 ´ ¼ ´ pn } + 1



Clearly, M is not divisible by any of the primes in P, because a remainder of 1 would always result. So, the number M is either: (1) a prime number itself that is evidently not in P and thus greater than pn, or (2) a composite number with a prime factor that, similarly, cannot be found in the set {p1, p2, p3, . . . pn} and thus is also greater than pn. Either way, there must always be a prime number greater than pn. So, the set of primes is not finite, but goes on infinitely. As happens in mathematics, Euclid’s proof opened up many new vistas for the imagination to explore. How can we tell if a number is prime? Is there a pattern to the occurrence of the sequence of primes? Is there a rule that will generate primes and only primes? One of the first to put forward a method for identifying primes among numbers was the Greek geographer and astronomer Eratosthenes (c. 275–194 bce) in the 200s bce. He constructed a ten-by-ten square listing the first 100 numbers. It is known figuratively as the Sieve of Eratosthenes (Figure 2.1). We start with 1; it is neither prime nor composite, so we can strike it out. The first prime number is 2. We keep it and we cross out every second number after 2 because such a number would have 2 as a factor. The next prime is 3; so, we keep it and we cross out every third number after 3, again because such a num­ ber would have 3 as a factor. Now, we move on to 4, but it is already crossed out, so we go on to 5, the next prime. We keep 5 and cross out every fifth number 1

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Figure 2.1  Sieve of Eratosthenes

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26 |  PR I M E N U M B ER S : TH E D NA O F MATH E MATI C S 1

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Figure 2.2  Primes in the sieve

after 5, for the same reason that it would have 5 as a factor. We continue this process, until only the primes are left—the numbers that were not crossed out (Figure 2.2). These can be thought to have passed through a sieve (strainer) that has caught all the primes. Eratosthenes’ sieve ends up straining 25 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. Needless to say, it would be a huge task to construct a sieve to catch the primes among just the first 10,000 numbers—a 100-by-100 sieve. So, it really is not a practicable method of prime number identification, but it shows ingenuity nonetheless. With the aid of modern computers, the sieve has been expanded considerably, with different arrangements of the numbers to see if a pattern may be concealed within them. But no underlying rule for generating all primes has yet been found. As Bellos (2014, 258) so aptly puts it: “That the primes are so easily defined, yet their distribution is so capricious, is one of the earliest, and deepest surprises in math.”

The Fundamental Theorem of Arithmetic In Book IX of the Elements, Euclid proved that every whole number can be written as a product of prime numbers in exactly one way. This has come to be known as the Fundamental Theorem of Arithmetic. At the start of this chapter, we determined that the prime factors 3 and 2 uniquely make up the number 12 (= 3 × 22).

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Th e F u ndam e ntal Th e o r e m o f Ar ithm etic  |  27

To find the distinctive prime factors of a number, we search first for the smallest prime that divides it leaving no remainder. For example, to find the prime factors of 220, we begin by dividing it by 2, the smallest prime in this case: 220 ÷ 2 = 110. We now divide the quotient (110) by 2: 110 ÷ 2 = 55. Now, 55 cannot be divided by 2 without leaving a remainder. So, we try the next prime, 3, finding that it also does not divide into 55 without leaving a remainder. But the next prime in order, 5, does: 55 ÷ 5 = 11. The quotient 11 is a prime number. So, the operation stops and we now have our unique prime factors for 220:

220 = 2 ´ 2 ´ 5 ´ 11 = 22 ´ 5 ´ 11

Euclid’s method will work every time. It is the first “algorithm”—a step-by-step set of procedures that never fails to produce the required result. It is also a model of factorization itself, since it breaks the operation down into its consti­ tutive steps. Incidentally, the reason why the number 1 is not classified as a prime is because of the Fundamental Theorem of Arithmetic (Benjamin 2015, 141). If it were, then the theorem would not be true, since it states that each composite number can be decomposed into a unique set of factors. The number 1 can be added to any set of factors ad infinitum. Consider again the number 12 and its unique prime factors: 12 = 22 × 3. If we include 1, then 12 can also be decomposed into: 12 = 1 ´ 22 ´ 3 12 = 1 ´ 1 ´ 22 ´ 3 = 12 ´ 22 ´ 3 12 = 1 ´ 1 ´ 1 ´ 22 ´ 3 = 13 ´ 22 ´ 3 12 = 1 ´ 1 ´ 1 ´ 1 ´ 22 ´ 3 = 14 ´ 22 ´ 3 ¼

= 1n ´ 22 ´ 3



The factor 1n can be added to any number, prime or composite, but this changes nothing—hence it does not alter the factorization process in any way. Euclid did not provide a proof of his theorem. That had to await German math­em­at­ ician Karl Friedrich Gauss (1777–1855), who proved it in his Disquisitiones Arithmeticae of 1801. The importance of the theorem is unmistakable; it asserts that every integer greater than 1 is either a prime number or the product of two or more prime numbers. Euclid also found a fundamental property of prime numbers through his theorem, called Euclid’s lemma: If a prime number divides

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28 |  PR I M E N U M B ER S : TH E D NA O F MATH E MATI C S the product of two positive integers, a and b, then it divides either one or both. Here are two examples: Example 1 a = 6 = 3 × 2 → unique prime factors: {2, 3} b = 10 = 5 × 2 → unique prime factors: {2, 5} ab = 60 = 6 × 10 = 3 × 2 × 5 × 2 = 3 × 5 × 22 → unique prime factors: {2, 3, 5} Conclusion: the prime factor 2 divides the product (ab) and both a and b; the prime factor 5 divides b and ab. Example 2 a = 18 = 32 × 2 → unique prime factors: {2, 3} b = 25 = 52 → unique prime factor: {5} ab = 450 = 2 × 32 × 52 → unique prime factors: {2, 3, 5} Conclusion: the prime factor 2 divides a and ab; the prime factor 5 divides b and ab.

Euclid’s work on prime numbers made it possible to unveil a host of patterns hidden within the integers. It also stimulated intensive research into solving the unique problems that the primes presented to the ancient mathematicians, including the possibility of finding a general rule to generate all the primes.

Searching for the primes As discussed (chapter 1), Euclid showed that the formula [2n−1 (2n−1)] would generate all the even perfect numbers when the expression (2n−1) is a prime number. For example, if n = 2, then (2n−1) = (22−1) = (4 − 1) = 3. Since this is a prime number, we can now use Euclid’s formula to generate the perfect number 6:

(

)

(

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é2n -1 2n - 1 ù = é22 -1 22 - 1 ù = é21 ( 4 - 1) ù = ( 2 ) ( 3 ) = 6 û ë û ë û ë

In the seventeenth century, the French priest and teacher, Marin Mersenne (1588–1648), believed that Euclid’s expression (2n−1) might be the elusive ­general rule for primes, if n was itself prime. Let’s look at a few cases:

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S e archin g fo r th e pr im e s  |  29





n = 2 ( prime )



(2



n = 3 ( prime )

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)

- 1 = 22 - 1 = 4 - 1 = 3 ( prime )



(2



n = 4 ( composite )

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- 1 = 23 - 1 = 8 - 1 = 7 ( prime )



(2



n = 19 ( prime )



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- 1 = 24 - 1 = 16 - 1 = 15 ( composite )

( 2 –1) = 2 n

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- 1 = 524,288 - 1 = 524, 287 ( prime )

Mersenne claimed that (231−1), (267−1), and (2257−1) were prime—since 31, 67, and 257 (the values of n) are prime. The first one was proven to be a prime in 1772, but the latter two turned out to be composite. So, Mersenne’s formula does indeed generate a significant number of primes, but it does not always do so. In 1996, computer programmer George Woltman started the Great Internet Mersenne Prime Search (GIMPS), with volunteers from across the globe who are still searching for larger and larger Mersenne primes. The hope is, presumably, that an algorithm can ultimately be found to generate all primes and, thus, that it will contain within it a general rule for doing so. The primes often show up in unexpected ways, shedding potential light on  their own mysterious nature. Consider a famous puzzle formulated in 1256 by Ibn Khallikan (1211–1282), a Kurdish Islamic lawyer and scholar, who was famous during his life for writing biographies. His puzzle is found in one of his biographies in which he discusses chess and exponential growth: How many grains of wheat are needed on the last square of a 64-square chess­ board if 1 grain is to be put on the first square of the board, 2 on the second, 4 on the third, 8 on the fourth, and so on in this fashion?

If one grain of wheat is put on the first square (1 = 20), two grains on the second (2 = 21), four on the third (4 = 22), eight on the fourth (8 = 23), and so on, it is obvious that 263 grains will have to be placed on the sixty-fourth square. The term 263 can also be represented as 264−1, and generally as 2n−1, where n is the

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Figure 2.3 Khallikan’s chessboard

Figure 2.4  Prime numbers on Khallikan’s chessboard

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number of squares. The successive numbers of grains on the chessboard is shown in Figure 2.3. If we take away one grain from each square, the result would be (2n−1), which is Mersenne’s formula. As it turns out, this can be used to test the primality of each square. For example, the third square has 22 or 4 grains on it, and if we take 1 away from it, (22 −1), we get 3, which is a prime number. Similarly, the fourth square has 23 or 8 grains on it, and if we take one away from it, (23−1), we get 7, which is also a prime number. The formula produces prime numbers on the squares shaded in Figure 2.4. This is truly a remarkable coincidental finding, which involves both of Euclid’s expressions for the even perfect numbers, namely [2n−1 (2n−1)]—the first one, 2n−1, indicates the number of grains on a square and the latter, (2n−1), is the one for testing the primality of a square. Now, if a second chessboard is placed next to the first, then the pile on the last square (= 128th square) of the second board contains 2127 grains. If we subtract the number 1 from this, (2127−1), we get a prime number. Ibn Khallikan’s chessboard is an example of

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S e archin g fo r th e pr im e s  |  3



serendipity or discovery by happenstance—a phenomenon that has always characterized research in mathematics and science. Pierre de Fermat (chapter 1) wrote to Mersenne in 1640, to inform him that his formula required modification. Fermat suggested the following alteration: n

22 + 1



Fermat’s formula has, however, generated only a handful of primes, including 3, 5, 17, 257, and 65,537 (n = 0, 1, 2, 3, 4, respectively). It is not known if it will generate other primes. These episodes in the search for an overarching prime number formula indicate that one must always be wary, as Beiler (1966, 219) has cogently argued, since they may appear to be foolproof, but often turn out to be faulty: A favorite example . . . is the remarkable formula, x2 + x + 41. As x assumes ­successive values from zero to 39, the function yields primes, but when x = 40, the resulting number 1681 is composite.

In 1963, mathematician Stanislaw Ulam was literally doodling during a boring academic meeting, when he started drawing a spiral pattern of consecutive numbers and circling the primes in it. He noticed that the prime numbers in  the spiral tended to cluster along vertical, horizontal, and diagonal lines (Figure 2.5). When Ulam programmed a computer to continue the spiral to 65,000, the same pattern of prime numbers falling on lines continued as well. What can we make of this astonishing result? No one really knows. However, some of the prime clusters generated by Ulam’s spiral have been used by mathematicians to develop formulas for predicting primes, albeit not all primes. The connection

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Figure 2.5  An Ulam prime number spiral (from Wikimedia Commons)

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32 |  PR I M E N U M B ER S : TH E D NA O F MATH E MATI C S between the spiral figure and prime numbers is enormously suggestive of something that lies just below the surface of understanding, waiting to come out to the light of reason, perhaps in some future serendipitous discovery.

The Riemann Hypothesis In 1859, German mathematician Bernhard Riemann (1826–1866) presented a paper to the Berlin Academy titled “On the Number of Prime Numbers Less Than a Given Quantity.” Known subsequently as the Riemann Hypothesis, the content of that paper has led to many important explorations and discoveries in number theory. The hypothesis is much too technical to discuss in any mean­ ingful way here. So, only a general characterization will be delineated (for excellent in-depth treatments of this hypothesis see: Derbyshire 2004; Du Sautoy 2004; Sabbagh 2004; Rockmore 2005; Wells 2005). On a number line, the primes become scarcer and scarcer as the numbers grow larger. Riemann suggested that the thinning out involves an infinite num­ ber of “dips” called “zeroes,” on the line, and that these zeroes encode all the information needed for determining if a number is prime (or not). So far no vagrant zero has been found, but at the same time no proof of the hypothesis has come forward to everyone’s satisfaction. From previous work, Riemann suspected that he might find a clue to answering his own question in a sequence that had been devised by Leonhard Euler:

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This can be re-written as shown below, since the negative exponent is another way of writing a reciprocal. The reciprocal of 2 is 1/2; the reciprocal of 3 is 1/3, and so on. These can be written as: 2−1 = 1/2, 3−1 = 1/3, etc. The sequence rewritten in this way is known as the zeta (ζ) function:

z ( s ) = 1 + 2 - s + 3- s + 4 - s + ¼ + n - s

It is known that finding the actual values of the zeta function whose cor­re­ spond­ing outputs are zeroes is the key to finding a prime-generating formula. Riemann believed that the zeroes appeared on a certain vertical line in the com­ plex plane (to be discussed in chapter 7). This has come to be known as the “critical line.”

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Epi lo g u e  |  33

The Riemann Hypothesis is connected to an intriguing suggestive ratio—as the numbers grow larger and larger, the proportion of primes decreases in a logarithmic pattern. There are four primes up to 10, twenty-five up to 100. If we let p(n) be the number of primes up to n, the ratio of p(n) to n, or p(n)/n, decreases as n increases. The potential connection between the Riemann Hypothesis and this pattern is intriguing. All that can be said is that the connec­ tion between the two is one of those mysteries that mathematicians have always sought to unravel. Indeed, if we were ever to crack the code of the primes, we might unravel many of the other mysteries of mathematics and, perhaps, of reality itself.

Epilogue There is a commonly held notion among mathematicians that all mathematics is trivial once it is understood (Hardy 1967). To put it another way, proofs make mathematical discoveries routine. But when they are not discovered, an unend­ ing quest for them motivates mathematicians. That has been the case with pri­ mality. In his delightful novel, Uncle Petros and Goldbach’s Conjecture (2000), the Greek writer Apostolous Doxiadis treats prime numbers as one of those “revelations” provided occasionally by God to mystify human beings, even if it is doubtful that, should a proof ever be revealed, it would change the world in any way. On the other hand, it might. The search for a prime number formula has been relentless. Just enumerating the people who have sought one would require an encyclopedia. What is ­striking is how many notions and ideas the search has generated. French ­math­em­at­ician Sophie Germain (1776–1831), for example, linked Fermat’s Last Theorem (chapter 1) with the primes around 1825. Germain claimed that if both p and (2p + 1) are prime, then p itself is prime, now known as a Sophie Germain prime. The first few Germain primes are as follows:

p = 2 ( prime ) ® ( 2 p + 1) = ( 4 + 1) = 5 ( Germain prime ) p = 3 ( prime ) ® ( 2 p + 1) = ( 6 + 1) = 7 ( Germain prime )

p = 5 ( prime ) ® ( 2 p + 1) = (10 + 1) = 11 ( Germain prime )

Even if this discovery did not generate a formula for primality, it shows how mathematics unfolds as a web of interconnected intellectual paths and ideas.

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34 |  PR I M E N U M B ER S : TH E D NA O F MATH E MATI C S Every single idea is caught up in a system of references to other ideas, unit­ ing mathematics from across the globe in a common pursuit of math­em­at­ ic­al truth. Consider twin primes. These are prime numbers that are two numbers apart, such as 5 and 7 and 29 and 31. How many are there? Although no proof has come forward, some breakthroughs have been made (Parker 2014, 151–152). In 1919, the Norwegian mathematician, Viggo Brun (1885– 1978), discovered that the sum of the reciprocals of twin primes converges to a specific sum, 1.902160583…, now known as Brun’s constant. For example, 3 and 5 are twin primes, so their reciprocals are 1/3 and 1/5; 5 and 7 are twin primes, so their reciprocals are 1/5 and 1/7; and so on. The Brun series is as follows:



(1/ 3 + 1 / 5) + (1/ 5 + 1 / 7 ) + (1/11 + 1 /13) + (1/17 + 1 /19 ) + ¼ = 1.902160583¼



One might ask what significance this series, and twin primes generally, might have beyond the fact of being interesting in themselves. Actually, biologists have discovered that periodical cicadas spend most of their lives sucking sap from tree roots. They come above ground to breed and develop wings. Thousands emerge, but only every 13 or 17 years, which are twin primes. This period between generations makes it impossible for predators to synchronize their own life cycles with those of the cicadas. Findings such as this one answer the above question rather emphatically.

Explorations 1.  Dudeney’s prime number magic square A magic square is a square arrangement of numbers in which the rows, the col­ umns, and the two diagonals add up to the same constant sum, known as the magic constant. Here is a difficult magic square puzzle devised by British puzzlemaker Henry E. Dudeney (1857–1930): Can you arrange the following nine prime numbers {1, 7, 13, 31, 37, 43, 61, 67, 73} into an order 3 magic square—a 3-by-3 square with nine cells (Figure 2.6)? The magic constant of the square is 111—the numbers in each row, column,

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E xplo r ati o ns  |  35

Figure 2.6  Magic square diagram

and diagonal add up to this number. Note that “1” is here considered to be prime, although this is not correct of course. Dudeney included it in all likeli­ hood because it is required to complete the square.

2.  Unravel the number A three-digit number, between 350 and 400, has three prime factors. If you add the digits of the actual number together you will get 12. Each digit is, itself, a prime number. What is the number and what are its prime factors?

3.  Prime number pattern Below are numbers that are all prime. They seem to show a pattern—namely that each successive one adds the digit 3 as shown. Does the pattern keep going?



31 331 3, 331 33, 331 333, 331 3, 333, 331 33, 333, 331

4.  Twin primes Recall the notion of twin primes as a pair of prime numbers that differ by 2. How many twin primes are there less than 100? Here are the first four: (3, 5), (5, 7), (11, 13), (17, 19).

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36 |  PR I M E N U M B ER S : TH E D NA O F MATH E MATI C S

5.  A prime number riddle Here is a riddle. Can you determine what the number is? The number is a prime less than 100; if you add 4 to it you will get the next prime number after it in sequence; and if you add its twin prime, you will get 40.

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3

Zero Place-holder and peculiar number

No number before zero. The numbers may go on forever, but like the cosmos they have a beginning. —Giuseppe Peano (1858–1932)

Prologue From about the first century to the early thirteenth century ce, little progress was being made in mathematics, not because of any lack of ingenuity, but because such progress was likely hampered by a cumbersome and inefficient numeral system in use at the time—the Roman one—which was based on seven alphabet letters having specific numerical values: I = one V = five X = ten L = fifty C = onehundred D = fivehundred M = onethousand

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38 |  Z E RO : PL AC E - H O LD E R AN D PE C U LIAR N U M B E R To grasp how unwieldy that system was, consider how the numeral standing for the number value of “two thousand two-hundred and fifty-three” was put together: MMCCLIII = two thousand two-hundred and fifty-three Now, let’s compare it with the one we would use commonly today, based on the decimal system: 2, 253 = two thousand two-hundred and fifty-three This is clearly much easier to read, after one has learned to use the decimal system, because the principle used to construct it is an efficient one—the pos­ ition of each digit in the numeral indicates its value as a power of ten. This is why it is called “decimal” (from Latin decem “ten”). Here is how the decimal numeral 2,253 is read—“one thousand” can be represented by 103 (because 103 = 10 ´ 10 ´ 10 = 1, 000), “one hundred” by 102 (because 102 = 10 ´ 10 = 100), “ten” by 101, and “one” by 100 (Figure 3.1). Now, imagine trying to carry out a simple arithmetical task, such as adding 2, 253 + 1, 337 , with Roman numerals. Here’s how it would look: MMCCLIII + MCCCXXXVII = MMMDXC The task is a daunting one, no matter how familiar we are with such numerals. It is further complicated by the fact that a smaller numeral appearing before a larger one indicates that its value is to be subtracted from the value of the larger one: for example, the numeral for “ninety” is represented by XC (one hundred, C, minus ten, X). Clearly, it would take quite an effort to carry out the above addition, keeping track of all the letter-to-number values, especially when we compare it with the minimal effort expended to perform it with decimal nu­merals: 2, 253 +1, 337 3, 590





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↓ two thousand ↓ 2 × 103

↓ two hundred ↓ 2 × 102

↓ fifty ↓ 5 × 101

↓ three ↓ 3 × 100

Figure 3.1  Structure of the number 2,253

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 Prolo g ue  |  39

The superiority of the decimal system over the Roman one lies in the fact that it is based on the “abacus principle,” whereby the position of a digit indicates its value in terms of powers of ten. As such, this system requires a symbol as a place-holder for a position that has “nothing” in it. That symbol is 0, which makes it possible to differentiate between numbers such as “eleven” (= 11), “one hundred and one” (= 101), and “one thousand and one” (= 1001) without the use of additional numerals. The 0 tells us, simply, that the position is “empty.” This chapter deals with the origins and significance of zero to mathematics. It is both a place-holder and a peculiar number in itself: it is the only number that is neither negative nor positive; it is neither prime nor composite; it represents the boundary between the negative and the positive numbers on a number line; and so on. Zero is the starting point on many scales, such as coordinate axes and thermometers. An automobile odometer—the device that tells us how far we have traveled—uses zero in a familiar way. After traveling 9 miles in a brand new car, the 9 on the odometer will be replaced by a 0 and the 1 digit will roll in to the left; when the car has traveled 99 miles, the 99 on the odometer will be replaced by 00 and the 1 digit will roll in to the left again, showing that we have traveled 100 miles; and so on. Ancient societies knew about the zero and its peculiar features. It appears as a pair of angled wedges in Sumerian mathematics. The Babylonians apparently left a blank space, as did the Chinese (Seife 2000, 15–18). In Greece, the zero was introduced by Hipparchus in the second century bce for use in astronomy, but it did not, strangely, find its way into Greek mathematics. Around the first century ce, the Maya used a small oval with an inner arc for zero. It was the Indian mathematician, Brahamagupta (c. 598–668 ce), who was seemingly the first to upgrade zero from a place-holder to a number. This paved the way for negative numbers and subsequently for the number line to become part of mathematics. Incidentally, the English word zero derives from ziphirum, a Latinized form of the Arabic word sifr (or zephyr) which, in turn, is a translation of the Hindu word sunya (void or empty). The common use of a circle symbol for zero can probably be traced to the Persian mathematician Muhammad ibn Musa al-Khwarizmi (c. 780–850 ce), who was among the first to discuss the importance of the decimal system invented by the Hindus. Since then, it has been called the Hindu-Arabic system. Aware of the efficiency of the decimal number system over the Roman one, the Italian medieval mathematician Leonardo Fibonacci (c. 1175–1250) wrote a key book on its advantages in 1202, titled appropriately, Liber Abaci

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40 |  Z E RO : PL AC E - H O LD E R AN D PE C U LIAR N U M B E R (“the book of the abacus”) (Devlin 2011). Fibonacci realized that a symbol for “emptiness” would provoke theological and philosophical derision. So he started off his book by reassuring readers that zero was only a sign that allowed for all numbers to be written distinctively (cited in Posamentier and Lehmann 2007, 11): The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0, which the Arabs call zephyr, any number whatsoever is written.

It was in 1889 when Italian mathematician Giuseppe Peano (1858–1932) defined zero as the first natural number (no matter what numeral system is used to represent it). He defined all other numbers as its successors, as will be discussed subsequently.

Negative numbers The most common numeral system in use in the world today is the decimal one. As mentioned, it was first developed by the Hindus in India in the third century bce, and then introduced into the Islamic world around the seventh or eighth century ce. This system reached Europe around 1000 ce through the efforts of Pope Sylvester II (c. 946–1003 ce). But it hardly got noticed at the time, until Fibonacci published his Liber Abaci. Negative numbers surfaced in China in the Nine Chapters on the Mathematical Art (chapter 1) as part of a color code. Red represented positive numbers and black negative numbers. The same practice also surfaces in the book-keeping practices and astronomical calculations of the Hindus in the eighth and ninth centuries. But it was not until the sixteenth century that negative numbers were explored systematically by Italian mathematician Gerolamo Cardano (1501– 1576). It is not coincidental that the term negative comes from the Latin negare (“to deny”), perhaps because their use implied a “denial” of some kind. But, after they came into circulation, the natural number system was enlarged, becoming a new powerful intellectual tool guiding mathematical discoveries that would have literally been unthinkable beforehand. Negative numbers are signed or vectorial numbers, that is, they indicate an opposite direction to the positive ones on scales and lines. This leads to the notion of absolute value—the value of a number regardless of its sign (plus or minus) and position on a number line, to the right of zero (positive) or to the left (negative). The absolute value of any number n is shown as |n|:

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N e gativ e number s  |  4



Number

Absolute Value

0 15 21 1/2 −15 −199 −4/5

|0| = 0 |15| = 15 |21| = 21 |1/2| = 1/2 |−15| = 15 |−199| = 199 |−4/5| = 4/5

The number line is a line-graph that displays how numbers are distributed with respect to each other. Its mid-point is zero and negative numbers are placed to the left and positive numbers to the right (Figure 3.2). This line can help us concretely visualize arithmetical operations with negative numbers. The numbers on the line are “signed” with (+) or (−)—the former is an instruction to move rightward from any point on the line, and the latter to move leftward, also from any point on the line. Consider the addition of two negative numbers, say (−2) and (−3). What does this mean? Looking at the number line, we can see that (−2) is two units to the left of zero. The addition of (−3) means that we move three more units to the left of (−2), landing on the (−5) point on the line. We could also start at (−3) and move two units to its left. The result would be same. Now, consider adding (+2) and (−3). The same movement procedure applies. We start at the (+2) point to the right of zero on the line, and then move three units to its left, landing on the (−1) point. We could also start on (−3) and move two points to the right, also ending at the (−1) point. The order doesn’t matter, which illustrates a fundamental principle, called the Commutative Law— numbers in an addition can be added together in any order. It also applies to multi­pli­ca­tion. To summarize: Example 1

(- 2) + (- 3) = ? (a) Start at (−2) (b) (−3) indicates to move leftward three units to (−5) (c) Therefore: (- 2) + (- 3 ) = (- 5) – –12 –3

–2

–1

√2 0

1

2

3

Figure 3.2  Number line

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42 |  Z E RO : PL AC E - H O LD E R AN D PE C U LIAR N U M B E R Or

(- 3) + (- 2) = ? (a) Start at (−3) (b) (−2) indicates to move leftward two units to (−5) (c) Therefore: (- 3) + (- 2) = (- 5) Example 2

(+ 2) + (- 3) = ? (a) Start at (+2) (b) (−3) indicates to move leftward three units to (−1) (c) Therefore: (+ 2) + (- 3) = (- 1) Or

(- 3) + (+ 2) = ? (a) Start at (−3) (b) (+2) indicates to move rightward two units to (−1) (c) Therefore: (- 3) + (+ 2) = (- 1)

Notice that the sign of the number with the larger absolute value (−3) is also the sign of the answer (end point). Let’s see if this is a general principle. Consider (- 3) + (+ 6) = (+ 3), in which the number with the larger absolute value is (+6). We start at point (+6) to the right of zero, and then we move three units to its left, as indicated by (−3). The end point is (+3), corroborating the principle. Similarly, we can start at (−3) and move rightward six units on the line, ending up at (+3). What about other arithmetical operations? Consider multiplying (+ 2) ´ (+ 3). What does it mean in terms of the number line? The operation tells us to go right two units from 0 three times in sequence. The end result of the operation is the point (+6). Now, multiplying (- 2) ´ (- 3) , which are both negative ­numbers, produces the answer of (+6). This can be broken down as follows.

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Analy tic g eometry  |  43



• Consider the absolute values of both factors—namely |2| and |3|. • The end point of multiplying these two is, of course, |6|. • On which side of 0 is |6| located? • It is located to the right at (+6), because the two signs of the numbers constitute a double negative. • To put it in concrete terms, if a number is negative, (−n), and it is ­multi­plied by a positive number, (+m), we would read the instruction as “keep going in the same direction m times.” We thus end up on a point to the left of 0. If it is multiplied instead by another negative number, (−m), we would read the instruction as “go in the opposite direction m times.” We now would end up on a point to the right of 0. What’s the upshot of all this? If the signs of two factors in a multiplication are the same—two positive or two negative—their product is positive. If the signs of the two are different, then their product is negative. An identical mode of analysis can be applied to the other arithmetical operations, and to all the real numbers, including fractions to irrationals. In all cases, the number line and the location of zero in the middle are what allow us to envision what is going on.

Analytic geometry The two notions of zero and number line led eventually to coordinate geometry. It was the French philosopher and mathematician René Descartes (1596–1650) who played a hunch that literally changed the mathematical world—a hunch that he described concretely in his 1637 work, La géometrie, published as an appendix to his treatise titled Disours de la méthode (Discourse on Method). A number line is, actually, a one-dimensional representation that shows a vec­ tor­ial relation (leftward or rightward) between positive and negative numbers and a one-to-one correspondence between a specific number and a point on the line. Descartes’ hunch was to take two number lines and make them cross at right angles with zero as the point of intersection. He called the horizontal number line the “x-axis,” the vertical one the “y-axis,” and their point of intersection the “origin.” This system is now called the Cartesian plane, in honor of Descartes. The plane (such as the surface of a piece of paper) can now be conceived as a system of points that are determined by their positions in relation to the two axes, called “coordinates.” For example, the paired coordinates for point P in Figure 3.3 are (3, 5). This means that point P is three units to the right of the y-axis, and five units directly above the x-axis.

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44 |  Z E RO : PL AC E - H O LD E R AN D PE C U LIAR N U M B E R y-axis 10

II

I 5

P(3, 5) x-axis

–10

5

–5 –5

III

10

(0, 0) origin

IV

–10

Figure 3.3  Coordinate system (from Wikimedia Commons)

Each point in the plane can now be identified by such ordered pairs of co­ord­ in­ates. The first one is called the “x-coordinate,” expressing the distance to the left or right of the y-axis. If it is to the right then it is positive and will thus lie in quadrants I or IV. If it is to the left, it is negative and will thus lie in quadrants II or III. The second one is the “y-coordinate,” expressing the distance above or below the x-axis. If it is above, it is positive; it will thus lie in quadrants I or II. If it is below, it is negative; it will thus lie in quadrants III or IV. The mathematics of the Cartesian plane came to be known as analytic geometry, and has been the basis of advanced mathematics ever since. Remarkably, all this would not have been possible without, first, the zero, then negative numbers, and finally the number line. Figure 3.4 shows other points in the Cartesian plane. y

(2, 3)

3 2

(–3, 1)

–3

1 (0, 0) –2

–1

–1 –2

Figure 3.4  Points in the Cartesian plane (from Wikimedia Commons)

(–1.5, –2.5)

–3

1

2

3

x

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A naly tic g eometry  |  45



With this system it is now possible to represent equations geometrically. Take the equation x 2 + y 2 = 4 . What figure does it represent? It turns out to be a circle with its center at the origin. By re-writing the equation as y = Ö (4 - x 2 ) we now have established a precise relationship between x and y. It is called a function—a term introduced by German mathematician Gottfried Wilhelm Leibniz (1646–1716). Let’s see where this leads geometrically: Let x = 0 x2 + y2 = 4 0 + y2 = 4 y = ± 2 ¬ Taking the square root of both sides Let x = 1 x2 + y2 = 4 12 + y 2 = 4 1 + y2 = 4

y 2 = 4 - 1 = 3 ¬ Transposing 1, changing its sign y = 1.73 ¬ Taking the square root of both sides Let x = - 1 -12 + y 2 = 4

1 + y2 = 4

y 2 = 4 - 1 = 3 ¬ Transposing 1, changing its sign y = 1.73 ¬ Taking the square root of both sides Let x = 2 x2 + y2 = 4

22 + y 2 = 4 4 + y2 = 4

y 2 = 4 - 4 ¬ Transposing 4, changing its sign y 2 = 0 ¬ Simplifying y = 0 ¬ Taking the square root of both sides Let x = - 2 x2 + y2 = 4

-22 + y 2 = 4

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46 |  Z E RO : PL AC E - H O LD E R AN D PE C U LIAR N U M B E R Table 3.1  Values for y = √ (4−x2) Values of x

0

0

1

Values of y

+2

−2

+1.73

−1

2

−2

…

+1.73

0

0

…

y 3 2

x2 + y2 = 4

1 –3

–2

–1

–1

1

2

3

x

–2

Figure 3.5  Function: y = Ö( 4 - x 2 ) (from Wikimedia Commons)

–3

4 + y2 = 4 y 2 = 4 - 4 ¬ Transposing 4, changing its sign y 2 = 0 ¬ Simplifying y = 0 ¬ Taking the square root of both sides Continuing on in this way (Table  3.1) and then plotting the values on the Cartesian plane, the figure of a circle emerges (Figure 3.5). Analytic geometry has realized the Pythagorean dream of amalgamating geometry and arithmetic in a systematic way. Recall that the Pythagoreans experimented with relations between numbers and rudimentary geometrical forms, discovering square and triangular numbers, among others (chapter 1). Analytic geometry has allowed us to determine what type of equation describes a geometrical figure and vice versa. And this has had enormous implications both within and outside of mathematics. Without it the calculus would have been unthinkable, or at least hard to visualize.

Division by zero If zero is a number, and not just a place-holder, then we must be able to perform arithmetical operations with it. The results for addition, subtraction, and

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D iv i s ion by zero  |  47



­ ulti­pli­ca­tion are a bit unusual, when compared with other numbers, but they m work out just the same: Addition:

1+ 0 =1 23 + 0 = 23 797 + 0 = 797 In general:

n+0=n Subtraction:

2 -0 =2

25 - 0 = 25

869 - 0 = 869 In general:

n-0=n Multiplication:

3´0 = 0

41 ´ 0 = 0

537 ´ 0 = 0 In general:

n´0 =0 Division by zero, however, is not allowed. Consider the proof below, which seems to show something contradictory. 1 . 2. 3. 4. 5. 6. 7. 8.

Assume that a = b . Multiply both sides of the equation by a: a 2 = ab . Subtract b2 from both sides: a 2 - b2 = ab - b2 . Factor both sides: (a + b) (a - b) = b (a - b). Divide both sides by (a - b) : (a + b) = b . Since we assumed that a = b , this can be rewritten as: b + b = b . Therefore: 2b = b, or 2b = 1b . Dividing both sides by the common factor b, we get: 2 = 1.

We have seemingly proved that 2 = 1, or have we? This fallacy arises because we started off by assuming that a = b , which means that (a - b) = 0:

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48 |  Z E RO : PL AC E - H O LD E R AN D PE C U LIAR N U M B E R a=b Subtract b from both sides:

(a - b) = (b - b) (a - b) = 0 When we divided the equation (a + b) (a - b) = b (a - b) by (a - b) , we were in effect dividing it by 0. This illustration brings out the reason for prohibiting division by 0 as a practical one—it is better to retain the zero for many reasons, as we have seen, rather than to throw it completely out because it can lead to a fallacy. Mathematical life goes on without division by zero. Division by zero, however, does have a place in the theory of limits. Consider the function y = 1/ x . As x approaches 0 from the right (in the Cartesian plane), y approaches positive infinity; and as x approaches 0 from the left, y approaches negative infinity, as can be seen in the graph in Figure 3.6 below—the curve in quadrant I shows how it increases to infinity upwards (positive infinity), and the curve in quadrant III shows how it increases to infinity downwards (negative infinity): Quadrant I:

y = 1/ x Let x approach 0 from the right:

x = 4, y = 1 / 4 = 0.25 x = 3, y = 1 / 3 = 0.33 x = 2, y = 1 / 2 = 0.5 x = 1, y = 1 / 1 = 1 x = 0.5, y = 1 / 0.5 = 2 x = 0.19, y = 1 / 0.19 = 5.26 . . . Conclusion: As x approaches 0 from the right, y becomes infinitely larger upwards (positive values) Quadrant III:

y = 1/ x

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D iv i s ion by zero  |  49



Let x approach 0 from the left:

x = - 4, y = 1 / - 4 = - 0.25 x = - 3, y = 1 / - 3 = - 0.33 x = - 2, y = 1 / - 2 = - 0.5 x = - 1, y = 1 / - 1 = - 1 x = - 0.5, y = 1 / - 0.5 = - 2 x = - 0.19, y = 1 / - 0.19 = - 5.26 … Conclusion: As x approaches 0 from the left, y becomes infinitely larger downwards (negative values)

This example illustrates the concept of limits—a concept that is key to the calculus, which can be defined simply as the mathematics of change. Zero is a point or value in the Cartesian plane that a function can be made to approach progressively, without ever reaching it. However, in so doing, we still get a result, such as the two curves in Figure 3.6, noting that they occur only in two quadrants. But does all this have any real interpretations? Sir Isaac Newton (1642–1727), one of the independent founders of the calculus, along with Gottfried Wilhelm Leibniz (1646–1716), answered this question insightfully

1 x

5 4 3 2 1 0 –1 –2 –3 –4 –5 –5

–4

–3

–2

–1

0

1

2

3

4

Figure 3.6  The function y = /x (from: Wikimedia Commons)

5

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50 |  Z E RO : PL AC E - H O LD E R AN D PE C U LIAR N U M B E R with their work on the calculus and its relation to physics. Zero does indeed have a physical interpretation—it is found in functions that represent a “flow,” not a discrete quantity. The calculus has thus shown that zero represents not what is, but how something moves and changes.

The zero exponent One of the more unexpected findings related to zero is that any number to the zero power is equal to 1, which seems truly odd:

10 = 1 20 = 1 40 = 1 50 = 1 …

0

n =1

The reason for this is a consequence of the arithmetical operations connected with exponential numbers, which will be discussed in more detail in chapter 5. Let’s take two identical digits that have the same exponent and divide them:

35 ¸ 35



We know from a law of exponents that:

35 ¸ 35 = 35 -5 = 30

But the result of dividing 35 by 35 is also 1, that is, 35 ¸ 35 = 1 , which is true of any two identical quantities:

5 ¸ 5 = 1

19 ¸ 19 = 1



37 ¸ 37 = 1



35 ¸ 35 = 1

…

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B inary di g it s  |  5



Since 35 ¸ 35 equals 30, as shown above, we have thus proved that: 30 = 1 (things equal to the same thing are equal to each other). In general, therefore, any number, n, raised to the power of zero equals 1: n0 = 1. This is truly an unexpected property of zero, although it can be proved logically. Now, what does the number 00 possibly mean? According to the proof above, it should be equal to 1. But not all mathematicians would agree, since 0 is not a typical ­number. The possibilities assigned to this number are: 1 . 00 = 1 (as per the rule n0 = 1 , with n = 0) 2. 00 = 0 3. The expression is undefined. The point to be made here is that the zero, which started off as a place-holder in numeral systems, has led mathematicians in directions that would have been unthinkable before accepting it as a number. Like the unexpected appearance of Ö 2, the 00 symbol may, or may not, develop theoretical roots on its own.

Binary digits Perhaps nowhere else has the zero played such a critical role as in computers, where two digits, 0 and 1, represent numbers, letters, and portions of pictures and sounds. These are called binary digits. The reason why the binary number system is the language of computers is that, in a computer, millions of tran­ sistors process charges by switching them from circuit to circuit. When a circuit is off, it corresponds to the binary digit 0. When a circuit is on, it corresponds to the digit 1. How do we read binary numerals? Every position has a value 2 times the value of the position to its right. For example, in the numeral 1101, the 1 on the right stands for 1 ´ 20 ; the 0 to its immediate the left stands for 0 ´ 21; the 1 to its left stands for 1 ´ 22 ; and the 1 at the far left of the numeral stands for 1 ´ 23 (Figure 3.7). The sum of the values of these digits is: eight + four + zero + one = thirteen. This means that 1101 stands for “thirteen:” It was German mathematician Gottfried Wilhelm Leibniz (mentioned above as a founder of the calculus) who developed the first arithmetical system based on binary numbers, although English philosopher Francis Bacon (1561–1626) had already used a similar system in 1605, when he published a cipher in which all letters are represented by A and B (rather than by 0 and 1). Bacon called it a “Bi-literary Alphabet” (Gaines 1989, 6; Figure 3.8).

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52 |  Z E RO : PL AC E - H O LD E R AN D PE C U LIAR N U M B E R 1

1

0

1

↓ 1 × 23 ↓ Value of “eight”

↓ 1 × 22 ↓ Value of “four”

↓ 0 × 21 ↓ Value of “zero”

↓ 1 × 20 ↓ Value of “one”

Figure 3.7  The binary digit 1101 Example of a Bi-literary Alphabet. Aaaaa, A, aabba, G, abbaa, N, baaba, T,

aaaab, B, aabbb, H, abbab, O, baabb, V,

aaaba, C, abaaa, I, abbba, P, babaa, W,

aaabb, D, abaab, K, abbbb, Q, babab, X,

aabaa, E, ababa, L, baaaa, R, babba, Y,

aabab, F, ababb, M, baaab, S, babbb, Z,

Figure 3.8  Bacon’s Bi-literary Alphabet (from Wikimedia Commons)

So, for example, the name BACON would be encrypted as: aaaab aaaaa aaaba abbab abbaa. Laurence Dwight Smith (1943, 21–22) makes the following ­relevant observation about Bacon’s cipher: An early admirer of his writes that Bacon’s system was “one of the most in­geni­ous methods of writing in cipher, and the most difficult to be deciphered, of any yet contrived.” Possibly Bacon had his tongue in his cheek when he declared that a perfect cipher was one “not laborious to write and read.” It is difficult to conceive of any more laborious to write than his bilateral cipher, which had to be printed by letter press in two different type faces, the difference between them being scarcely discernible; once enciphered, the message could be de­ciphered only by the most complicated process, that strained not only the patience but also the eyes.

But even before Bacon, the Chinese had actually come up with a binary system of their own, called the I Ching. It was devised during the Shang dynasty of ancient China (c. 1766–1027 bce) and has traditionally been used for div­in­ation. It is organized around sixty-four hexagrams and eight trigrams, constructed with two lines—a yin (broken line) and a yang (unbroken line). The lines are converted into numbers and then into symbolic answers to spiritual questions. It is said that Leibniz was inspired by the I Ching to devise his system of binary arithmetic. He gave the hexagrams a numerical value up to 26 (= 64); and the trigrams, which are half hexagrams, the value of 23 (= 8). Leibniz p ­ ublished his

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E pilo g ue  |  53

system in 1679, in a text titled Explanation of Binary Arithmetic. A comparison of the binary and decimal systems will show how much more difficult the binary one is to read for those of us whose eyes are accustomed to decimal numbers: Decimal Binary 0 0 1 1 2 10 3 11 4 100 5 101 6 110 7 111 8 1000 9 1001 10 1010 11 1011 12 1100 13 1101 14 1110 15 1111 16 10000 17 10001 18 10010 19 10011 20 10100

It was English mathematician George Boole (1815–1864) who translated binary arithmetic into a system of logic that became the theoretical foundation for designing computers. It is mind-boggling to think that the simplest of all number systems, made up of two symbols, may provide insights into how things work or, at least, how we can make them work to simulate tasks that seem complex on the surface but which, below, reveal a simple yes-no, or yin-yang, structure.

Epilogue The incorporation of 0 into the number system added the idea of direction, or vectoriality, to numbers. The first description of the number line comes from John Wallis’ (1616–1703) 1685 book, A Treatise of Algebra, in which zero is

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54 |  Z E RO : PL AC E - H O LD E R AN D PE C U LIAR N U M B E R described as the symbol that divides numbers into two categories—positive and negative. In so doing, he changed the subsequent history of mathematics. Bellos (2014, 172) describes the event eloquently as follows: By replacing the idea of quantity with the idea of position, Wallis argued that negative numbers were neither “unuseful” [nor] “absurd,” which turned out to be something of an understatement. It took a few years for Wallis’s idea to enter the mainstream, but in retrospect it is the most successful explicatory diagram of all time. It has endless practical applications, from graphs to thermometers. We have no conceptual difficulties in imagining negative numbers now that we can show them on a line.

Mathematics could not have been the powerful scientific tool that it has become without zero and negative numbers. The zero is, however, a peculiar number, as we have seen, producing “faults” in the system, such as the preclusion of div­ ision by zero. But then, the faults themselves are often the source of new ideas and discoveries. Mathematics reveals that not everything can be perfect or regular, as the Pythagoreans wanted it to be.

Explorations 1.  A contradiction Examine the following proof. 1. You are given that: a = b + c (c ¹ 0) 2. Multiply both sides by (a - b): a (a - b) = b (a - b) + c (a - b) 3. The result is: a 2 - ab = ab - b2 + ac - bc 4. Transpose (+ac) to the left side: a 2 - ab - ac = ab - b2 - bc 5. Factor both sides: a (a - b - c) = b (a - b - c) 6. Divide both sides by (a - b - c) : a = b Step (6) now shows that a is equal to b. But in (1), a is equal to both b and c together, which means that it is greater than b. Can you explain this contradiction?

2.  Binary arithmetic Using binary numbers, can you tell what the answers to the following ­problems are?

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 E xploration s  |  55

(a)  1100 + 111 = ? (b)  1011 - 1001 = ? (c)  110 ´ 100 = ?

3.  Classic snail problem A classic problem that involves the use of the number line conceptually turned up in an arithmetic textbook written by Christoff Rudolf and published in Nuremberg in 1561: A snail is at the bottom of a 30-foot well. Each day it crawls up 3 feet and slips back 2 feet. The snail has the ability to stick to the walls of the well and, thus, does not slide down to the bottom at the end of a day when it stops to rest. At that rate, when will the snail be able to reach the top of the well?

4.  Firefighter puzzle Here is another classic puzzle that also requires thinking of numbers as if they were on the number line. During a warehouse fire, a firefighter stood on the middle rung of a ladder, pumping water into the burning warehouse. A minute later, she stepped up three rungs, and continued directing water at the building from her new pos­ ition. A few minutes after that, she stepped down five rungs, and from her new position continued to pump water into the building. Half an hour later, she climbed up seven rungs and pumped water from her new position until the fire was extinguished. She then climbed the remaining seven rungs up to the roof of the warehouse. How many rungs were on the ladder?

5.  Trick problem Here is a trick problem. Consider the properties of zero before attempting to solve it. Do you think that the product of the first ten digits is between 100 and 1,000, or is greater than 1,000?

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4

π (Pi) A ubiquitous and strange number

If we take the world of geometrical relations, the thousandth decimal of pi sleeps there, though no one may every try to compute it. —William James (1842–1910)

Prologue In the Old Testament (II Chron. 4:2) we read the following: “Also he made a molten sea of ten cubits from brim to brim, round in compass and five cubits the height thereof, and a line of thirty cubits did compass it round about.” This tells us that the Hebrews were not only aware of the fact that the ratio of the circumference of a circle to its diameter was a constant one, no matter the size of the circle, but took the ratio to be 3. The ratio is known by the Greek letter π (“pi”)— first proposed by Welsh mathematician William Jones (1675–1749). The symbol became widespread after its adoption by Leonhard Euler in 1737. Knowledge of π is found throughout antiquity. The Babylonians and Chinese, like the Hebrews, determined 3 to be its value, while the Egyptians estimated it to be around 3.16. Pi is defined formally as the ratio of the circumference (C) of a circle to twice the radius (r), which means, of course, that the circumference of any circle is π times twice the radius:

C = 2pr p = C / 2r

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This is an amazing discovery, if we stop and think about it. It tells us that no matter the size of the circle, the ratio of the circumference to the diameter (twice the radius) never changes. This might strike us as obvious, but it takes genius to make the obvious, well, obvious. Like the Pythagorean theorem, it established something that people knew practically and gave it an abstract form, so that it could be used and studied further. And it has produced incredible results. Pi is, in fact, a ubiquitous number. It was (and continues to be) used not only to calculate the areas of circles but the volumes of spheres and cones; and it appears in all kinds of mathematical formulas and functions, such as equations that describe the motion of a pendulum, the vibration of a string or the meandering patterns of a river. Like the prime numbers, it seems to be an elemental constituent of reality. It is truly a remarkable discovery. This chapter takes a look at this enigmatic number—yet another great mathematical idea traced back to antiquity. The appearance of π in human history has obviously not been a trivial matter. In the movie π: Faith in Chaos (1998), by American director Darren Aronofsky, a brilliant mathematician, Maximilian Cohen, teeters on the brink of insanity as he searches for an elusive secret numerical code hidden in the seemingly random digits of π, whose first digits are: 3.1415926535. . . . For the previous ten years, Cohen was, in fact, on the verge of his most important discovery— unlocking a pattern that he suspected was hidden in the seemingly random numbers related to stocks on the market. He believed that the key to this was π. As Cohen verges on a solution, an aggressive Wall Street firm set on financial domination and a Kabbalah sect intent on unlocking the secrets hidden in their ancient holy text approach him, as he races to crack the code. In stubborn reaction, he throws his solution irretrievably away. We are left in a quandary and even a state of angst, without Cohen’s solution. Like the Pythagoreans, Aronofsky’s movie treats π as a symbol in a secret code of the universe. In his novel Contact (1985), the late scientist Carl Sagan also suggested that the creators of the universe had buried a message deep within π for us to figure out over time. What is our attraction to this number? Is it perhaps the fact that a circle is the most perfect form known to human beings? And why does π appear in many domains of Nature and physics (see Beckmann 1971; Blatner 1997; Eymard and Lafon 2004; Posamentier 2004)? It always seems to cropping up somewhere, reminding us that it is there, and defying us to understand why. Very much like the universe itself, the more technologically advanced we become and as our picture of π grows ever more

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58 |  Π (PI): A U B I Q U ITOU S AN D STR AN GE N U M B ER sophisticated, the more its mysteries grow. In the end, Cohen loses his sanity over this mystery and maybe we do as well, metaphorically speaking.

Value An early proof for the value of π is found in an Egyptian manuscript dated 1650 bce, titled the Ahmes Papyrus, after the Egyptian scribe Ahmes who copied it, or the Rhind Papyrus, after Scottish lawyer and antiquarian A. Henry Rhind, who purchased it in 1858 while vacationing in Egypt. The papyrus had been found a few years earlier in the ruins of a small building in Thebes in Upper Egypt. It is a copy of an older anonymous work that was written in the 1800s bce. Problem 48 in the text is the relevant one, which can be paraphrased and simplified somewhat as follows: Compare the area of a circle with diameter 9 to that of its circumscribing square, which also has a side length of 9.

From this, it is possible to determine a value for π. Here is how. Inserting the circle in a square of length 9 and then trisecting each side of the square, as shown in Figure 4.1, produces nine smaller squares within it (each 3 × 3). We also draw the diagonals in the corner squares. Such modifications produce an octagon, which can be assumed to be close enough in area to the circle for the practical purposes of the problem.

Figure 4.1  Ahmes’ Octagon (from Wikimedia Commons)

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The area of the octagon is equal to the areas of the five inner squares (which form the outline of a cross) plus half the areas of the four corner squares (= the areas of two squares). Its area is thus equal to the sum of the areas of seven small squares. The area of one small square is, 3 × 3 = 9 square units. The total area of seven such squares is, therefore, 9 × 7 = 63 square units. Let’s assume the circle’s area to be 64, for the sake of argument, since it is a little larger than the area of the octagon. Its diameter is 9, and is equal to one of the sides of the square (as stipulated). We can now estimate the value of π as follows: Area of circle:

= πr2 = 64

Diameter:

=9

Radius (r):

= 9/2

So, r

= (9/2)2 = 20.25

π

2

= 64/20.25 = 3.16049…

Although this is not the exact value for π, it is close enough. What is pertinent here is the method used to determine it. A strikingly similar one was used in the fifth century bce, by the Greek mathematicians Antiphon and Bryson of Heraclea, contemporaries of Socrates. They inscribed a polygon inside a circle and another slightly bigger one just outside. The circumference of the circle would thus lie in between the perimeters of the two polygons. The same idea was used by Archimedes (c. 287–212 bce) much later. The difference between the perimeters of the polygons and the circumference of a circle, he argued, could be made as small as one desired by progressively increasing the number of sides of the polygons. The limiting figure of such an incremental procedure was the circle (Figure 4.2). This type of proof is called an iterative algorithm (an algorithm based on repeating something over and over). One of the best known is the one by

Figure 4.2  Archimedes’ Polygon Method for the Value of π (from Wikimedia Commons)

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60 |  Π (PI): A U B I Q U ITOU S AN D STR AN GE N U M B ER Chinese mathematician Liu Hui (third century ce). He calculated π to between 3.141024 and 3.142708 with a 96-gon, recalling Archimedes. There are many practical uses of π in geometry, science, and engineering. Below are some examples (h = height): Area of a circle

= πr2

Volume of a cone

= 1/3 πr2h

Area of a sphere

= 4πr2

Volume of a sphere

= 4/3 πr3

Volume of a cylinder

= πr2h

Lateral area of a cylinder

= 2πrh

Surface area of a cylinder

= 2πrh + 2πr2

Transcendental numbers Pi is an irrational number, since like √2 (chapter 1) it cannot be written as a simple fraction, p/q. A popular approximation is 22/7. It is close, but not quite close enough. π = 3.1415926535… 22/7 = 3.1428571428…

Let’s compare π and √2. The latter is found in the equation for the hypotenuse of an isosceles right triangle with sides of unit length (as we saw in chapter 1): √2 = √(12 + 12). Moreover, it is found as a root (solution) to the following ­equation:

x2 – 2 = 0 x2 = 2 x= 2

For this reason, it is called algebraic, defined as a number found in the solution of an equation. Pi is not algebraic; that is, there is no equation to be found in which it occurs, making it a distinctive type of number called, more specifically, transcendental. The first mention of such numbers can be traced to a paper by Leibniz in 1682. A little later, Euler put forth the first definition as a number that is not a root of any polynomial. A polynomial is an expression that consists

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–3

–2

–1

0

1

2 √2

3 e π

Figure 4.3  Real number line

of variables (letters or symbols) and numbers. An example is x2 – 2x + 7. The German mathematician Ferdinand von Lindemann (1852–1939) was among the first to prove that π was transcendental in 1832 even though French math­ em­at­ician Joseph Liouville (1809–1882) had put forth a sketch of a proof early in his career, publishing it only later in 1844. As it has turned out there are an infinite number of transcendental numbers. So, π is hardly an exception; it is part of a rule. Indeed most real numbers—numbers that can be located on the number line (chapter 3)—turn out to be transcendental. On this line, we can locate numbers such as √2, π, and e (discussed in chapter 6) (ℝ = symbol for real numbers); see Figure 4.3. It is amazing to consider that numbers such as √2 and π, which have cropped up essentially by happenstance, have so much information hidden in them. Their discovery has changed the world. The evolution of technology, engineering, science, and human intelligence itself would have been vastly different without numbers such as these. Perhaps Carl Jung’s (1972) notion of syn­chron­icity applies here—which the psychologist defined as the simultaneous occurrence of events that appear related but have no discernible causal connection. As he (1972, 91) emphasized: “When coincidences pile up in this way, one ­cannot help being impressed by them—for the greater the number of terms in such a series, or the more unusual its character, the more improbable it becomes.” A number such as π entices us to flesh out some deeper meaning in it, as Jung suggested.

Manifestations Take a piece of cardboard and a needle. Mark, say, 10 parallel lines on the cardboard, spacing them a little more than a needle length apart. So, if the needle is 40 mm long, then the space between the lines must be more than 40 mm, say, 45 mm. Toss the needle in the air so that it falls on the cardboard randomly, as many times as you want.

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62 |  Π (PI): A U B I Q U ITOU S AN D STR AN GE N U M B ER The object of the experiment is to determine the relation between the number of tosses of the needle and the number of times the needle touches a line on the cardboard. If you keep track of the outcomes, you will find that as the number of tosses of the needle increases, the ratio of the number of tosses to the number of times the needle touches the line approaches π. This is known as Buffon’s Needle Problem, first posed in the eighteenth century by the French naturalist Georges-Louis Leclerc, Comte de Buffon (1707–1788). Margaret Willerding (1967, 120) recounts a historical anecdote whereby in 1901 “a scientist made 3408 tosses of the needle and claimed that it touched the lines 1085 times,” and thus the “ratio of the number of tosses to the number of times the needle touched the lines, 3408/1085, differs from π by less than 0.1 per cent.” Again, as suggested above, this is a synchronic event—an event that is somehow connected to π that manifests itself serendipitously and which requires a human brain to interpret. This applies as well to the manifestations of π in mathematical equations. Here are three examples:

(1) (2) ( 3)

p/2

= 2 /1 ´ 2 / 3 ´ 4 / 3 ´ 4 / 5 ´ 6 / 5 ´ ¼

p/4

= 1 /1 – 1 / 3 + 1 / 5 – 1 / 7 + 1 / 9 - ¼

p2 / 6 = 1 /12 + 1 / 22 + 1 / 32 + 1 / 42 + ¼

The top one was devised by John Wallis; it shows that π depends on the product of the terms in a particular infinite series. The middle one is traced to the Indian mathematician, Madhava, in the fourteenth century. In 1671, Scottish math­em­ at­ician James Gregory discovered this same formula in an independent ­manner. It was studied a little later by Leibniz. The last one was formulated by Euler in 1764. As Mackenzie (2012, 44) points out: These equations reveal that the number pi is not merely a geometric concept. Three of the great tributaries of mathematics merge in these formulas: geometry (the number pi), arithmetic (the sequence of odd numbers, and the sequence of squares 12, 22, 32, . . .), and analysis of the infinite (in this case, infinite sums). Archimedes would have been flabbergasted to see formulas like these.

An architectural appearance of formula (1) above is in the Great Pyramid at Giza. The ratio of the length of one side to the height is approximately π/2. As Blatner (2014: 11) observes: “It can be shown that any pyramid with this framework will automatically approximate pi.” The use of π in this structure typically produces a majestic aesthetic effect on whoever views it. As Berggren,

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Borwein, and Borwein (1997) have aptly put it, in a fundamental way the fascination with π and its manifestations across domains is emblematic of mathematics itself: To pursue this topic (pi) as it developed throughout the millennia is to follow a  thread through the history of mathematics that winds through geometry, analysis and special functions, numerical analysis, algebra, and number theory. It offers a subject which provides mathematicians with examples of many ­current mathematical techniques as well as a palpable sense of their historical development.

The manifestations and scientific uses of π are literally innumerable. A few examples will suffice to grasp the enormous importance of π in probing the universe:

• It has been used, since antiquity, to study the rotation of the Earth and its orbits.

• It is used in equations and formulas for exploring planets and their densities and atmospheres (called exoplanets) outside the solar system.

• It is used in formulas for calculating the trajectories of spacecrafts— known as pi transfer.

• It has been discovered in sound and light waves known as sine waves. • It is used in calculating vibrations, pendulum swings, and frequencies in everything from ocean waves to ultrasound graphs. • It has been found in DNA, such as in the double helix spiral which is held together by so-called pi bonds. In effect, π appears in anything that has circularity, curvature, or roundness (even if contorted in some way), from a rainbow, the moon, the sun, to the pupil of the eye. It is truly a ubiquitous and mysterious number, found in every­ thing from an orbiting star to a heart pulse, that may, as Aronofsky’s movie suggested, be the key to unraveling the meaning of the universe. It governs all our lives, even though we cannot see it directly in them. We can only see it in a ratio connected to a circle and as a point on the real number line. Yet, it is present everywhere. This whole line of discussion and illustration related to a single mathematical idea, π in this case, brings us once again to the nature of mathematics—is it out there or in us? As modern physics attempts to develop a “theory of everything” by using increasingly abstract mathematics, as financial markets succumb to the mystery of mathematical measures of risk, and as biology employs algorithms

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64 |  Π (PI): A U B I Q U ITOU S AN D STR AN GE N U M B ER to unlock the genetic code, it does not seem far-fetched to imagine that numbers do indeed hold the key to decoding the universe, as the Pythagoreans suspected. Does the cosmos make mathematics, or does mathematics make the cosmos? Since the same mathematical notions, such as π, surface in people across time speaking different languages, the answer seems to veer towards the possibility that mathematics is already in the universe, and that we are dis­cover­ing its existence in bits and pieces all the time.

Epilogue As a key to unlocking some of the secrets of the universe, π has captured our fancy since the dawn of history, and our fascination has grown throughout the ages. If one charts the estimates for π throughout the centuries, it can be seen that they increase logarithmically (Figure 4.4). In 2011, using a computer, mathematicians Shigeru Kondo and Alexander Yee calculated π to a mind-boggling ten trillion decimal places. It took the computer 191 days to produce the output. Pi even has its own cult followers, who celebrate it on Pi Day, which takes place on March 14 (with numerical representation of 3/14) and which is the birthday of Albert Einstein. Pi Day started in 1988, at the Exploratorium in San Francisco, a museum of science, art, and human perception—an intellectual

Record approximations of pi

1014

Number of decimal digits

1012 1010 108 106 104 100 1 2000 BCE

250 BCE

480 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000 Year

Figure 4.4  Approximations of π through the centuries (from Wikimedia Commons)

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and aesthetic collage that characterizes the history of mathematics itself. Arthur Benjamin (2015, 198) describes the celebration as follows: A typical Pi Day event might consist of mathematically themed pies for display and consumption, Einstein costumes, and of course π memorization contests. Students generally memorize dozens of digits of π, and it is not unusual for the winner to have memorized over a hundred digits.

The first Pi Day was organized by physicist Larry Shaw, who came to be named with the epithet, Prince of Pi. In March of 2009, the US House of Representatives passed a non-binding resolution recognizing March 14 as National Pi Day. So captivating has π become to some that there is now even an artistic movement called π art, which involves generating graphic images based on this number. Pi has also stimulated people to devise creative mnemonic techniques, such as the following one that was published in a 1914 edition of Scientific American: See, I have a rhyme assisting my feeble brain, its tasks ofttimes resisting.

Replacing each word by the number of letters it contains yields π to 12 decimal places: See

=3

I

=1

have

=4

a

=1

rhyme

=5

assisting

=9

my

=2

feeble

=6

brain

=5

its

=3

tasks

=5

ofttimes

=8

resisting

=9 ——————— 3.141592653589

There are other interesting piems, as such memory aids are called—a word formed by combining pi and poem. This fascination with π has spilled over into popular culture generally. It is found in TV programs ranging from an episode

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66 |  Π (PI): A U B I Q U ITOU S AN D STR AN GE N U M B ER from the original Star Trek series (1966–1969) when the character Spock commands an evil computer to compute π completely, which it cannot do because, as Spock quips, “the value of π is a transcendental figure without resolution,” to several episodes of the The Simpsons. It appears in movies such as Torn Curtain (1966), The Net (1995), Twilight (2008), and of course Aronofsky’s movie mentioned at the start of this chapter. Maybe, just maybe, concepts such as π may be anchored in our genes, given that they are found universally across cultures and eras and continue to fas­cin­ ate us (as the pop culture examples show). However, although these concepts may predate the evolution of human consciousness, they are not understood nor do they exist outside of the human mind. As the American philosopher and mathematician Charles S. Peirce aptly put it, the human mind has “a natural bent in accordance with nature” (volume 6, 1931–1958, 478). A world in which π is not known is an intellectually impoverished one. What we now know about certain objects in the world, like the sun and the tides, would be much more rudimentary without it. As Kasner and Newman (1940, 89) aptly put it, without π “our ability to describe all natural phenomena, physical, biological, chemical or statistical, would be reduced to primitive dimensions.”

Explorations 1.  A way to calculate π Here is an exploration for determining the value of π called the Monte Carlo Method (named after the famous gambling spot). Consider Figure 4.5, in which the square is 2 units in length, and hence so is the diameter.

2 units

Figure 4.5  Calculating π

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Here are some relevant aspects about this diagram: Area of the square: 2 × 2 = 4 square units Diameter of the circle: 2 units Radius: 1 unit Area of the circle: πr2 = π (12) = 1π = π Ratio of two areas (area of circle / area of square): π/4 = 0.7854

Now using a technique that is analogous to the Buffon Needle one above, with eyes closed and pencil in hand, randomly put dots on the figure above. Do this many times. After this, calculate the ratio of the number of times the dots land inside and outside the circle. What is the percentage of the number of times the dots landed inside?

2.  A circular walk This is a straightforward problem that exemplifies concretely how π is used to calculate circular distances. If you decide to walk around a circular garden that has a diameter of 200 feet, how far have you walked?

3.  A reverse puzzle This next puzzle is a reverse puzzle to the previous one. A circular garden is enclosed by a circular fence measuring 500 ft. (approximately). You want to get to the center of the garden, from an opening in the fence. How many feet will you walk?

4.  String around a circle Here is an interesting problem that involves the use of π. A string is wound perfectly around a circular object twice—that is, the string goes exactly two times around the object. The diameter of the object is 14 cm. Find the length of the string.

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5.  Pythagoras meets π Here’s a problem that requires the use of both π and the Pythagorean theorem: In the diagram in Figure 4.6 below the radii shown meet at right angles. The length of the hypotenuse of the right-angled triangle so formed is 9 in. What’s the circumference of the circle?

A 9

O

Figure 4.6  Circumference problem

B

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5

Exponents Notation and discovery

I will not vote against the truths of the multiplication table. —James A. Garfield (1831–1881)

Prologue Consider the following multiplication problem:

10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 = ?

As it stands, it requires a cumbersome effort just to read, let alone contemplate the answer. Is there a more efficient way to represent such a problem? The answer came in the 1500s with the invention of exponential notation, which was devised as a type of shorthand to facilitate the cumbersomeness of reading repeated multiplications of the same digit, such as the one above. Unless some­ one is a computational wizard, the mind boggles at such a tedious task. But the use of “15” in superscript form, which stands for the times the number 10 is to be used as a factor, greatly reduces the effort of deciphering the task at hand: 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 ´ 10 = 1015 The superscript saves space and lessens the mental energy required to process the relevant information. Such notation is especially critical for writing larger

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70 |  E XPO N E NT S : N OTATI O N AN D D I S COV E RY and larger numbers. The term googol was introduced by the American math­ em­at­ician Edward Kasner, after his 9-year-old nephew is said to have invented it (Kasner and Newman 1940). This is the number written as 1 followed by 100 zeros. It would take a considerable amount of effort and space simply to write it out. However, by using the notation of 10100, we can now grasp the number more concretely. For the sake of illustration, here is what a googol would look like when written out completely: 10100 = 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 A googolplex is an even larger number, defined by Kasner as 10 to the googol power, or 10 multiplied by itself a googol number of times

10

googol

100

= 1010

It is impossible to envision the number this represents, let alone write it out completely. It has, in fact, been estimated that there are more zeroes in a googolplex than there are particles in the known universe. The late astron­ omer Carl Sagan (1934–1996) once remarked that writing out a googolplex would be physically impossible, because there is simply not enough space in the universe. Yet, a simple notational device allows us, at the very least, to give it form. Exponential notation has achieved more than a way to represent large num­ bers. Right after its introduction in the sixteenth century it took on a life of its own. Mathematicians started to play with exponential notation in an abstract way, discovering new facts about numbers. For example, they discovered that n0 = 1, thus fleshing out a property of zero that was previously unknown, as discussed in chapter 3. It also led to an “arithmetic of exponents,” with its own properties, such as the following: Example:

(na )(nb ) = na + b (22 )(23 ) = 4 ´ 8 = 32



22 + 3 ® 25 = 32



(na )(ma ) = (nm)a

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Example:

(22 )(32 ) = 4 ´ 9 = 36



(2 ´ 3)2 = (6)2 = 36



(na ) ¸ (nb ) = na -b (n ¹ 0)

Example:

53 ¸ 52 = 125 ¸ 25 = 5 53 -2 ® 51 = 5 (na )b = nab

Example:

(42 )3 = (16)3 = 4096



4(2)(3) ® 46 = 4096

This kind of arithmetic led, shortly after, to the concept of logarithms—a topic that will be broached later in this chapter. Logarithms have hardly been just useful notation devices for carrying out complex arithmetical tasks. As often happens when new ideas come into play, they allow for all kinds of unexpected discoveries to foment and develop. This chapter will deal with exponents and logarithms, given their significance to the history of mathematics—a history often characterized by problems of notation that have led serendipitously to new ideas and branches.

Exponential notation The first recorded use of exponential notation can be traced to the 1544 book Arithemetica Integra, written by English mathematician Michael Stifel (1487– 1567). It was René Descartes, however, who saw the true significance and math­ ematical implications of such notation. Consider again a multiplication problem, such as multiplying the number 3 fifteen times:

3 ´ 3 ´ 3 ´ 3 ´ 3 ´ 3 ´ 3 ´ 3 ´ 3 ´ 3 ´ 3 ´ 3 ´ 3 ´ 3 ´ 3 = 14, 348, 907

This layout is clearly unwieldy for the mind to process. As discussed, it can be rendered more understandable if written as 315. In this notation, 3 is called the

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72 |  E XPO N E NT S : N OTATI O N AN D D I S COV E RY base, and the superscript, 15, the exponent or power. The exponent tells us the number of times that the base is to be multiplied. In general, nm indicates that any number, n, is to be multiplied m times:

nm = n ´ n ´ n ´ n¼´ n   



m factors

The concept of exponent leads logically to its converse of root. For example, we know that:

42 = 16 Or



4 ´ 4 = 16

The root of 42 is thus 4. It is also called the “square root” of 16, written as √16 = 4. Another square root of 16 is −4, since the product of two negative ­numbers is positive, as we have seen previously (chapter 3): −4 × −4 = 16. This may seem like a trivial fact; but as will be discussed in chapter  7, it led  to  the discovery of imaginary numbers. The number inserted within the symbol √ is called the index. This tells use what kind of root is required (square, cube, etc.). The number enclosed by the root symbol is called the  radicand. For example, the cube root of 27 would be represented as follows:

3

27 = 3

The largest number symbolized by the ancient Greeks was what they called a myriad, or 10,000. A myriad-myriad was equal to 100 million. But they had no efficient numeral system to represent large numbers. It was in China, around 190 bce, that mathematicians had devised a way of simplifying the representa­ tion of large numbers. They showed that any number could be written in a reasonably accurate fashion with a few digits and then multiplied by 10 a given number of times—that is, by what we now call powers of ten. The Chinese also used the same technique for representing small numbers, dividing numbers by reciprocal powers of 10.

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Exponential arithmetic The first aspect of exponential notation is that it can be used with all kinds of number bases—integers (positive and negative), fractions, decimals, and so on:

(-4)3 = (-4) ´ (-4) ´ (-4) = - 64 5 = (1) ´ (1) ´ (1) ´ (1) ´ (1) = 1 (1) 4 (.03) = (.03) ´ (.03) ´ (.03) ´ (.03) = 0.00000081

Let’s look at a few examples for the sake of illustration. Multiplying two powers with the same base is equivalent to adding their exponents:

34

´

35



¯

¯

¯

(3 ´ 3 ´ 3 ´ 3) ´ (3 ´ 3 ´ 3 ´ 3 ´ 3) = 39



In general, if a is any base and n and m exponents with any value, then: an ´ a m = an



+m



Dividing is equivalent to subtracting exponents. Consider 35 ÷ 33. The answer is 32:

35 ¸ 33 can be written as:



(3 ´ 3 ´ 3) ´ 3 ´ 3



(3 ´ 3 ´ 3) Canceling, we get:



( 3 ´ 3 ´ 3) ´ 3 ´ 3



( 3 ´ 3 ´ 3)



This leaves: 3 ´ 3 = 32



So: 32 = 35 ¸ 33 = 35 -3



In general: an ¸ a m = an - m

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74 |  E XPO N E NT S : N OTATI O N AN D D I S COV E RY Any number raised to the power of 0 is equal to 1. The proof is simple. Take the base 3 with exponent 5, or 35. Divide this by itself. The result is, of course, “1”, as can be seen by the method of cancellation:

35 ¸ 35



Is the same as:



3´3´3´3´3 3´3´3´3´3



Which equals 1

We know that 35 ÷ 35 = 35−5 = 30. It follows by the axiom of equality (things equal to the same thing are equal to each other) that 30 = 1. In general: n0 = 1 (where n is any number). A number raised to a negative power indicates that it is a reciprocal. Consider the following fraction:

1_ 33

We know that any number to the power of 0 is equal to one. So, let’s replace the numerator with 30:

30 33

Dividing the numerator by the denominator is equivalent to 30 ÷ 33, which, as we saw above, is equivalent to 30−3, or 3−3. This shows that indeed 3−3 = 1/33. In general:

a -n = 1/ an

There are a number of other laws of exponential arithmetic, which need not concern us here. The purpose of the foregoing discussion has been to show that exponents were not only a means for simplifying the representation of a certain kind of multiplication, but after their invention, they led to a new way of doing arithmetic that was essential to the arithmetic of large numbers. Moreover, exponential notation expanded the study of algebra, leading to precise new definitions. For example:

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Pascal’ s Trian g le  |  75

 (14)x

y

4x 2x

(12)x

0

x

Figure 5.1  Examples of exponential functions (from Wikimedia Commons)

1. Equations where the variable or variables are to the power of 1 are called first degree: 3x + 1; 5x + 2y; etc. 2. Expressions where the variable or variables are to the power of 2 are named accordingly the second degree: 3x2 + 1; 5x2 + 2y; etc. 3. Expressions where the variable or variables are to the power of 3 are called expressions to the third degree: 3x3 + 1; 5x3 + 2y; etc. 4. In general, expressions where the variable is to the power of n are called expressions to the nth degree. These were known in antiquity, but the lack of an appropriate notation some­ what impeded the development of advanced algebra. The new notation pro­ vided mathematicians with a powerful descriptive tool that has had implications throughout mathematics. One of the more fascinating discoveries associated with exponents is the so-called exponential function. This is a function of the form y = ax (for example, y = 2x). All such functions, when plotted onto the Cartesian plane have a parabolic shape as in Figure 5.1. These functions show up in many places, from the calculation of compound interest to the measurement of anything that grows or decays, such as bacteria. They are used in radiocarbon dating, which has been of great importance to archeology.

Pascal’s Triangle After the spread of exponential arithmetic, mathematicians started looking more closely at certain types of expressions. French mathematician Blaise Pascal (1623–1662) was working with the expression (a + b)n, known as a ­binomial (consisting of two terms), when he came across a hidden pattern in its expansion, n = {0, 1, 2, 3, 4, 5, . . .}. Here are the first eight expansions:

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76 |  E XPO N E NT S : N OTATI O N AN D D I S COV E RY (a + b)0

=

1

1

=

a+b

2

(a + b)

=

a 2 + 2ab + b2

(a + b)3

=

a 3 + 3a 2b + 3ab2 + b3

(a + b)4

=

a 4 + 4a 3b + 6a 2b2 + 4ab3 + b 4

(a + b)5

=

a 5 + 5a 4 b + 10a 3b2 + 10a 2b3 + 5ab 4 + b5

(a + b)6

=

a 6 + 6a 5b + 15a 4 b2 + 20a 3b3 + 15a 2b 4 + 6ab5 + b6

(a + b)7

=

a 7 + 7a 6b + 21a 5b2 + 35a 4 b3 + 35a 3b 4 + 21a 2b5 + 7ab6 + b7

(a + b)



In the top row, the numerical coefficient in (a + b)0 is 1; in the next row down the numerical coefficients of (a + b)1 = a + b = 1a + 1b are both 1; in the next one down the numerical coefficients of (a + b)2 = a2 + 2ab + b2 = 1a2 + 2ab + 1b2 are 1, 2, and 1; and so on. Extracting these coefficients from the other terms in the expressions, and leaving them in their respective rows, Pascal noted that they could be arranged into the shape of a triangle (Figure 5.2). Now, surprisingly, the triangle itself harbors patterns that can be broken down reductively as follows: 1. Leaving aside the vertex 1 of the triangle, each row begins and ends in 1. 2. The other numbers in a row are the sum of the two numbers right above it—for example, each of the 10s in the sixth row down equals the sum of 4 and 6 above them. 3. In fact, each number in the triangle is the sum of the two directly above it. 4. It is this structure that makes it an infinite triangle, since we could continue to generate numbers within it with this simple rule ad i­nfinitum. 5. As Sir Isaac Newton noted, the exponent in each expression in the binomial distribution indicates the row in Pascal’s triangle where the numerical coefficients occur: for example, in the expansion of (a + b)4,

Figure 5.2  Pascal’s Triangle

1

1

1 7

1 6

1 5 21

1 4 15

1 3 10 35

1 2 6 20

1 3 10 35

1 4 15

1 5 21

1 6

1 7

1

1

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Lo garith m s  |  77

which is a4 + 4a3b + 6a2b2 + 4ab3 + b4, the numerical coefficients (1, 4, 6, 4, 1) coincide with the fourth row of numbers in Pascal’s triangle (if we do not include the top 1). Similarly, the numerical coefficients in the expansion of (a + b)5, a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 coin­ cide with the fifth row of numbers in the triangle (1, 5, 10, 10, 5, 1); those in the expansion of (a + b)6 with the sixth row; and so on. 6. The sum of the digits in any row, n, of the triangle is a power of 2, that  is, 2n. For instance, the sum of the digits in the fifth row is: 1 + 5 + 10 + 10 + 5 + 1 = 32, which is 25 (Chamberlain 2015, 164–165). There are many other hidden patterns in this remarkable triangle. For the sake of historical accuracy, it should be mentioned that the triangle appears in a Sanskrit text named the Chandas Shastra, written between 500 and 200 bce, and in a 1303 Chinese work Si Yuan Yu Jian written by Zhu Shijie (1206–1320). There is no evidence to suggest that Pascal was aware of any of these texts—a fact that raises a truly deep enigma with regard to discovery. Why is it that ­people in different parts of the world and at different times are able to come up with similar ideas? No real answer to this question can be given—it can only be observed. To contextualize Pascal’s amazing discovery, Solomon (2008, 80) makes the following relevant remarks: To be fair to Pascal, he did not claim to have invented the triangle. He certainly did not name it after himself. He actually referred to it as the “arithmetic ­triangle” and used it to calculate probabilities.

Logarithms Exponential notation was the spark for logarithms, which have been used in many areas of mathematics, science, and statistics, leading to many discoveries. A logarithm is the actual power or exponent to which a base, usually 10, must be raised to produce a given number: If nx = a, the logarithm of a, with n as the base, is x; symbolically, logn a = x. For example, 103 = 1,000; therefore, log10 1,000 = 3.

To get a sense of the usefulness of logarithms, suppose we wanted to calculate the number of ancestors in any previous generation. We have 2 parents, so there are 2 ancestors in the first generation. This can be expressed as 21 = 2. Each of

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78 |  E XPO N E NT S : N OTATI O N AN D D I S COV E RY our parents has 2 parents, and so there are 2 × 2 = 22 = 4 ancestors in the ­previous generation. Each of the four grandparents has 2 parents, and so there are 4 × 2 = 2 × 2 × 2 = 23 = 8 ancestors in the generation before. The calculation continues according to this pattern. In which generation do we have 1,024 ancestors? This can be rephrased as follows: For which value of n is 2n = 1,024? We can find the answer by multiplying 2 a number of times until we reach 1,024. But if we know that log2 1,024 = 10, because 210 = 1,024, we can get the  answer much more quickly and efficiently. This tells us that in the tenth ­previous generation we have 1,024 ancestors. Note that in this case the base is not 10, but 2. The invention of logarithms is traced to Scottish mathematician John Napier (1550–1617), who devised the first ones for carrying out tedious complex arithmetical operations in his 1614 work, Mirifici logarithmorum canonis ­ descriptio (“Description of the Marvelous Canons of Logarithms”), coining the term logarithm as a combination of Greek logos (reason) and arithmos ­(number). In it, Napier showed how simple it is to carry out arithmetical operations with logarithms, since each number can be represented with a logarithmic value and all such values can be included in a table. For example, to multiply any two numbers: 1. Find their logarithms in the appropriate tables. 2. Add the two logarithms. 3. Look the result up in the antilogarithm tables (an antilogarithm is the number itself). Napier’s logarithms were based on powers of 0.999999; their translation into tables with 10 as the base was due to English mathematician Henry Briggs (1561–1630) in 1624. Briggs’ improvements and refinements of Napier’s ­techniques helped the notion of logarithms gain broad acceptance (Solomon 2008, 67). The number of discoveries, implications, and ideas that logarithms have spawned, within and outside of mathematics, is mindboggling. One of the most widespread uses of logarithms is in so-called log scales. Perhaps the best-known contemporary example is the Richter scale, which is used to measure earth­ quake sizes, developed by Charles Richter in 1935 (see Figure 5.3). The scale shows how the magnitude of an earthquake is measured in logarithmic terms. Other log scale phenomena include the decibel scale of sound and the pH scale of acidity. The remarkable thing to consider is that all such discoveries and uses of logarithms came about after the invention of a simple notation—exponents.

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1 × 1020

1 × 1020

1 × 1018

1 × 1018

1 × 1016

1 × 1016

1 × 1014

1 × 1014

1 × 1012

1 × 1012

10

1 × 1010

1 × 10

1 × 108

1 × 108

Energy/J

Frequency/100r.y

E pi lo g u e  |  79

1,000,000

1,000,000

10,000

10,000

100

100 1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

Richter Magnitude Scale

Figure 5.3  A Richter Scale (from Wikimedia Commons)

Notations and re-arrangements (such as the layout of Pascal’s triangle) contain more hidden information about real phenomena than is initially apparent. It is upon reflecting on the hidden information that we come to extract more mean­ ing from it than we would otherwise be likely to be able to do.

Epilogue All of mathematics is encoded in symbolic notation. The Pythagorean equation c2 = a2 + b2, for example, is essentially a convenient notation of saying the same thing as the sentence “the square on the hypotenuse is equal to the sum of the squares on the other two sides.” In so doing, it takes the semantics in the lin­ guistic sentence out, leaving only the structural (symbolic) outline of the infor­ mation. It is this feature that makes it cognitively powerful, since we can now find many more meanings and applications for it, in addition to the original geometrical one. For example, one can now ask what integers fit the equation and, further, if there are other exponents for which the equation holds in gen­ eral, cn = an + bn. From this deliberation on the notation itself, detached from its original geometrical meaning, has come much subsequent mathematical con­ templation leading to such intriguing ideas as Fermat’s Last Theorem, as we saw. In other words, notation suggests meanings that could not otherwise be contemplated.

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80 |  E XPO N E NT S : N OTATI O N AN D D I S COV E RY The ancients did not have exponential notation at their disposal, so they used other devices to accomplish many of the same things as the new notation allowed so efficiently. In his book, The Sand Reckoner, Archimedes discussed the notion of powers in a roundabout way. However, it did not allow him to develop such techniques as exponential laws and logarithms. Einstein’s equation E = mc2 has imprinted in it a lot of information about physical reality that could not be expressed in any other way. It says, in a nut­ shell, that the speed of light is constant and thus that it constrains physical real­ ity. What happens if there is a universe where this formula does not hold? It would be unimaginable, even though it is possible. Above all else, we would need a notation to devise formulas for such a universe. As the philosopher Ludwig Wittgenstein (1922) put it, “Whereof one cannot speak, thereof one must be silent.” As the language of Nature, mathematics breaks the silence peri­ odically. E = mc2 speaks volumes, to belabor the point somewhat.

Explorations 1.  Square numbers Here is a tricky problem, involving the exponent 2. Find three numbers, less than or equal to 10, which when raised to the power of 2 and then added together equal 150.

2.  Exponential arithmetic Here is an age problem that requires exponential arithmetic. Alexander loves his grandmother. Here’s how to figure out her age. It is between 50 and 100, divisible by 8. Each of the grandmother’s sons has as many sons (her grandsons) as brothers. The combined number of her sons and grandsons equals her age. How old is she?

3.  Generational logarithm Above, the generation to which 1,024 ancestors belonged was computed using logarithmic method. Using the same method, to which generation do 256 ancestors belong?

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 E xplorations  |  8

4.  A tough nut This is a tough nut that became a meme on the Internet not too long ago: Can you figure out the value of x in √(x + 15) + √x = 15?

5.  Laws of exponents Here is a straightforward exploration what will allow you to review some of the  laws of exponents discussed in this chapter. Simply provide the missing exponents in each of the following, as indicated by n in each case.

(a) (2n ) (212 ) = 32, 768



(b) (37 ) (2n ) = 279, 936



(c) (5n ) ¸ (55 ) = 25



(d) (92 )n = 81

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6

e A very special number

You have no idea how much poetry there is in the calculation of a table of logarithms. —Carl Friedrich Gauss (1777–1855)

Prologue As discussed in chapter 4, π is a transcendental number. Another significant transcendental number is e, which is equal to 2.71828. . . . The relation and importance of these two numbers is brought out eloquently by Ian Stewart (2008, 101): The number e is one of those strange special numbers that appear in math­em­ at­ics, and have major significance. Another such number is π. These two num­ bers are the tip of an iceberg—there are many others. They are also arguably the most important of the special numbers because they crop up all over the math­ ematical landscape.

In correspondence between Gottfried Leibniz and Christian Huygens, there is mention of a constant with value 2.71828, which they represented with the let­ ter b. It was Leonhard Euler who used e in a 1731 letter to Christian Goldbach, and then again in his 1736 book, Mechanica. From then on, this has been the symbol used as the standard one in mathematics. The mathematical definition of e is: the limiting value of the series represented by the expression (1 + 1/n)n, as n becomes large without bound. A few values are given in Table 6.1.

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 Pro lo g u e  |  83

Table 6.1  Values for (1 + 1/n)n n

(1 + 1/n)n

Value

1

(1 + 1/1)1

2.0

2

(1 + 1/2)

2.25

5

(1 + 1/5)

10

(1 + 1/10)

2.59

100

(1 + 1/100)100

2.70

…

…

…

10,000

(1 + 1/10,000)10,000

2.71815

2

2.49

5 10

Another formula used by Euler to determine the value of e (and there are a significant number of them) is the following one. Note that “!” means factorial, the product of an integer and all the integers below it: 4! = 4 × 3 × 2 × 1, 3! = 3 × 2 × 1, etc. e = 1 / 0! + 1 / 1! + 1 / 2! + 1 / 3! + 1 / 4 ! + 1 / 5! + 1 / 6! + 1 / 7 ! ¼ Equivalent to: e = 1 + 1 + 1 / 2 + 1 / 6 + 1 / 24 + 1 / 120 + 1 / 720 + 1 / 5040 ¼ Equivalent to: e = 1 + 1 + 0.5 + 0.16 + 0.042 + 0.008 + 0.00138 + 0.000198 ¼ Now, one may ask: What possible significance does such a number have? As it turns out, it forms the base of natural logarithms, which have had broad impli­ cations and applications; it appears in functions associated with the calculus; it surfaces in formulas for various kinds of curves; it crops up frequently in prob­ ability theory; it appears in the calculation of compound interest; and it has been found in natural and social phenomena such as the following: 1 . radioactive decay 2. the growth of bacterial colonies 3. epidemiological patterns of all kinds 4. patterns in the accumulation of money 5. rates of change in many physical phenomena Even though the value of 2.71828 appeared in an appendix in John Napier’s 1618 book on logarithms (chapter 5), the first awareness of its mathematical

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84 |  E : A V ERY S PEC IAL N U M B ER significance is by English mathematician and clergyman William Oughtred (1574–1660), who defined it as the base of the natural logarithm. Here’s what this means, recalling logarithms from the previous chapter: The natural logarithm of a number such as 7.389 is defined as: loge (7.389), also written as: ln (7.389), which is approximately 2.

There is little doubt that this number has enormous significance, although the reasons why this is so remain as obscure as they are for π. This chapter deals with e. So many books, articles, research papers, websites, etc. have been writ­ ten on this subject that it would be presumptuous and even pretentious to claim originality and even depth of coverage here (see bibliography for relevant titles). The purpose is simply to link e to other great ideas, showing how math­ ematical discovery forms a chain—a chain constructed with a handful of great ideas that appear across time and space, finding form and explanation in the writings and musings of individual mathematicians.

Mathematical connectivity The number e is connected to many other concepts, structures, and patterns within mathematics, suggesting that it may be yet another kind of code for establishing interrelationships among mathematical entities. A significant property of e is that it is associated with various infinite series, thus linking it to the concept of mathematical infinity (to be discussed in chapter 8). The Swiss mathematician Jacob Bernoulli (1654–1705) came up with the value of e by studying compound interest. He noted that a 100% increase when paid out annually results in the sum doubling year after year. However, splitting the 100% over shorter periods results in more than doubling the return at the end of the year. Bernoulli estimated what the payout would be if the interest was calculated continuously over infinitely small periods—which can be repre­ sented as an exponential function. The answer he discovered was that it grew at the annual rate of 2.71828…, which is the value of e. Soon after, similar findings emerged from diverse fields that involve growth or change, including epidemi­ ology, the study of bacterial infections, and the like. Incidentally, Bernoulli’s estimation of e involved thinking of e as resulting from an infinite series. As one compounds more and more interest, the value of one’s investments approach a limit. Bernoulli thus defined e in terms of an infinite series that converges to a limit—that is, we can get closer and closer to the limit, but never quite reach it.

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Math e matica l co nn e ctivity  |  85



Euler was the first mathematician to study e in depth, coming up with the formula for an infinite series mentioned above in his 1748 book, Introductio in analysin infinitorum (“Introduction to Infinite Analysis”). Euler also noticed something intriguing, namely that ex, now called an exponential function, had intriguing properties. One formula for this function is the so-called Taylor Series, formulated by British mathematician Brook Taylor in 1715: Equivalent to:

e x = 1 + x / 1! + x 2 / 2! + x 3 / 3! + x 4 / 4 ! ¼



e x = 1 + x + x 2 / 2 + x 3 / 6 + x 4 / 24 ¼ One of the most surprising findings regarding this function is that in differen­ tiation none of the terms disappear, which they normally do. Rather, they repeat ad infinitum. Differentiation is an operation in the calculus. It allows us to determine the derivative of a function, which measures the slope of the graph of the function at a particular point on the graph. In other words, the derivative is a measure of change in the value of the function as the independ­ ent variable changes. This is yet another exceptional finding, implying that ex is its own derivative and, thus, that the graph of ex describes its own rate of change. Figure 6.1 is the graph showing how the slope rises dramatically. So, more technically, if f(x) = ex, then dy/dx = ex (where dy/dx is the symbol for differentiation). The negative exponential function, f(x) = e−x, shows a slope y 20

15

10

5

–3

–2

–1

1

2

Figure 6.1  The Exponential Function e-x (from Wikimedia Commons)

3

x

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86 |  E : A V ERY S PE C IAL N U M B ER in the opposite direction, thus visually demonstrating the relation between positive and negative exponents in the study of functions—another amazing discovery (see Figure 6.2). Yet another serendipitous appearance of ex is in the logarithmic spiral shown in Figure 6.3. In this spiral, the distance from the center O is 1/ex where x is the angle that results from the spiral turn. As x gets larger, 1/ex becomes smaller. It was Jacob Bernoulli who discovered this remarkable property, and, given that spirals occur throughout nature, this is a rather important one. So surprised was Bernoulli by this finding that he called it a spira mirabilis, a miraculous spiral. Research on this figure has revealed a host of other hidden patterns. Nature is replete with logarithmic spirals having a similar mathematical structure.

30

20

10

–4

–2

2

4

Figure 6.2  The Exponential Function, e-x (from Wikimedia Commons)

Figure 6.3  The Logarithmic Spiral (from Wikimedia Commons)

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Math e matica l co nn e ctivity  |  87

Bernoulli was so thrilled by his discovery that he wanted to be commemorated in death with it. As it turned out, an error was made, as Freiberger and Thomas (2014, 92) quip: Bernoulli was so fascinated by this self-similarity that he planned to have a logarithmic spiral engraved on his tombstone, along with the words (in Latin) ‘Although changed, I shall arise the same.’ Unfortunately, though, the wrong type of spiral, an Archimedean one whose loops are an equal distance apart, ended up adorning his grave. Poor Bernoulli, he must have turned (loga­rith­ mic­al­ly) in his grave. Perhaps he should have gone for a simpler shape, the trusty old triangle.

The function ex is a special one—called the natural exponential function. In gen­ eral, an exponential function is one that has the form: f(x) = ax, that is, a func­ tion in which the base is constant and the variable (x) is an exponent. Exponential functions have had many implications, not only in pure math­em­ at­ics, but in in the following fields: 1. The study of radioactivity, since radioactive substances decay at an exponential rate. 2. Population analysis, since if unchecked, a population (organisms, ­people, cells, etc.) will grow at a rate proportional to itself. 3. In electrical engineering, since the theory of alternating current relies on the exponential function. The connection between domains of science and exponential functions is something that is truly amazing, showing that mathematics and science are intrinsically intertwined. Indeed, mathematics is rightfully called the language of science. It has also been found that logarithmic functions occur in many areas of physics and biology. For example, the logarithm of measures of living tissue (skin area, weight), lengths of inert appendages (hair, claws), and certain physiological measures such as blood pressure all display a normal distribution. In economic theory, changes in the logarithms of exchange rates, as well as price and stock market indices also display the same type of distribution. The natural logarithm is also used by Google to give every page on the World Wide Web a score (PageRank), which is a rough measure of importance based on a logarithmic scale.

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88 |  E : A V ERY S PE C IAL N U M B ER

Euler’s identity Working with various exponential functions, Euler discovered an intriguing formula, known appropriately as Euler’s formula, where “i” is an imaginary number (see chapter 7): e ij = cos j + i sin j The sine (shortened to sin) of an acute angle is defined in terms of a right tri­ angle: it is the ratio of the length of the side that is opposite a given angle to the length of the longest side of the triangle (the hypotenuse). The cosine (short­ ened to cos) is the ratio of the length of the adjacent side to a given angle divided by the hypotenuse. The above formula can be plotted on the complex number plane as shown in Figure 6.4 below—complex numbers will be examined in the next chapter (Im = imaginary; Re = real): Now, let’s bring π into the picture. It is known in trigonometry that:

sin p = 0

cos p = - 1

Let’s replace φ with π in the formula above with these values:

e ij = cos j + i sin j e i p = cos p + i sin p



Im i

eiφ = cos φ + i sin φ

sin φ φ 0 cos φ

Figure 6.4  Euler’s identity (from Wikimedia Commons)

1 Re

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Epi lo g u e  |  89

ei p = - 1 + (i ´ 0 )



ei p + 1 = 0



This is called Euler’s identity. It connects some of the most significant numbers in the history of mathematics—e, i, π, 1, and 0. In 1990, the Mathematical Intelligencer periodical took a survey of its readers, conducted by math­em­at­ ician David Wells, asking them to vote on the “most beautiful theorem in math­ em­at­ics.” Euler’s identity came out as the winner by a large margin. It certainly is easy to see why mathematicians perceive it as having an inherent beauty, since it connects some of the most important numbers in mathematics. There are many interesting and bizarre aspects of this equation, starting with the fact that it is really not an equation in the traditional sense, as Henshaw (2014, 58) aptly observes: Generally, the term “equation” implies something that it can be solved. For example, we can solve x + 2 = 4 and get the result x = 2. But Euler’s identity isn’t like that. There’s nothing to solve for. It’s just a statement of fact, like saying that 2 + 2 = 4. And so, just as 2 + 2 = 4 is a fact, eiπ + 1 = 0 is also a fact.

Epilogue The fact that e is a descriptor of physical reality, especially in the domains of growth, decay, change, and probability once again raises the question left by Pythagoras’ legacy: Why is mathematics such a powerful and unexpected lan­ guage of truth? Also, as we have seen in this chapter: Why is change in natural phenomena logarithmic and described by exponential functions? Even practical matters are governed by such mathematics. Consider the fol­ lowing: Bank A offers 100% interest on money. In one year, a sum of $1000 becomes $2000. Bank B pays 50% twice a year. In this case, $1000 becomes $1500 after six months and this becomes $2250 after twelve months. This is clearly the better deal. Why is this so?

The formula for compound interest is the following one: (1 + r/n)n r = annual interest rate (using 1 rather than 100%)

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90 |  E : A V ERY S PE C IAL N U M B ER n = number of yearly periods In the yearly model, r = 1 and n = 1: (1 + 1/1)1 = 2.0 In the half-yearly model, r = 1 and n = 2 (twice a year): (1 + 1/2)2 = 2.25 In the monthly model, r = 1, n =12 (twelve times a year):

(1 + 1 / 12 )

12

= 2.613



The higher the value of n the greater the yield, and this shows why Bank B gave the better deal. Recall the formula for e at the start of this chapter:

(1 + 1 / n ) As can be seen, the compound interest formula is based on this very formula. When the interest rate is n, the formulas become the same. As Euler’s identity seems to suggest, there may be some deeper meaning in e, as it relates to other numbers, of which we are not yet aware. All this suggests that mathematics is itself a conundrum. Its discoveries, such as the Pythagorean theorem, e, π, and so on, cannot be tied down to a specific meaning, even if they emerge in a context, including practical calculations such as those related to compound interest. n

Explorations 1.  A possible series for e The number e has been determined by a number of series, as illustrated in this chapter. Determine if any of the following approaches e (2.71828); that is, if the sum of the terms might approach e as a limit.

(a ) 1 + 1 / 2 + 1 / 3 + 1 / 4 + 1 / 5 + 1 / 6 + 1 / 7 + 1 / 8 ¼

( b) 1 - 1 / 2 + 1 / 3 - 1 / 4 + 1 / 5 - 1 / 6 + 1 / 7 - 1 / 8 ¼ (c) 1 ´ 1 / 2 ´ 1 / 3 ´ 1 / 4 ´ 1 / 5 ´ 1 / 6 ´ 1 / 7 ´ 1 / 8 ¼

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E xplo r ati o ns  |  9

2.  Another series Here is another series. It was introduced in this chapter: 1 / 0! + 1 / 1! + 1 / 2! + 1 / 3! + 1 / 4 ! + 1 / 5! + ¼ Note that in this case 1/0! = 1. Does this series allow us to estimate the value of e to any degree of accuracy?

3.  Plotting a function Plot the function, y = f(x) = x2 and then describe its general shape. In Table 6.2 below are some values:

Table 6.2  Values for f(x) = x2 x

0

1

−1

2

−2

…

f(x) = x2

02 = 0

12 = 1

–12 = 1

22 = 4

−22 = 4

…

4.  Compound interest The notions in this chapter are used in calculations of compound interest, as dis­covered by Jacob Bernoulli (above). Here is a problem that is similar to the one above. You are offered a new part-time job as a pizza delivery person, working only on weekends. Your boss gives you a choice in salary options as follows: Option A: $4,000 for your first year of work, and a raise of $800 for each year after the first. Option B: $2,000 for your first six months of work, and a raise of $200 every six months thereafter. Which offer is the better one?

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92 |  E : A V ERY S PE C IAL N U M B ER

5.  The exponential function ex Below are a few values for ex:

x x x x

= 1, e x = 2, e x = 3, e x = 4, e x

= 2.718 = 7.389 = 20.085 = 54.598

What power of x would produce the value of 1096.633? What does this imply generally about change (growth, decay, etc.) that is described by this function?

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7

i Imaginary numbers

Imaginary numbers are a fine and wonderful refuge of the divine spirit, almost an amphibian between being and non-being. —Gottfried Leibniz (1831–1881)

Prologue Consider the solution of the following quadratic equation, which is an equation with a variable, x in this case, which is squared:

x2 + 1 = 0



x 2 = -1



x = Ö - 1 or, moreprecisely, ± Ö - 1

(

)

What kind of number is √−1? In a way that parallels the unexpected discovery of √2 by the Pythagoreans (chapter 1), when this number surfaced as a solu­ tion to a quadratic equation, mathematicians asked themselves what it could possibly mean. Not knowing what to call it, René Descartes named it, ­logically enough, an imaginary number. Euler introduced the letter “i” to stand for this number. Like the irrationals, the discovery of i led to new ideas and discoveries. One of these was complex numbers—numbers having the form

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94 |  I : I MAG I N ARY N U M B E R S (a + bi), where a and b are real numbers and i is √−1. Incredibly, complex numbers turn out to have many applications. They are used, for instance, to describe electric circuits and electromagnetic radiation and they are fun­ damental to quantum theory in physics. Returning to the equation above, x2 + 1 = 0, if i is used like any other number, the equation has the following solution: x = ±i. In his treatise on equations, Ars magna (The Great Art), Italian math­em­at­ ician Gerolamo Cardano (1501–1576) was among the first to come across the square root of negative numbers (Salem, Testard, and Salem 1992, 48–49). In  one part, he was examining two numbers, x and y, which add up to 10 and, when multiplied, come to 40:

(1) x + y = 10 ( 2 ) xy = 40

The solution is shown below:

From (1): y = 10 − x Substituting in (2): x (10 − x) = 40 Expanding and transposing: −x2 + 10x − 40 = 0 Multiplying by −1: x2 − 10x + 40 = 0 x = 5 ± √−15

Cardano decided to abandon exploring the implications of this result calling the solution fictitious. On the other hand, Cardano’s contemporary and com­ pat­riot Rafael Bombelli (1526–1572) saw such numbers as simply that—num­ bers that show up as the roots of some equations, formulating the first coherent theory of complex numbers in his Algebra of 1572. So, the values of Cardano’s equation could be written as follows:

x = 5 ± i Ö 15

This chapter will look at the implications and significance of imaginary numbers, which constitute yet another of the great ideas of mathematics that have not only changed the course of mathematics but also of human history. Contemporary science and engineering would be much more limited ­without them.

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Q uad rati c equati o ns  |  95



Quadratic equations Quadratic equations were known throughout the ancient world, from Babylon to China and India. The concept of “variable” and the use of symbols to repre­ sent numbers is traced to Greek mathematician Diophantus (c. 250 ce), who lived in Alexandria, Egypt, where he developed the techniques that later earned him the title “father of algebra.” He also discussed methods of solving quadratic equations—similar methods were elaborated a few centuries later by the Hindu scholar Aryabhata (c. 476–550 ce). It is useful here to rapidly review quadratic equations, which are defined as equations in which the power of one of the variables is 2. The general form of the quadratic equation is as follows:

ax 2 + bx + c = 0

The a and b are the numerical coefficients of x2 and x respectively. The letter c represents the constant in the equation. Thus, for example, if a = 3, b = 2, and c = 5, then: ax2 + bx + c = 3x2 + 2x + 5 = 0. Consider the simple equation below, noting that in this case, a = 1, b = 0, and c = −16:

x 2 - 16 = 0

The solution is determined as follows:

x 2 - 16 = 0



x 2 = 16



x = + 4 or - 4, shortened to ± 4

Now, consider the following equation:

x 2 + 8 x + 15 = 0

We know that the expression (x2 + 8x + 16) represents a perfect square because it can be rewritten as (x + 4)2. So, let’s add 16 to both sides of the equation above, in order to produce a perfect square on the left side. This does not alter the values of the equation since the same number has been added to both sides. Note that 15 is moved to the right side, becoming −15:

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96 |  I : I MAG I N ARY N U M B E R S

x 2 + 8 x + 16 = - 15 + 16



x 2 + 8 x + 16 = 1





(x + 4) = 1 Ö ( x + 4 ) = Ö1 (x + 4) = 1



x = -3

2

2



Or:

-( x + 4) = 1



x = -5

Solution set:

x = {-3, - 5}

The most common method for solving a quadratic equation is to use the ­following formula:

x=

-b ± b2 - 4ac 2a



Without going into a step-by-step explanation, we plug these values in the equation above and we end up with the same solution: x = {−3, −5}. The Babylonians were among the first to develop an algorithm for solving quadratic equations, as is evidenced by a Babylonian tablet preserved in the British Museum in which the following problem is posed (Solomon 2008, 11–12): The area of a square added to the side of a square comes to 0.75. What is the side of the square?

The solution unfolds as shown in Table 7.1, with corresponding contemporary algebraic notation added to show the reasoning involved. The graphs of quadratic functions show parabolic curvature. Where the parabola touches or crosses the x-axis is where the two solutions lie. For ex­ample, the graph in Figure 7.1 shows the solution set of x2 − x − 2 = 0, namely x ={ −1, +2}, which are the points on the x-axis crossed by the curve:

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Co mple x numbers  |  97

Table 7.1  Babylonian Algorithm for Quadratic Equations Babylonian Tablet

Contemporary Notation

I have added the area and the side of my square: 0.75.

x2 + x = 0.75

You write down 1, the coefficient.

Coefficient of x is 1

You break half of 1.

Half of 1 is 0.5

You multiply 0.5 and 0.5.

(0.5)2 = 0.25

You add 0.25 and 0.75.

0.25 + 0.75 = 1

This is the square of 1.

√1 = 1

Subtract 0.5, which you multiplied.

1 – 0.5 = 0.5

0.5 is the side of the square

x = 0.5

y 3 2 1 x –3

–2

y = x2–x–2

–1

0

1

2

3

–1 –2 –3

Figure 7.1  Graph of y = x2 − x −2 (from Wikimedia Commons)

Complex numbers Squaring a negative number produces a positive number (chapter 3). So, what kind of number is √−1? All we can say is that it does not behave like any of the real numbers:

i = Ö -1



i 2 = (Ö - 1) ´ (Ö - 1) = - 1

The first mention of such numbers is traceable, actually, to the great twelfthcentury Hindu mathematician Bhāskara (1114–1185). This was followed

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98 |  I : I MAG I N ARY N U M B E R S c­ enturies later by Cardano and Bombelli, as mentioned above. It was JeanRobert Argand (1768–1822) who showed how imaginary numbers and real numbers could be interconnected, followed by Carl Friedrich Gauss (1777–1855), who introduced the term, complex number in 1831. For example, every real number can be represented as a complex number, by simply letting the imaginary part be 0. So, for instance, 5 is: a + bi = 5 + 0i = 5



In 1797 Gauss proved something revolutionary—namely, that complex num­ bers could solve any equation built from real numbers. This means that every equation has a full set of solutions among the complex numbers. When imaginary numbers were first discovered, it was not clear how they fit into the number system or how they could be represented on the Cartesian plane. This conundrum led to the ingenious invention of a diagram by Argand that made it possible to show the relation of imaginary numbers to real ones. Figure 7.2 is the representation of (a + bi) on the Argand plane (Im = imaginary number line; Re = real number line). To add two complex numbers, say (a + bi) and (c + di), we add each part (real and imaginary) separately:

( a + bi ) + ( c + di ) = ( a + c ) + (b + d )i

Below is an example:

( 4 + 3i ) + ( 5 + 6i ) = ( 4 + 5) + ( 3 + 6 )i = 9 + 9i

To multiply complex numbers we multiply them in the usual algebraic fashion:

( a + bi ) ( c + di ) = ac + ( ad )i + (bc )i + (bd )i

2



Im b

Figure 7.2  Graph of a + bi (from Wikimedia Commons)

0

a+bi

a

Re

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F un damental Th e o rem o f Algebra  |  99



Below is an example. Note that i2 = −1, so 18i2 = (18) (−1) = −18:

( 4 + 3i ) ( 5 + 6i ) = 20 + ( 24 )i + (15)i + 18i

2

= 20 + ( 24 ) i + (15 ) i - 18 = ( 2 + 39i )

The other operations can also be specified in the same way, using the formulas (a + bi) and (c + di). The point is, as Bombelli and others suggested, that com­ plex arithmetic is just like any type of arithmetic. The difference is, of course, the addition of i to it.

Fundamental Theorem of Algebra The study of complex numbers led to an extraordinary discovery, known as the Fundamental Theorem of Algebra, which states that the field of complex num­ bers is algebraically closed—that is, there is no polynomial expression that does not have at least one complex root. It stipulates that an equation of degree n has n roots. So, for example a quadratic equation will have two roots, a cubic equa­ tion three, a quartic equation four, and so on. Let’s look at a few for the sake of illustration: Linear (one root): x−5=0 x = {5} Quadratic (two roots): x2 − 4 = 0 x = {+2, −2} Cubic (three roots): 2x3 + 3x2 −11x − 6 = 0 x = {2, −1/2, −3) Quartic (four roots): 3x4 + 6x3 − 123x2 − 126x + 1080 = 0 x = {3, 5, −4, −6}

This theorem was already implicit in the work of Descartes and Albert Girard (1595–1632). The first attempt to prove it is traced to French mathematician Jean d’Alembert (1717–1783) in 1746. But his proof contained some infelicities, which were corrected by Gauss, who presented his corrections in 1799. Gauss published a complete proof in 1816, building on ideas established by Euler.

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00 |  I : I MAG I N ARY N U M B E R S The Fundamental Theorem is part of a broader theory of equations. It was Evariste Galois (1811–1832) who took a closer look at equations after the the­ orem was established, showing that solutions to equations could be carried out with permutations, leading to the emergence of group theory. Solutions to the quadratic, the cubic, and quartic have been found. However, no solution to the quintic (equation with a variable to the power of five) has ever emerged. Galois found that the groups of permutations associated with the quadratic, cubic, and quartic share a property that the quintic does not. This is the reason why no formula can be devised to solve the quintic.

Epilogue The number i came from solving quadratic equations and had no meaning ­initially—much like √2 had no meaning for the Pythagoreans. After being labeled an imaginary number, it took on a mathematical life all its own, opening up a new way for expanding, classifying, and thus understanding numbers. Where were these numbers before they were discovered? As Ian Stewart (2013, 313) observes, the concept of existence in any treatment of mathematics raises sev­ eral key questions, the most obvious one being the definition of existence itself: The deep question here is the meaning of “exist” in mathematics. In the real world, something exists if you can observe it, or, failing that, infer its necessary presence from things that can be observed. We know that gravity exists because we can observe its effects, even though no one can see gravity . . . However, the number two is not like that. It is not a thing, but a conceptual construct.

The irrational and the imaginary numbers did not “exist” until they cropped up in the use of the Pythagorean theorem and in solving quadratic equations respectively. Were they waiting to be discovered? This question is clearly at the core of the nature of mathematics. This “story” can be told over and over within the field—transfinite numbers, graph theory, and so on. These did not “exist” until they crystallized in the conduct of mathematics, through ingenious nota­ tional modifications, diagrammatic insights, various explorations with math­em­ at­ic­al signs, and so on. Coming up with a theory of discovery in mathematics is an intractable task, since so many unknown factors are involved. Consider the following question: What would happen if we add j and k to i in the Argand plane? The result is numbers with four parts. These were described by Irish mathematician William Rowan Hamilton (1806–1865) in

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E pi lo gue  |  0

1843. He called them quaternions. They have led to a whole set of new rules of arithmetic shown in the multiplication table in Table 7.2. Hamilton showed, in effect, that complex numbers were a subset of quater­ nions. Just as complex numbers can be shown as points on a plane, so quater­ nions can be considered to be points in a graph such as the one in Figure 7.3. As it turns out, quaternions have uncovered the “existence” of many previ­ ously unknown number patterns and have had significant applications in phys­ ics. Quaternion theory has become especially significant for describing rotation in space. And it is now an essential tool in computer graphics, molecular mod­ eling, and space flight. Serendipity is likely to be a product of the imagination, as the late writer and semiotician Umberto Eco argued in his book Serendipities: Language and Lunacy (1998). The term was coined by English writer Horace Walpole who came across an ancient Persian tale, The Three Princes of Serendip, that sug­ gested serendipity as a principle of life. The tale goes somewhat as follows. Three princes from Ceylon were journeying in a strange land when they came upon a man looking for his lost camel. The princes had never seen the animal, but asked the owner: Was it missing a tooth? Was it blind in one eye? Was it

Table 7.2  Multiplication of quaternions ×

1

i

j

k

1

1

i

j

k

i

i

−1

k

−j

j

j

−k

−1

i

k

k

j

−i

−1

i –k

j

–1

1 –j

–k –i

ij = k ji = –k ij = –ji

Figure 7.3  Quaternion graph (from Wikimedia Commons)

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02 |  I : I MAG I N ARY N U M B E R S lame? Was it laden with butter on one side and honey on the other? Was a preg­ nant woman riding it? Incredibly, the answer to all their questions was “yes.” The owner accused the princes of having stolen the animal since, clearly, they could not have had such precise knowledge. But the princes merely pointed out that they had observed the road, noticing several patterns in it: for example, the grass on either side was uneven, suggesting a lame gait; there were places where chewed food seemed to emerge from a gap in the animal’s mouth; there were uneven patterns of footprints, the signs of awkward mounting and dismounting typical of someone who was pregnant; and there were differing accumulations of ants and flies, which congregate around butter and honey. Their questions were really prompted by inferences based on these observations. Knowing the world in which they lived, they were able to make concrete connections between the observations and what happened. Since Serendip was Ceylon’s ancient name, Walpole coined the word serendipity, to designate how we come about our discoveries, such as mathematical and scientific ones, in a similar way.

Explorations 1.  Imaginary numbers To get a sense of what i means, figure out the following:

(1) i 3 (2) i6 (3) i0

2.  A square root Express √−9 as an imaginary number.

3. Conjugates A conjugate number is a complex number with the opposite middle sign. So, the conjugate of the number (a + bi) is (a − bi), and vice versa. What happens when you multiply conjugates?

( a + bi ) ( a - bi ) = ?

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 E xplo rati o ns  |  03

4.  Conjugate arithmetic Now, apply what you discovered in exploration 3 to multiply the following:

( a ) ( 3 + 2i ) ´ ( 3 - 2i ) = ?



( b ) ( 5 + 3i ) ´ ( 5 - 3i ) = ?

5.  Powers of complex numbers Can you figure out the following?

( a ) ( 4 - 5i )

2





( b ) ( 3 - 3i )

3



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8

Infinity A counterintuitive and paradoxical idea

I cannot help it—in spite of myself, infinity torments me. —Alfred de Musset (1810–1857)

Prologue The ancient philosopher Zeno (c. 490–430 bce), who lived in the Greek colony of Elea in southern Italy, was a staunch defender of his teacher, Parmenides (who lived circa 514 bce), devising a series of artful arguments, called paradoxes, in sup­ port of his mentor’s view that motion and change are impossible. As an e­ xample of the kind of clever reasoning that Zeno deployed, consider the following: If we throw a stone from point A to point B, one mile away, the stone will never reach B because first it must reach the halfway point, then from there another half distance, and after that another half distance, and so on ad infinitum. The stone will thus come close to B, but never reach it.

One way to represent this visually is in Figure 8.1. A

B

1/2

1/4

1/8

…

Figure 8.1  Successive half distances of /2, /4, /8 . . .

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Z e n o ’ s par ad oxe s  |  05



This shows that the stone must first traverse half the distance to the finish line, or 1/2 (mile). Then, from mid position, it must traverse half of the remaining distance between itself and the finish line, which is a distance of 1/4 (mile). But thereupon, the stone must once more cover half of the new remaining dis­ tance between itself and the finish line, which is a distance of 1/8 (mile). Although the successive half distances between the stone and the finish line would become increasingly (indeed infinitesimally) small, the stone would come very close to the finish line, but would never cross it. The successive dis­ tances that it must cover form an infinite sequence, each term of which is half the one before: {1/2, 1/4, 1/8, 1/16, . . . .}. The sum of the terms in this sequence— {1/2 + 1/4 + 1/8 + 1/16 + . . . .}—will never reach 1 (mile), the whole distance to be covered, but will approach it ever so closely. This is, actually, a version of Zeno’s Dichotomy Paradox, showing how a lin­ ear distance consists of points that dichotomize (divide) it infinitely. His para­ doxes are examples of one of the most enigmatic and counterintuitive concepts in mathematics—infinity. This chapter will deal with this topic, which is one of the most intriguing of all mathematical notions.

Zeno’s paradoxes Zeno is believed to have contrived at least 40 paradoxes, but only eight have survived. They have come down to us indirectly—some have come through Aristotle, which he included in his Physics. The four paradoxes concerning motion make up his most famous surviving ones (Salmon 1970, Mazur 2008). They were labeled dialectical (capricious and specious) by Aristotle, who tried to dismiss them, but could not really do so. In the original version of the Dichotomy Paradox, Zeno argued that a runner would never reach the end of a race course. If the length of the race course is represented by a line, with unit length, the successive stages of the runner’s location can be shown in Figure 8.2.

1/2

1/4

1/8

1/16

Figure 8.2  Zeno’s Dichotomy Paradox (from Wikimedia Commons)

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06 |  I N F I N IT Y As mentioned, these form an infinite series with each term in it half of the previous one. The paradox raised profound issues about time, space, and infin­ ity, leading over the centuries to the concept of limits, which, in turn, inspired the calculus. Another of Zeno’s famous paradoxes is the Achilles and the Tortoise Paradox, again recounted in Aristotle’s Physics (VI: chapter 8). It says that Achilles, the fleetest runner of antiquity, will never overtake a tortoise in a race if the tortoise is given a head start. Below is a paraphrase: Achilles decides to race against a turtle. To make the race fairer, he allows the turtle to start at half the distance away from the finish line. In this way, Achilles will never surpass the turtle. Why?

In order for Achilles to surpass the tortoise, he must first reach the halfway point, which is the tortoise’s starting point. But when he does, the tortoise will have moved forward a bit (since it is also moving). Achilles must then reach this new point before attempting to surpass the tortoise. When he does, how­ ever, the tortoise has again moved a little bit forward, which Achilles must also reach again, and so on, ad infinitum. In other words, although the distances between Achilles and the tortoise will continue to get smaller, Achilles will never surpass the tortoise. Of course, in reality Achilles will do so, because motion is also a factor of time (and other factors), not only a matter of travers­ ing discrete points in space. The other two surviving paradoxes are called the Arrow and the Stadium Paradoxes. The former asserts that an arrow in flight is really at rest as well. Imagine an arrow in flight. At any given moment, the arrow has an exact loca­ tion and so it is not moving (with respect to that specific location). Moreover, it cannot move forward to where it is not from that point, because no time elapses for it to do so. In other words, it is at rest at every point and every moment. The Stadium Paradox involves three rows of identical marchers in a stadium. The middle row is stationary (S1, S2, S3); the bottom or lowest row (L1, L2, L3) and the top row (R1, R2, R3) move in parallel at exactly the same speeds, but in opposite directions. Let’s assume they are positioned initially as follows: R1

R2

R3

S1

S2

S3

L1

L2

L3

Note that R3, S2, and L1 are in line with each other vertically in this con­fig­ ur­ation. Now, suppose the Ls move to the left one spot and the Rs to the right one spot, as the Ss remain motionless:

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Th e Liar Par ad ox  |  07



R1

R2

R3

→

S1

S2

S3

←

L1

L2

R1

R2

R3

S1

S2

S3

L1

L2

L3

L3

Result:

As can be seen, R3, which has moved one space to the right (along with the other Rs) is vertically over S3 and L3. In the bottom row, the movement of the Ls one space to the left, has led to L1 becoming aligned vertically under S1 and R1. But at the same time L1 is now two spaces away from being under R3, even though it was lined up with it previous to the move. How is it possible for them to be two spaces apart in a single move? In other words, how is it possible that the Ls and the Rs move at the same rate with respect to the Ss but twice as fast with respect to each other? Zeno’s paradoxes portray movement in terms of discrete points on a number line, where a move from A to B is done in separate (discrete) steps. But the gap in between is continuous. So, to resolve the paradoxes a basic distinction between discrete and continuous is required—a gap that was filled with the calculus much later.

The Liar Paradox While Zeno’s paradoxes dealt with motion, space, and change; other Greek ­philosophers, including Eubulides of Miletus (fourth century bce), devised a different kind of paradox, known as the self-contradicting liar. In its most famous formulation, it is attributed to a Cretan named Epimenides in the sixth century bce about whom almost nothing is known, but about whom various myths and legends were spun, which need not concern us here. All Cretans are liars. I am a Cretan. Do I speak the truth?

If we assume the statement to be true, we can then conclude that Epimenides, being himself a Cretan, was lying, as the statement (“All Cretans are liars”) so declares. But, if he is a liar, then his statement cannot be true, as we had assumed that it was. It can only be a lie, according to his nature. We have reached a

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08 |  I N F I N IT Y c­ ontradiction. Therefore, our assumption is not correct. The statement must be false—Cretans are not liars, but truth-tellers. Epimenides is thus a truth-teller, being a Cretan himself. But, then, why did he make a false statement, declaring that Cretans are liars, which includes himself, a supposed truth-teller? It is clearly impossible to determine if Epimenides spoke the truth. This paradox has fascinated logicians, mathematicians, and philosophers throughout history. The English philosopher Bertrand Russell (1872–1970) found it to be especially troubling. His version of it, known as the Barber Paradox, has animated many debates in twentieth-century logical and math­ em­at­ic­al circles. We will return to it in the next chapter. For some reason, circu­ lar statements such as this paradox have a bizarre appeal, as the famous British puzzle-maker, Henry E. Dudeney (1958, 15), perceptively observed: A child asked, “Can God do everything?” On receiving an affirmative reply, she at once said: “Then can He make a stone so heavy that He can’t lift it?”

The child’s purported question is similar to a philosophical conundrum: What would happen if an irresistible moving body came into contact with an im­mov­ able body? As Dudeney went on to observe, such bizarre paradoxes arise only because we take delight in inventing them. In actual fact, “if there existed such a thing as an immovable body, there could not at the same time exist a moving body that nothing could resist.”

Galileo’s and Cantor’s paradoxes In the sixteenth century the Italian scientist Galileo Galilei (1564–1642) came up with a paradox that seemed to pose a challenge to common sense, much like  those of Zeno did in antiquity. As early as 1638, he observed that there are  as many square numbers as there are whole positive integers. As absurd as  this might seem, this can be shown by putting the set of square integers, {2, 4, 9, 16, 25…} in a one-to-one correspondence with all the whole numbers {1, 2, 3, 4, 5…}: 1

2

3

4

5

6

7

8

9

10

11

12

13

14

…

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

…

1

4

9

16

25

36

49

64

81

100

121

144

169

196

…

No matter how far one continues along this linear one-to-one correspond­ ence, there will never be a gap between the top line and the bottom one. This

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Galileo ’ s an d Ca nto r ’ s par ad oxe s  |  09



suggests that the “number” of elements in the set of all positive whole integers and the “number” of those in one of its proper subsets, the square integers, is the same. If one stops to think about it, this is indeed an astonishing para­ doxical result. It implies the preposterous possibility that there may be as many square integers as there are whole numbers, even though the squares are themselves only a part of the set of integers. As Clark (2007, 68) aptly puts it: “We are so accustomed to thinking of finite collections that our intuitions become disturbed when we first consider infinite sets like the set of positive integers.” This one-to-one comparison technique no doubt inspired German math­em­ at­ician Georg Cantor’s (1845–1918) truly incredible demonstrations about in­fin­ite sets around 1870. Like Galileo, he showed that the set of the whole or counting numbers, also called cardinal numbers, can be matched against any of its subsets, such as the even numbers: 1

2

3

4

5

6

7

8

9

10

11

12

13

14

…

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

…

2

4

6

8

10 12

14

16

18

20

22

24

26

28

…

The even numbers are said to have the same “cardinality,” or same number of  elements of the whole numbers, because they can be put in a one-to-one ­correspondence with them. Remarkably, from these types of counterintuitive and paradoxical demonstrations have come many insights into the nature of  numbers, sets, and mathematical infinity. Cantor’s proofs were, in fact, ­earth-shattering in mathematical circles when he first made them public. Like the paradoxes of Zeno, they revolutionized thinking about numbers, infinity, and sets. Consider the set of rational numbers. As we saw (chapter  1), these are ­numbers that can be written in the form p/q where p and q are integers (and q ≠ 0)—so, 2/3, 5/8, 4/7 are rational numbers. The cardinal (whole) numbers are, themselves, a subset of the rationals—every integer p can be written in the form p/1: 5 = 5/1, 6 = 6/1, etc. Terminating decimal numbers are also included, because a number such as 3.579 can be written in p/q form as 3579/1000. Finally, all repeating decimal numbers are defined as rational, although the demonstration of this is beyond the scope of the present discussion. For ex­ample, 0.3333333. . . can be written as 1/3. Amazingly, Cantor demonstrated that the rationals also have the same cardinality as the set of integers. His method of proof is ingeniously elegant and simple. First, he arranged the set of all rational numbers as shown in Figure 8.3.

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0 |  I N F I N IT Y 1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 … 2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 … 3/1 3/2 3/3 3/4 3/5 3/6 3/7 3/8 … 4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8 … 5/1 5/2 5/3 5/4 5/5 5/6 5/7 5/8 … 6/1 6/2 6/3 6/4 6/5 6/6 6/7 6/8 … 7/1 7/2 7/3 7/4 7/5 7/6 7/7 7/8 …

…

…

…

…

…

…

…

…

8/1 8/2 8/3 8/4 8/5 8/6 8/7 8/8 … …

Figure 8.3  Cantor’s diagonal proof (from Wikimedia Commons)

In each row the successive denominators (q) are the integers {1, 2, 3, 4, 5, 6, . . .}. The numerator (p) of all the numbers in the first row is 1, of all those in the second row 2, of all those in the third row 3, and so on. In this way, all numbers of the form p/q are covered in the array. Every fraction in which the numerator and the denominator have a common factor is included in the array. If these are deleted, then every rational number appears once and only once in it. Now, Cantor set up a one-to-one correspondence between the positive inte­ gers and the numbers in the array as follows: he let the cardinal number 1 cor­ respond to 1/1 at the top left-hand corner of the array; 2 to the number below (2/1); following the arrow, he let 3 correspond to 1/2; following the arrow, he let 4 correspond to 1/3; and so on, ad infinitum. The path indicated by the arrows, therefore, allows us to set up a one-to-one correspondence between all the car­ dinal numbers and all the rational numbers (eliminating the numbers that are repetitions): 1

2

3

4

5

6

7

8

9

10

11

12

13

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

↕

1/1

2/1

1/2

1/3

3/1

4/1

3/2

2/3

1/4

1/5

5/1

6/1

5/2

…

…

There is only one inescapable conclusion—there are as many rational num­ bers as there are whole (cardinal) numbers. One cannot help but be impressed by the imaginative way in which Cantor constructed this proof. There was some doubt about the validity of his method, but most mathematicians now see it as a valid proof. Cantor classified those numbers with the same cardinality as belonging to the set “aleph null”, or ℵ0 (ℵ is the first letter of the Hebrew alphabet). He called ℵ0 a transfinite number. Amazingly, Cantor found that there are other ­transfinite

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Galileo ’ s an d Ca nto r ’ s par ad oxe s  |  

numbers. These are sets of numbers with a greater cardinality than the integers. He labeled each successively larger transfinite number with increasing sub­ scripts {ℵ0, ℵ1, ℵ2, . . .ℵn}. Now, one may ask: How can there be different transfinite numbers? Cantor’s proof is again remarkable for its simplicity and originality. Suppose we take all the possible numbers that exist between 0 and 1 on the number line and lay them out in decimal form. Let’s label each number as follows: {N1, N2,…}. Here are a few possibilities:

N1 = .4225896¼ N2 = .7166932¼ N3 = .7796419¼ ¼

How could we possibly construct a number that is not on that list? Let’s call it C. To create it, we do the following: (1) for its first digit after the decimal point we choose a digit that is greater by one than the first digit in the first place of N1; (2) for its second digit we choose a digit that is greater by one than the second digit in the second place of N2; (3) for its third digit we choose a digit that is greater by one than the third digit in the third place of N3; and so on: N1 = .4225896… The constructed number, C, would start with 5 rather than 4 after the decimal: C = .5… N2 = .7166932… The constructed number would have 2 rather than 1 in that position: C = .52… N3 = .7796419… The constructed number would have 0 rather than 9: C = .520… …

Now, the number C = .520. . . is different from N1, N2, N3, . . . because its first digit is different from the first digit in N1; its second digit is different from the second digit in N2; its third digit is different from the third digit in N3, and so on, ad infinitum. We have in fact just constructed a different transfinite number than ℵ0. It appears nowhere in the list above.

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Hilbert’s infinite hotel paradox German mathematician David Hilbert (1862–1943) would explain Cantor’s approach to infinity to his students with an imaginary scenario that can be paraphrased as follows: Imagine being a receptionist at a hotel that has an infinite number of rooms. One day it happens that the hotel is fully booked. However, it is a policy of the hotel to never turn anyone away. A new guest arrives, so obeying the policy, the receptionist simply asks all the guests to move to the room number that is one more than their current one. Since there is an infinite number of rooms, the new guest can now be accommodated in room number 1; the guest who was in that room has moved to room number 2, and so on. In effect, we cannot fully book the rooms in an infinite hotel.

Now, what would happen if an infinity of new guests arrived? George Gamow explains what would happen in his 1947 book, One, Two, Three . . . Infinity: Let us imagine a hotel with an infinite number of rooms, and all the rooms are occupied. To this hotel comes a new guest and asks for a room. “But of course!” exclaims the proprietor, and he moves the person previously occupying room N1 into room N2, the person from room N2 into room N3, the person from room N3 into room N4, and so on. . . . And the new customer receives room N1, which became free as the result of these transpositions. Let us imagine now a hotel with an infinite number of rooms, all taken up, and an infinite number of new guests who come in and ask for rooms. “Certainly, gentlemen,” says the proprietor, “just wait a minute.” He moves the occupant of N1 into N2, the occupant of N2 into N4, the occupant of N3 into N6, and so on. . . . Now all ­odd-numbered rooms have become free and the infinite number of new guests can easily be accommodated in them.

Epilogue To sum up, Zeno’s counterintuitive infinity paradoxes made an intellectual impact, leading eventually to new mathematics. When the calculus was first proposed, it too seemed nonsensical. Indeed, the Irish prelate and philosopher George Berkeley (1685–1753) charged that it was a useless science because it dealt with small, meaningless quantities. But the calculus survived such attacks because it produced answers to the classical problems related to the nature of

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E xplo r ati o n s  |  3

change. It is truly mind-boggling to think that the train of thought initiated by seemingly trivial paradoxes invented by an ancient philosopher led to the establishment of scientific ideas that have allowed human beings to learn so much about the universe and about themselves. The paradoxes were not the only ways in which infinity caught the interest of ancient mathematicians. It showed up implicitly in the methods to determine π, as we saw in chapter 4, whereby Archimedes inscribed and circumscribed a the­or­­ etic­al­ly infinite number of polygons in a circle to do so. Euclid’s proof of the infin­ ity of primes (chapter 2) is another case of reasoning about infinity in an ingenious way. Without mathematics, infinity would have remained an obscure concept and might not even have crystallized at all as something useful or even real. A paradox, such as the one by Galileo, was likely sparked by a happenstance occurrence. This might have then have suggested a hypothesis—what would hap­ pen if the whole positive numbers are matched to square numbers, one-by-one? Once this was demonstrated by Galileo to produce a paradoxical result, a “thought form” emerged. This likely inspired Cantor a few centuries later to develop a new view of numbers and to establish the theory of sets as a powerful method of mathematical exploration. Mathematical discovery is, ironically, itself a paradox.

Explorations 1.  Einstein’s paradox The following puzzle seems to harbor a paradox. It was apparently devised by Albert Einstein: A group of sportsmen, having pitched camp, set forth to go bear viewing. They walk 15 miles due south, then 15 miles due east, where they sight a bear. They return to camp by traveling 15 miles due north. What was the color of the bear?

2.  Cantorian method Using Cantor’s simple method, show that the following sets have the same car­ dinality, that is, the same number of members in them: (a) the odd numbers (b) successive powers of 10, starting with 101. (c) cubic numbers (numbers that are raised to the power of three).

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3.  Alexia’s paradox Here’s a version of the Liar Paradox. The clever Alexia likes to confuse people by saying: “I come from a town where everybody, including myself, is a liar.” Did Alexia tell the truth?

4.  Lateral thinking puzzles Einstein’s paradox above illustrates what psychologists describe as lateral thinking, which involves reaching an answer by an indirect and creative approach, usually by viewing the situation or the information given in a new and unusual light. The term was introduced by Edward De Bono, a Maltese-born British psychologist. Like any paradox of infinity, lateral thinking puzzles involve a type of thinking called “outside-the-box.” Here are a few classic ones: (a) A truck stuck under a low bridge. What’s the easiest way to get it out? (b) A man walks into a bar and asks for a glass of water. The bartender reaches under the counter, takes out a gun, and aims it at the man. The man says thank you and leaves. What happened? (c) This one is by Lewis Carroll: A monkey hangs on a rope that goes up over a pulley. On the other end is a weight, which exactly balances the monkey. Everything is balanced, and is initially stationary. Then, the monkey tries to climb the rope. What happens?

5.  Cantor again Take the first transfinite number ℵ0: (a)  What happens when you add “1” to it? ℵ0 + 1 = ? (b)  What happens if you double it? ℵ0 + ℵ0 = 2ℵ0 = ?

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9

Decidability The foundations of mathematics

Logic is the beginning of wisdom, not the end. —Leonard Nimoy (1931–2015)

Prologue Consider the following problem: Find five consecutive odd numbers that add up to 64.

Let’s start by considering the sum of the first five odd numbers in sequence and then of five other consecutive odd numbers starting with 19:

1 + 3 + 5 + 7 + 9 = 25 19 + 21 + 23 + 25 + 27 = 115



If we continue adding sets of five consecutive odd numbers we will find that the sum always seems to end in 5, and thus is an odd number. It seems impossible for five consecutive odd numbers to add up to an even sum, such as 64. Here is a proof. 1. If the first odd number in a sequence of five consecutive odd numbers is represented by (2n + 1), one more than an even number (2n), the one after it can be represented with the expression (2n + 3), the third with (2n + 5), the fourth with (2n + 7), and the fifth with (2n + 9).

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6 |  D E C I DAB I LIT Y: TH E FO U N DATI O N S O F MATH E MATI C S 2. Adding up five consecutive odd numbers yields the following result: (2n + 1) + (2n + 3) + (2n + 5) + (2n + 7) + (2n + 9) = (10n + 25). 3. Now, consider the expression (10n + 25). The term 10n is a number ending in 0 because any number, n, multiplied by 10 will invariably produce a digit ending in 0: 1 × 10 = 10, 2 × 10 = 20, 15 × 10 = 150, and so on. 4. The second term in the expression is 25. This is added to the previous digit ending in 0. This means that the result will always end in the digit 5: 10 + 25 = 35, 20 + 25 = 45, 150 + 25 = 175, and so on. 5. So, the expression (10n + 25) represents an odd digit ending in 5, no matter what n is. 6. Thus, five consecutive odd numbers added together cannot produce the sum of 64, which is an even number.

The problem posed is thus decidable; that is, it can be determined with a viable proof—in this case it is determined to be impossible. This mode of reasoning takes us into the heart of mathematics. Obviously, if something cannot be solved or proved, we should not be wasting our time trying to find a solution or proof; that is, if it can be shown to be undecidable, that is the end of the matter. This is a key theoretical principle on which computer science is built. If something can be programmed to produce a certain output, then the algorithm used shows that it is decidable in the first place; if it does not produce an output, then it is undecidable. This chapter deals with the question of decidability and its basis in logic. In his Elements, Euclid began with accepted mathematical truths (axioms and postulates). From these, he proved 467 propositions, using contradiction, induction, deduction, and other kinds of logical strategies. The whole edifice of mathematics was built on Euclid’s axiomatic-propositional-logical foundation, or at least so it seemed until 1931. That was the year in which Austrian-born American mathematician Kurt Gödel (1906–1978) proved that within any ­formal system of logic there are propositions (statements) that can be neither proved nor disproved. Since then, mathematicians have embraced a much more flexible way to do mathematics, without discarding the Euclidean system.

Consistency As Aristotle and other ancient philosophers believed, mathematics and logic are complements of the same mental process. Aristotle himself provided a ­the­or­et­ic­al model of what logical consistency entails. He called it the syllogism. It has three parts—the major premise, the minor premise, and the conclusion:

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Major premise: Minor premise: Conclusion:

All humans are mortal Barbara is human. Barbara is mortal.

The major premise states that a category—humans—has (or does not have) a certain characteristic—mortality—and the minor premise that a particular elem­ent is (or is not) a member of that category. The conclusion then affirms (or negates) that the element in question has that categorical characteristic. This abstract structure became the basis for proving or disproving many prop­ os­itions in mathematics as being consistent and thus decidable. Euclid, as mentioned, constructed the entire theoretical edifice of mathematics on logic, whereby propositions are connected to each other and can be shown to be provable or not (logically). He started with definitions, axioms, and postulates, from which he derived theorems and propositions. Some of the axioms included the following: 1. Things equal to the same thing are also equal to each other. 2. If equals are added to equals, then the wholes are equal. If equals are subtracted from equals, then the remainders are equal. 3. Things that coincide with one another equal each other. 4. The whole is greater than the part. These are self-evident. But they do not always hold, as we saw with Cantor’s proofs in the previous chapter, which violate the fourth axiom above. Moreover, one of Euclid’s axioms, known as the Parallel Postulate, exposed a fault in his logic. It can be paraphrased as follows: If two straight lines are parallel to each other, they will never meet.

From the start, the postulate seemed to be more of a proposition (something to be proved) than an axiom. So, during Euclid’s time, and for centuries thereafter, mathematicians attempted to prove that the postulate could be derived from Euclid’s other axioms and postulates, but they could not. In the 1800s, math­ emat­icians finally demonstrated that it cannot be proved within the Euclidean system of proof. This led to the creation of mathematical systems in which the postulate was replaced by other ones. And from these “non-Euclidean geom­ etries” emerged, such as those developed by Nikolai Lobachevski (1792–1856) and Bernhard Riemann (chapter 2). Is there a world where no lines are parallel? The answer is the surface of a sphere on which all straight lines are great circles. It is, in fact, impossible to draw any pair of parallel lines on the surface of a sphere, since they would meet at the two poles.

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8 |  D E C I DAB I LIT Y: TH E FO U N DATI O N S O F MATH E MATI C S Because an important use of geometry is to describe the physical world, we might ask which type of geometry, Euclidean or non-Euclidean, provides the best model of reality. Some situations are better described in non-Euclidean terms, such as aspects of the theory of relativity. Other situations, such as those related to building, engineering, and surveying, seem better described by Euclidean geometry. In other words, it all depends on which reality we are dealing with. Euclidean mathematics is still around because it is practical for our common needs.

Axiomatic structure In 1889, the Italian mathematician Giuseppe Peano (1858–1932) developed nine axioms, which have remained central to arithmetic and number theory generally. His intent was to get mathematicians to take nothing for granted. Peano started by establishing the first natural number (no matter what numeral system is used to represent it), which is zero. The other axioms are successor ones. 1 . 0 is a natural number. 2. For every natural number x, x = x. 3. For all natural numbers x and y, if x = y then y = x. 4. For all natural numbers x, y and z, if x = y and y = z, then x = z. 5. For all x and y, if x is a natural number and x = y, then y is also a natural number. That is, the set of natural numbers is closed under the previous axioms. 6. For every natural number n, S(n) is also a natural number: S(n) is the ­successor to n. 7. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0. 8. For all natural numbers m and n, if S(m) = S(n), then m = n. 9. If K is a set such that 0 is in K, and if every natural number n is in K, then S(n) is in K, and K contains every natural number.

These axioms seem self-evident, but self-evidence in logic, as we have seen in this book, cannot be taken for granted. Overall, the axioms tell us what it means to be a number, formally. A little before Peano, English mathematician George Boole (1815–1864) put forth a system of logic based on binary digits (as briefly discussed in chapter 3) to ensure that axiomatic structure would be consistent with what he called the “laws of thought.” To test an argument, Boole always converted statements into symbols, divesting them of real-world reference.

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x

y x ∧y

x

y x∨y

x ¬x

Figure 9.1  Boolean algebra (from Wikimedia Commons)

Then, through rules of derivation, he showed that it is possible to determine what new formulas may be derived from the original axioms. Boolean algebra, as it is called, came forward to help mathematicians solve problems in logic, probability, and engineering. Boole’s ambitious purpose was not only to unite mathematics and logic, but also to break down proofs and formulas into their bare structure, making the 1 of the binary system stand for true and the 0 for false. Instead of the typical signs used in addition, multiplication and the other operations of arithmetic, he used the conjunction (∧), disjunction (∨), and complement or negation (¬), in order to show that these were skeletal patterns of thought, rather than just different ways of doing arithmetic. These operations can be expressed either with truth tables or Venn diagrams, which involve sets, such as x and y. Figure 9.1 shows a few Venn diagrams for these operations. American engineer Claude Shannon (1916– 2001) was developing switching circuits in the 1930s when he decided to apply Boolean algebra to the operation of the circuitry. In so doing, he achieved control on the basis of an off (0)-­versus-on (1) structure, thus laying the foundation for modern-day digital computing. Shannon’s logic gates, as he called them, represented the action of switches within a computer’s circuits, now consisting of millions of transistors on a single microchip. Clearly there is a lot more to logical axioms for mathematics than just a system of consistency for deciding if something is a number or if something falls within a system and is thus likely to be decidable. Following on Boole’s and Peano’s ­coattails, at the First International Congress of Mathematicians of the twentieth century in Paris, David Hilbert asked if all science could not be broken down into similar groups of fundamental axioms. The question is still an open one.

Undecidability The paradoxes discussed in the previous chapter posed a threat to the axiomatic structure of mathematics established by Euclid. Recall the Liar Paradox, in which the reasoning went around in circles— rather than leading to a conclusion, it leads nowhere.

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20 |  D E C I DAB I LIT Y: TH E FO U N DATI O N S O F MATH E MATI C S In the nineteenth century—the century of Peano and Boole—this paradox surfaced again, threatening the logical basis on which mathematics was built. German philosopher Gottlob Frege (1879), in response, suggested that such paradoxes could be avoided by considering form separately from content, following in Boole’s footsteps. In this way, one could examine the consistency of the propositions (premises) without having them correspond to anything in the real world. Frege’s approach was developed further by Ludwig Wittgenstein (1921), who used symbols (H = human, M = mortal, B = Barbara): Major premise: Minor premise: Conclusion:

∀ H is M (All H is M) B ∈ H (B is a member of H) B ∈ M (B is a member of M)

Between 1910 and 1913, and using a similar abstract system of symbols, British philosophers Bertrand Russell (1872–1970) and Alfred North Whitehead (1861–1947) aimed to develop, once and for all, a set of axioms, definitions, propositions, and other logical constructs that would provide a solid im­mut­ able foundation for mathematics, free from paradoxes. This resulted in their monumental work, Principia Mathematica, published in parts in 1910, 1912, and 1913. The project was ignited by Russell’s unease over the Liar Paradox. So frustrated was he by its undecidability that he formulated his own version, called the Barber Paradox: The village barber shaves all and only those villagers who do not shave themselves. So, shall he shave himself?

The barber is “damned if he does and damned if he doesn’t,” as the colloquial expression goes. If he does not shave himself, he ends up being an unshaved villager. But this goes contrary to the condition that the barber must shave anyone who does not shave himself. So, we must conclude that the barber should shave himself. But in so doing he has shaved someone in the village who shaves himself, rather than someone who does not—himself! It is not possible, therefore, for the barber to decide whether or not to shave himself. Russell argued that the paradox arises because the barber is himself a member of the village. So, he concluded, in a consistent logical system paradoxes are eliminated by disallowing such statements from a member of a set—they may be made only by those outside the set. Russell thus introduced the concept of metalanguage— a language “apart from” other language—that was intended to immunize lo­gic­al systems against such paradoxes. But this notion did not put an end to the question of consistency or decidability within logical systems.

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In 1931, Kurt Gödel showed why systems of propositional logic, such as the one by Russell and Whitehead, break down. It came to be known as the incompleteness theorem. Before Gödel, it was taken for granted that every pro­position within a logical system could be either proved or disproved within it. But Gödel startled the mathematical world by showing that this was not the case. He argued that a logical system invariably contains a proposition within it that is “true” but “unprovable.” Gödel’s argument is far too complex to be taken up in an in-depth manner here. For the present purposes, it can be condensed as follows: Consider a mathematical system that is both correct—in the sense that no false statement is provable in it—and contains a statement “S” that asserts its own unprovability in the system. S can be formulated simply as: “I am not provable in system T.” What is the truth status of S? If it is false, then its opposite is true, which means that S is provable in system T.  But this goes contrary to our assumption that no false statement is provable in the system. Therefore, we conclude that S must be true, from which it follows that S is unprovable in T, as S asserts. Thus, either way, S is true, but unprovable in the system.

The axioms and postulates of the founders of mathematics were designed to  make mathematics a consistent logical system. But, as Gödel showed, the system will always be imperfect and mathematicians will have to live with this fact. Crilly (2011, 11–12) puts it as follows: The Greeks assumed the truth of their axioms, but today’s mathematicians expect only that axioms be consistent. In the 1930s Kurt Gödel rocked math­ em­at­ics when he proved his incompleteness theorem, which held that there were some mathematical statements in a formal axiomatic system that could neither be proved or disproved using only the axioms of the system. In other words, mathematics could now contain unprovable truths that might just have to stay that way.

There are many open questions in mathematics that tantalize us, but may be undecidable. Logic tells us that something can be either true or false, but not both. Aware of this verity, Aristotle clarified the connection between contradiction and falsity in his principle of non-contradiction, which states, simply, that an assertion cannot be both true and false. Therefore, if the contradiction of an assertion can be derived logically from the assertion it can be concluded that a false assumption has been used. The discovery of contradictions at the foundations of mathematics, however, has not blocked the course of mathematical discovery. On the contrary, it has spurred it on relentlessly.

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22 |  D E C I DAB I LIT Y: TH E FO U N DATI O N S O F MATH E MATI C S The English philosopher Thomas Hobbes (1588–1679) claimed that logic was the only attribute that kept human beings from retrogressing—a view developed further by the French philosopher René Descartes (1596–1650), who refused to accept any belief, even the belief in his own existence, unless he could “prove” it to be logically true. Descartes also maintained that logic was the only effective way to solve all human problems, most of which were caused by the emotions and the passions. Logic had the ability to tame them. In their insightful book, Descartes’ Dream, Davis and Hersh (1986, 7) encapsulated Descartes’ vision as “the dream of a universal method whereby all human problems, whether of science, law, or politics, could be worked out rationally, systematically, by logical computation.” German mathematician and philosopher Gottfried Wilhelm Leibniz saw logic as a language of thought, recalling the origin of logic in the Greek word lógos (meaning both “word” and “thought”). Leibniz characterized this language as a characteristica universalis (a “universal language” of mind), claiming that it could be used to great advantage for the betterment of humanity because “errors” in thinking could be reduced to errors in logic and thus easily fixed. But what is logic? Is the logic that applies to proving theorems in mathematics the same logic that we use to solve everyday practical problems? The phil­oso­ pher Charles Peirce (in his writings, 1931–1958) differentiated between two kinds of logic—logica utens (a practical logic) and logica docens (a theoretical or learned logic). The former is a rudimentary logic-in-use that everyone possesses without being able to specify what it is. Peirce distinguished it from the latter, which he defined as a sophisticated and tutored use of logic practiced by ­mathematicians, scientists, detectives, and medical doctors. Because everyone possesses logica utens, no special training is required to understand what most everyday problems are about or what to do in order to solve (or resolve) them.  However, understanding formal logical structures or theories requires logica docens.

Epilogue The ancient Greeks had actually come to a truly insightful understanding of the nature of mathematics by observing how arguments and proofs unfolded, coming to the conclusion that various forms of logic were necessary to do mathematics. In some cases, however, their methods of proof broke down, as in the case of the paradoxes of infinity.

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In the end, maybe logic is not as crucial to mathematics as is pattern-detection. Consider the following computations. The task is to figure out if they conceal any general pattern. 2 ´ 9 = 18 and 1 + 8 = 9 3 ´ 9 = 27 and 2 + 7 = 9 4 ´ 9 = 36 and 3 + 6 = 9 5 ´ 9 = 45 and 4 + 5 = 9 ¼ 12 ´ 9 = 108 and 1 + 0 + 8 = 9 123 ´ 9 = 1107 and 1 + 1 + 0 + 7 = 9 1, 245 ´ 9 = 11, 205 and 1 + 1 + 2 + 0 + 5 = 9 12, 459 ´ 9 = 112,131 and 1 + 1 + 2 + 1 + 3 + 1 = 9 ¼ Upon close scrutiny, this layout reveals that the digits of any multiple of 9, when added together, produce 9. Now, this revelation, while interesting in itself, is not all there is to it. It has, in fact, had concrete implications for facilitating computation and carrying out arithmetical operations, which need not concern us here. Patterns such as this abound throughout mathematics. Mathematics is  the  “science of patterns.” Gödel made it obvious to mathematicians that ­math­em­at­ics was made by them, and humans are fallible, and so the exploration of “mathematical truth” would go on forever as long as humans were around. All we can truly rely on is the discovery of the patterns within this truth. As Cantor showed, mathematical structure has many dimensions and shapes to it, and may likely never conform to one and only one system of logic.

Explorations 1.  Smullyan’s Gödelian puzzle The late American logician Raymond Smullyan (1919–2017) devised an ­in­geni­ous puzzle version of Gödel’s argument as follows: Let us define a logician to be accurate if everything he can prove is true; he never proves anything false. One day, an accurate logician visited the Island of Knights and Knaves, in which each inhabitant is either a knight or a knave, and  knights make only true statements and knaves make only false ones.

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24 |  D E C I DAB I LIT Y: TH E FO U N DATI O N S O F MATH E MATI C S The  logician met a native who made a statement from which it follows that the native must be a knight, but the logician can never prove that he is! What was the statement?

2.  Gardner’s box logic puzzle Logic takes many forms. Here is one that involves figuring out the contents of a set of boxes. The originator was the late Martin Gardner (1914–2010), who wrote a famous puzzle column for Scientific American. Here is a paraphrase of his original puzzle: There are three closed boxes on a table which contain nickels, separately. A’s label shows 10¢, B’s label shows 15¢, and C’s label shows 20¢. However, they are labeled incorrectly. Someone takes the contents out of box B, labeled 15¢, two nickels, and puts the nickels out in front of the box. Can you tell the ­contents of each box?

3.  Dudeney’s logic puzzle This is a classic puzzle invented by Henry E. Dudeney that involves using rudimentary deductive logic: In a certain company, the positions of programmer, analyst, and accountant are held by Amy, Sharma, and Sarah, but not necessarily in that order. The accountant, who is an only child, earns the least. Sarah, who is married to Amy’s brother, earns more than the analyst. What position does each person fill?

4.  A derivative of Gardner’s puzzle Here is a logic puzzle, similar to Gardner’s. A gold coin is in one of three boxes, each of which has an inscription written on it, as in Figure 9.2. Can you tell where the coin is if, at most, only one of the inscriptions is true?

A The coin is in here.

B The coin is not in here.

Figure 9.2  Coins in a box puzzle

C The coin is not in A.

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5.  Deception logic We are all susceptible to deception, especially if we do not pay attention to the actual contents of a statement. Here’s an example. A farmer had seven daughters, and they each had a brother. How many ­children did he have?

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10

The algorithm Mathematics and computers

Computers are like Old Testament gods; lots of rules and no mercy. —Joseph Campbell (1904–1987)

Prologue As discussed previously (chapter 3), a decimal numeral such as 2,234 can be easily deconstructed in terms of the values of the digits according to their ­pos­ition. This type of analysis is worth revisiting here (Figure 10.1). Now, can we devise a rule such that a computer can generate such numerals ad infinitum? The rule would need to break down the numeral construction ­technique into its skeletal-symbolic form: D ® Nn ´ 10n -1 + Nn -1 ´ 10n -2 + Nn -2 ´ 10n -3 + ¼ N3 ´ 102 + N2 ´ 101 + N1 ´ 100

2

2

3

4

two thousand

two hundred

thirty

four

2 × 103

2 × 102

3 × 101

4 × 100

Figure 10.1  Decimal structure of 2,234

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This says that a decimal numeral (D) is composed of digits (N1, N2, N3,…,Nn−2, Nn−1, Nn) that have values in powers of 10 according to their position in the sequence, as indicated below. 1. 2. 3. 4. 5.

The digit N1 is the first one to the right. Its value is N1 × 100 The next digit to its left, N2, has the value N2 × 101 The third digit leftward, N3, has the value N3 × 102 And so on, until we reach the last digit to the left, Nn, with value Nn × 10n−1. We note that the power of 10 to produce the value of each digit is one less than the subscript of the digit, which indicates its position starting at the right and moving left.

We now have specified the make-up of a decimal numeral with a formal representation that can be converted into a computer algorithm, which will then run a program to generate numerals ad infinitum. All we have to do is input a value for n and the output will always be a decimal numeral. Mathematics has formed a partnership with computer science. Devising algorithms and turning them into computer programs is akin to following a manual for assembling some object. A computer can be used to attack the problem of decidability, discussed in the previous chapter, on the basis of the validity (or not) of some algorithm. This final chapter will deal with this key notion, completing the chain of ideas described in this book that started with Pythagoras.

Algorithms An algorithm is a procedure for solving a problem in a finite number of steps. The instructions for each step must be precise. The first algorithm of math­em­ at­ic­al history is, as we saw, Euclid’s algorithm, which allows us to find the greatest common divisor of any two whole numbers, A and B, in a simple, repeatable and almost mechanical way (chapter 2). The term algorithm comes from the Latinized name of Muhammad ibn Musa  al-Khwarizmi (c. 780–850 ce), a Persian mathematician, whose text, Compendious Book on Calculation by Completion and Balancing (c. 813–833 ce), established the methods for solving linear and quadratic equations in a step-by-step fashion, hence the adoption and adaptation of his name to refer to the concept of algorithm. Al-Khwarizmi’s book is also the source of the word algebra, from al-jabr, meaning “completion, which occurs in the Arabic title of his book: Kitab al-jabr wa’l-muqabala.

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28 |  TH E ALGORITHM : MATH E M ATI C S AN D COM PUTE RS Preparing a program begins with a complete breakdown of the operation that the computer is intended to process. This includes what information must be inputted, what system of instructions and types of algorithmic rules are involved, and what form the required output should take. The initial step is to prepare a flowchart that represents the steps needed to complete the task. This is itself a model of the task, showing all the steps involved in putting the instructions together into a coherent program. Each step in the chart gives options and thus allows for decisions to be made. The flowchart is converted into a program that is then typed into a text editor, a program used to create and edit text files. The flowchart in Figure  10.2 shows how to build a computer program for Euclid’s algorithm. entry Euclid’s algorithm for the greatest common divisor (gcd) of two numbers INPUT A, B

1

2

yes

B = 0? no

3

yes

A > B? no ( < or = )

Figure 10.2  Flowchart for Euclid’s algorithm (from Wikimedia Commons)

4

B←B-A

5

GOTO 2

6

A←A-B

7

GOTO 2

8

PRINT A

9

END

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Figure 10.3  Example of computer code (from Wikimedia Commons)

This breaks down the steps described in the Fundamental Theorem of Arithmetic (see chapter 2) in a precise and machine-readable way. It stipulates what we do with pencil and paper. Before a computer can run, special programs must translate the algorithm into a machine language composed of numbers, called a code. The basic code is based on binary digits (see Figure 10.3). When the binary numbers are too long for computers to store and process, they are converted into hexadecimal numbers, from 1 to F.  A hexadecimal number is a number to the base 16; using 16 distinctive symbols, including 0 to 9, and A to F, to represent values. This numeral system describes locations in ­computer memory because it can stand for every byte (eight bits of information).

Computability Computability is the process of determining the existence or lack of an algorithm to solve a problem. In 2009 a computer program was devised that was able to run the so-called Grover reverse phone book problem (Elwes 2014, 289), which had been considered to be intractable. A phone book is a list of names organized in alphabetical order to which a unique number is assigned. So, looking up a name in it is a straightforward (finite-state) task; and this allows us to locate the required phone number. However, if we have a phone number and want to locate the person to whom it belongs we are faced with a much more difficult problem to solve. This is the essence of the reverse phone book problem. Today, a simple Google algorithm will allow us to solve the reverse number problem (in most cases). As Elwes (2014, 289) observes, the problem was first solved in 1996: In 1996, Lou Grover designed a quantum algorithm, which exploits a quantum computer’s ability to adopt different states, and thus check different numbers, simultaneously. If the phone book contains 10,000 entries, the classical

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30 |  TH E ALGORITHM : MATH E M ATI C S AN D COM PUTE RS a­ lgorithm will take approximately 10,000 steps to find the answer. Grover’s algorithm reduces this to around 100. In general, it will take around √N steps, instead of N.  The algorithm was successfully run on a 2-qubit quantum ­processor in 2009.

Computers are adept at solving all kinds of problems that involve pattern and structure. They do so by computing all possibilities for a problem, rather than providing a theoretical explanation for the solution. Consider the so-called Eight Queens Problem, whereby eight queens must be placed on an 8-by-8 chessboard in such a way that none of the queens is able to capture any other queen (with the normal rules of chess). In other words, a solution requires that no two queens share the same row, column, or diagonal. There are ninety-two distinct solutions to the puzzle, although if rotations and reflections of the board are taken into account, then it has 12 unique solutions. Figure 10.4 shows one solution. Solving this problem requires quite a bit of imagination, and finding all ­possible solutions would require a considerable amount of effort. Now, this problem can be solved easily with a recursive algorithm by specifying its components precisely. The significance of this to mathematics is not the fact the computer allows us to resolve the problem rapidly, but the fact that it forces us to deconstruct the structure of the problem into its parts, as we did for the ­construction of the decimal numeral algorithm at the start. In other words, by writing the program for solving the problem, we are able to glean insights into its nature. The algorithm is thus a theoretical model of the problem. Computer modeling is thus a means for understanding mathematical structure and to determine decidability or undecidability (chapter  9). Is there an algorithm for computing prime numbers? Can a rule be written to generate the digits in π? And so on. There may be no algorithm to resolve such questions, 0 0 1 2 3 4 5 6 7

Figure 10.4  Eight Queens Problem (from Wikimedia Commons)

1

2

3

4

5

6

7

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but the effort to devise one is worth it just the same, because it impels us to think about the problem in specific ways. Consider a standard 9-by-9 Sudoku puzzle. Solving it is a fairly simple task. The complexity arises when the grid size increases. By augmenting the grid to 10,000-by-10,000 the solution becomes gargantuan in terms of effort and time. Computer algorithms can easily solve complex Sudoku puzzles, but start having difficulty as the degrees of complexity increase. The idea is, therefore, to devise an algorithm to find the shortest route to solving complex problems such as this one. If we let P stand for any problem with an easy solution, and NP for any problem with a difficult complex solution, then the question of decidability can be represented in computer terms. If P were equal to NP, P = NP, then problems that are complex (involving large amounts of data) could be tackled easily as the algorithms become more efficient. The P = NP problem is the most important problem in computer science and formal mathematics. It seeks to determine whether every problem whose solution can be quickly checked by computer can also be quickly solved by computer (Fortnow 2013). But there is a caveat in all this. As discussed in the previous chapter, Kurt Gödel showed that in any logical system there is always some statement that is true, but not provable in it. The implication is that logical systems are, by their very nature, undecidable. This includes computational systems. Alan Turing (1912–1954)—one of the greatest theorists of computability in history—­ reinforced Gödel’s argument by proving that no algorithm can be devised that will eventually bring a program to a halt when it runs with the input. It is called the “halting problem.” Turing assumed that the halting problem was decidable, constructing an algorithm that halted if and only if it did not halt, which is a contradiction. However, even if writing an algorithm does not produce the required output, the process of writing it in itself leads to a deeper understanding just the same—that is, by examining the “fault” in the algorithm we can better understand the mathematical principle needed to make it run better. When nothing works, then we will have to go back to the principle and ­re-examine it both logically and imaginatively.

Epilogue The discussion of algorithms leads to the question that has been a subtext throughout this book: Is mathematics discovered or invented? Plato’s view that mathematics is discovered rather than invented means that we never should find

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32 |  TH E ALGORITHM : MATH E M ATI C S AN D COM PUTE RS faults within it. But, as we have seen, there are faults in mathematics. But then, if mathematics is faulty, why does it lead to demonstrable discoveries, both within and outside of itself? Discoveries emerge by happenstance through contemplation and manipulation of mathematics. As this goes on, every once in a while, something crops up that leads to new insights, disrupting the previous system. Plato divided reality into two realms, one inhabited by invisible ideas or forms, and another by concrete familiar objects. The latter are imperfect copies of the ideas because they are always in a state of flux. Thus, Plato rejected any philosophy that claimed to explain knowledge on the basis of sensory experience—in his view, true knowledge is based on innate ideas. In his Republic, he portrayed humanity as imprisoned in a cave where it mistook shadows on the wall for reality. Only the person with the opportunity to escape from the cave— the true philosopher—had the perspicacity to see the real world outside. The shadowy environment of the cave symbolizes the realm of physical appearances. This contrasts with the perfect world of ideas outside. A circle, for instance, is an ideal form. An object existing in the physical world may thus be called a “circle” insofar as it resembles that idea. Pythagoras spent most of his life in the city of Crotone, a Greek colony in southern Italy, where he established his society to study the mathematical nature of reality. Followers were initiated by secret rituals and they took an oath to live by the tenets of the so-called Golden Verses, which included the following two, which are as important today as they were in ancient times:

• Search for the just measure, which is the measure that does not cause pain. • Be kind with your words and useful with your work. The Pythagoreans assigned symbolic value to numbers, thus blending numerology and numeration:

• 1 symbolized reason • 2 stood for the undefined feminine spirit in all humans • 3 was the sum of 1 and 2 and thus stood for the masculine in all of us • 5 was the sum of 2 + 3 and thus the most powerful of all, blending the symbolism of the two numbers. Pythagoras also claimed that the distances between the planets showed the same ratios as those produced by the harmonious sounds emanating from plucked strings. If the planets were close to each other, their movements ­produced lower tones; if they were farther away, they produced higher tones.

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The tones blended together to produce the “music of the spheres.” In effect, Pythagoras saw a connection among movement, objects, and harmonious music. The connection was specified by a common language—mathematics. The Pythagoreans were also infamous for several eccentric beliefs, such as the fear of white cockerels and touched beans. The people of Crotone eventually came to distrust them. This allowed Pythagoras’ enemies to slaughter the members of his society. It is not known if Pythagoras was also killed in the uprising. The Pythagoreans saw mathematics as a secret code that held all truths about the world. Galileo’s well-known assertion that the book of Nature is written in the language of mathematics extended the Pythagorean view. Is mathematics truly the language of the spheres, allowing us to “discuss” the features of the universe in the form of geometric figures, numbers, and the like? And if so, does it mean that mathematics is an investigative tool of the brain, rather than a system in and of itself that the brain has allowed us to produce? From the Pythagorean practice of giving sacrifice to the gods for mathematical discoveries to the seventeenth century practice on the part of the Japanese of giving sangaku (the Japanese word for “mathematical tablet”) to the spirits for dis­ cover­ing mathematical proofs, there seems to be a universal feeling across the world that discoveries reveal the world to us in bits and pieces. This is why the ancients believed that a causal connection existed between earthly matters and the stars. Those who could use numbers to calculate forthcoming events, such as the next planting season, garnered great power unto themselves, becoming wizards, mathematicians, and astronomers. So, what is mathematics? This is a question that cannot be answered directly. Mathematics is best exemplified with its great ideas, such as the ten chosen for this book (among many others). As Pythagoras maintained, it is likely to be the code that can unlock the meaning of reality. Pythagoras’ legacy continues to motivate mathematics and its relation to science, philosophy, and the arts. Perhaps Galileo said it best: “Mathematics is the language in which God has written the universe.”

Explorations 1.  Measuring algorithm Below is a problem penned in the fifteenth century by French mathematician Nicolas Chuquet (c. 1445–1488), which entails devising a step-by-step procedure:

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34 |  TH E ALGORITHM : M ATH E M ATI C S AN D COM PUTE RS You have two jars holding 5 and 3 pints respectively, neither jar being marked in any way. How can you measure exactly 4 pints from a cask, with an unspecified quantity of liquid in it, given that you are allowed to pour liquid back into the cask at any step?

2.  Movement algorithm Consider the following game, which also involves devising a step-by-step ­procedure (see Figure 10.5): There are six checkers in a row on a table, three colored white and three colored black, with one space between the two sets. Change the positions of the ­checkers by moving only one checker at a time. A checker may be moved over one adjacent checker into an empty space; or else, it may be moved one space into an empty space. You are not allowed to move a checker backward: that is, the white ones can only move to the right, and the black ones to the left:

3.  Alcuin’s river crossing puzzle Writing algorithms implies deconstructing a problem into a step-by-step structure. As an example, consider a famous puzzle, traced to the English scholar and ecclesiastic Alcuin (735–804 ce), who devised fifty-six puzzles that he put  into an instructional manual, titled Propositiones ad acuendos juvenes (“Problems to Sharpen the Young”). A certain traveler needed to take a wolf, a goat, and a head of cabbage across a river. However, he could only find a boat which would carry himself and one other at a time. How did he get all of them across unharmed and intact?   Note: if left alone the goat will eat the cabbage and the wolf will eat the goat.

4.  A complex version Alcuin’s puzzle is a fairly simple one to solve. Increasing the variables (items to be brought across) and the conditions makes the river crossing algorithm a more complex one. Here is an example. The puzzle is of Russian origin: Three soldiers have to cross a river without a bridge. Two boys with a boat agree to help the soldiers, but the boat is so small it can support only one soldier or

Figure 10.5  Checkers problem

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two boys. A soldier and a boy cannot be in the boat at the same time for fear of sinking it. Given that none of the soldiers can swim, it would seem that in these circumstances just one soldier could cross the river. Yet all three soldiers ­eventually end up on the other bank and return the boat to the boys. How do they do it?

5.  Possibility versus impossibility One of the themes of this chapter (and others) is about possibility or impossibility. A simple puzzle invented by philosopher Max Black in his 1946 book Critical Thinking encapsulates in a nutshell what this entails. It is called the mutilated chessboard problem. If two opposite corners of a checkerboard are removed, can the checkerboard be covered by dominoes? Assume that the size of each domino is the size of two adjacent squares of the checkerboard. The dominoes cannot be placed on top of each other and must lie flat.

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ANSWERS

Chapter 1 1. First, we can draw a diagram that will visually represent the information presented by the puzzle, with the vertical line PQ representing the pillar, which is 15 cubits high. The top point P represents the perch on which the peacock sits. We also draw a line from the pillar’s base (Q), at a right angle, to a point S, representing the snake’s initial position. The line QS is 45 cubits long, for this is “a distance of thrice the pillar’s height” (15 × 3 = 45): P

15

Q

45

S

Figure A.1  Bhāskara’s problem

We are told that the snake is moving along the ground towards the pillar and that the peacock stoops on the snake obliquely. So, let’s mark a point on the line QS, R, where the peacock meets the snake. Both animals, Bhāskara tells us, travel an equal distance to point R, although their paths are at an unknown angle to each other. Therefore, x can stand for both the distance from P (the peacock’s perch) to R, and the distance from S (the snake’s

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38 | An sw e r s s­ tarting point) to R, and we can label the two equal line segments accordingly. We can then label the distance from the point where the two meet (R) to the pillar’s base (Q) as (45 − x), the distance between the pillar and the snake when it reaches R: P

15

x

Q

45–x

R

x

S

45

Figure A.2  Bhāskara’s problem solved

The diagram now makes it obvious that the Pythagorean theorem can be applied to the right triangle PQR in order to determine the length x. This produces the equation x2 = 152 + (45 − x)2. Solving it, we get x = 25. Therefore, the distance QR, or (45 − x), is 45 − 25 = 20. The peacock thus meets the snake at a distance of 20 cubits from the base of the pillar, Q. 2. Lay out the three identical triangles, labeled A, B, and C, in such a way that they form a fourth smaller one of the same shape in the center, labeled D, as shown in the diagram below. Note that D is “upside down” with respect to the three triangles. Three vertices of the three t­ riangles can now be joined to produce a larger triangle (shown by the dotted lines): A D

B

C

Figure A.3  Solution to Abu al-Wafa’s puzzle

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3. In the diagram below the largest circle is constructed to be tangent to the given three. Note that one of the given circles is tangent to the larger one within it:

Figure A.4  Solution to Apollonius’ problem (from Wikimedia Commons)

4. First, we draw the rectangular floor. On it, we let x stand for its width and 2x for its length (twice the width): A

2x

x

B

x (bug)

C

2x

D

Figure A.5  Representation of the “bug” problem

The optimal path for the bug is the diagonal path to the opposite corner: A

2x

x

B

x (bug)

C

2x

Figure A.6  Solution to the “bug” problem

D

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40 | An sw e r s The diagonal is, clearly, the hypotenuse of right-angled triangle BCD, with sides of length x and 2x feet. So, if we can determine what these lengths are, we could then use the Pythagorean theorem to determine the length of the diagonal. We are told that the area of the floor is 32 square feet. The area of a rectangle is the product of the length times the width. In this case, the length is 2x and the width x. Multiplying these together, we get: Area of floor:

(2x ) ´ ( x ) = 32 2x 2

= 32

2

= 16 =4

x x

We now know that the width is 4 feet. Since the length is twice this, it is 8 feet. These are the lengths of the sides of right-angled triangle BCD. We can  now use the Pythagorean theorem to determine the length of its ­hypotenuse BC: BC 2 = 42 + 82 BC 2 = 16 + 64 BC 2 = 80 BC = Ö 80 = 8.94 Thus, the optimal path for our bug is 8.94 feet. 5.  Seven acute triangles can be drawn as shown below:

1 2

3

4

6 5

7

Figure A.7  Solution to Gardner’s problem

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Chapter 2 1.  Here is the solution to Dudeney’s puzzle. 67

1

43

13

37

61

31

73

7

Figure A.8  Solution to Dudeney’s magic square problem

2. The number is 372. The digits of 372, when added together, produce 12: 3 + 7 + 2 = 12. Each one is a prime number. The prime factors of 372 are shown below: 372 = 22 ´ 3 ´ 31 3. This puzzle is meant as a caveat. When it comes to prime numbers one must always be wary and careful with generalizations. The answer is that it does not keep going. The next number built with the same pattern (inserting a 3 to the previous number) is 333,333,331. This is a composite number: 17 × 19,607,843 = 333,333,331. 4. There are eight twin primes under 100: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73). 5.  The answer is 19. It is less than 100; if you add 4 to it, you get 23, which is the next prime number in sequence; and if you add its twin prime, 17, to this, you will get 23 + 17 = 40. Note that the twin number pair is (17, 19).

Chapter 3 1. In (6), (a − b − c) was used as a divisor on both sides. Now, if we re-write (1) as follows, we will see that this expression equals 0: 1. a = b + c

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42 | An sw e r s Transposing the symbols on the right side over to the left we get:

( a - b - c ) = 0 So, the contradiction was caused because we divided by 0 in (6). 2. A simple way to solve these problems is to convert the binary digits to decimal digits:

( a ) 1100 + 111 = 12 + 7 = 19 or 10011 in binary digits ( b ) 1011 - 1001 = 11 - 9 = 2 or 10 in binary digits ( c ) 101 ´ 100 = 5 ´ 4 = 20 or 10100 in binary digits 3. This puzzle is an ingenious play on the number line. Since the snail crawls up 3 feet, but slips back 2 feet, its net distance gain at the end of every day is, of course, 1 foot up from the day before. To put it another way, the snail’s climbing rate is 1 foot up per day. At the end of the first day, therefore, the snail will have gone up 1 foot from the bottom of the well, and will have 29 feet left to go to the top (remembering that the well is 30 feet in depth). If we conclude that the snail will get to the top of the well on the twenty-ninth day, we will have fallen into the puzzle’s hidden trap. Here is a chart detailing the linear path up and down of the snail, much like going forward and backward on a number line: Day 1:

Goes up to the 3-foot mark and slides down to the 1-foot mark.

Day 2: Starts at the 1-foot mark, goes up to the 4-foot mark and slides down to the 2-foot mark. Day 3: Starts at the 2-foot mark, goes up to the 6-foot mark and slides down to the 3-foot mark. … Day 26: Starts at the 25-foot mark, goes up to the 28-foot mark and slides down to the 26-foot mark. Day 27: Starts at the 26-foot mark, goes up to the 29-foot mark and slides down to the 27-foot mark.

Consider the start of day 28. The snail finds itself at 27-foot mark from the bottom. This means that the snail has 3 feet to go to the top on that day.

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It goes up the three feet, reaches the top, and goes out, precluding its slippage back down. So, it took the snail 27 full days and half of day 28 (during ­daylight). For the sake of historical accuracy, it should be mentioned that a version of same puzzle, with different details of course, is found in the third section of Leonardo Fibonacci’s Liber Abaci of 1202. He states it as follows: A lion trapped in a pit 50 feet deep tries to climb out of it. Each day he climbs up 1/7 of a foot: but each night slips back 1/9 of a foot. How many days will it take the lion to reach the top of the pit?

4. This puzzle also plays on the number line. At the beginning, we do not know what rung the firefighter is on, except that it is the middle rung. So, we label her starting rung as 0, since each rung above and below 0 can be compared to a point above or below the zero point, given that there will be as many rungs above it as there are below it. Let the middle rung be 0 (like the mid-point on a number line). We are told that the firefighter went up three rungs from the 0 rung: 3 above 2 above 1 above 0

Figure A.9  Partial solution to firefighter problem: part 1

We are then told that she stepped down five rungs. So, from rung 3 above 0, she went down five rungs, ending up at rung number 2 below the mid-point: 3 above 2 above 1 above 0 1 below 2 below

Figure A.10  Partial solution to firefighter problem: part 2

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44 | An sw e r s Next, we are told that that the firefighter climbed up seven rungs (from rung 2 below). So, she started from rung 2 below and went up seven rungs from there. She thus ends up at rung 5 above the zero: 5 above 4 above 3 above 2 above 1 above 0 1 below 2 below

Figure A.11  Partial solution to firefighter problem: part 3

Finally, we are told that the firefighter climbed up another seven rungs (from rung 5 above) to the roof. This means that she started from rung 5 above and climbed up another seven rungs to rung 12 above her starting point: 12 above 11 above 10 above 9 above 8 above 7 above 6 above 5 above 4 above 3 above 2 above 1 above 0 1 below 2 below

Figure A.12  Partial solution to firefighter problem: part 4

Rung 12 above is the top part of the ladder, because from that rung the firefighter stepped onto the roof. Now, let’s complete the ladder. We know that it has twelve rungs above the 0 rung. Since the 0 rung is the middle rung, a complete ladder will, of course, also have twelve rungs below the 0 rung:

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An sw e r s

| 45

12 above 11 above 10 above 9 above 8 above 7 above 6 above 5 above 4 above 3 above 2 above 1 above 0 1 below 2 below 3 below 4 below 5 below 6 below 7 below 8 below 9 below 10 below 11 below 12 below

Figure A.13  Completed solution to firefighter problem

In sum, the ladder has 12 rungs above the 0 rung, 12 below it, and the 0 rung itself. This makes, of course, 25 rungs in total. 5. The product of the first ten digits is zero, because there is a zero in the set— {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Any number or sequence of numbers multiplied by 0 will equal 0.

Chapter 4 1. If you perform the experiment a large number of times you will get around 78.54%, or 0.7854, which is the ratio of the two areas.

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46 | An sw e r s 2. The distance traveled is the circumference of the circle. The formula for the circumference of a circle is: C = p d In this case: d = 200

C = 200p C = 200 ´ 3.141592¼ C = 628.3185¼ So, you have traveled 628 ft. (approximately). 3. The fence covers the circumference of the circle. Its length, 500 ft. (approximately), is therefore the circumference of the circle. Walking from an opening point to the center is the equivalent of walking the length of the radius, since the radius is any line from the center to the circumference (and vice versa). So: C = 2p r 500 = 2p r 250 = p r r = 250 / p r = 79.577¼ So, you will walk a little more than 79 ft. 4. The string goes around twice, making it twice the circumference of the object. The diameter is 14 cm. So: C = p d C = 14p C = 43.9822¼= 44 ( approximately ) Twice this is: 88 (approximately)

The length of the string is thus around 88 cm. 5. First, note that the radii of a circle are equal. If we let the length of the radius be r, then AO = r and OB = r. Now, using the Pythagorean theorem: 9 2 = r 2 + r 2 = 2 r 2

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An sw e r s

| 47

81 = 2r 2 81 / 2 = r 2 r 2 = 40.5 r = 6.36 ( approximately ) This is the value of the radius. So, plugging this into the circumference ­formula, we get: C = 2p r C = 2p ´ 6.36 C = 40 ( approximately ) So, the circumference is approximately 40 in.

Chapter 5 1.  The answer is shown below: 150 = 102 + 72 + 12 = 100 + 49 + 1 2.  This can be solved as follows. 1. Let x stand for the number of sons the grandmother has. Then each of her sons has (x − 1) brothers, that is, one less (himself) than the total number of the sons. For example, if she has 8 sons, then one of the sons would have 7 brothers. 2. Each of the sons has, himself, as many sons (the grandmother’s grandsons) as he has brothers. Since each son has (x − 1) brothers, then each son also has (x − 1) sons of his own. 3. Now, since the grandmother has x sons, and each one has (x − 1) sons of his own, altogether she will have x (x − 1) grandsons. The combined number of sons and grandsons is, therefore, x + x (x − 1). This equals the grandmother’s age, which is a number between 50 and 100: x + x (x − 1) = 50…100. 4. In other terms: x + x (x - 1) = 50¼100 x + x 2 - x = 50¼100 x 2 = 50¼100

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48 | An sw e r s 5. The number we are looking for, therefore, is a square number between 50 and 100. There are three possibilities: 82 = 64, 92 = 81, and 102 = 100. 6. We are told that the number is divisible by 8. So, the answer is 64. This is the grandmother’s age. 3. Since 28 = 256, then log2 (256) = 8. This means that there will be 256 ancestors in the previous eighth generation. 4. The answer is x = 49 1. Since √(x + 15) + √x = 15, then: 2. √(x + 15) = 15 − √x 3. Square both sides: x + 15 = (15 − √x)2 = 225 − 30√x + x 4. So, x + 15 = 225 − 30√x + x 5. Therefore, 15 = 225 − 30√x + x − x 6. Simplifying: 15 = 225 − 30√x 7. Therefore, 15 − 225 = −30√x 8. So, −210 = −30√x 9. Divide by −30: 7 = √x 10. Square both sides: x = 49 5.  (a) (23 ) (212 ) = 215 = 32, 768 (b) (37 ) (27 ) = 67 = 279, 936 (c) (57 ) ¸ (55 ) = 52 = 25 (d) (92)1 = 92 = 81

Chapter 6 1.  Only (a) seems to converge to e. Summing the given terms as decimals we get: 1 + 0.5 + 0.3 + 0.25 + 0.2 + 0.16 + 0.14 + 0.125 +¼ = 2.675¼ 2.  Here are the values of the first six terms: 1/0! = 1 (defined in this way)

1 / 1! = 1 / 1 = 1

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| 49

1 / 2! = 1 / ( 2 ´ 1) = 1 / 2 1 / 3! = 1 / ( 3 ´ 2 ´ 1) = 1 / 6 1 / 4 ! = 1 / ( 4 ´ 3 ´ 2 ´ 1) = 1 / 24 1 / 5! = 1 / ( 5 ´ 4 ´ 3 ´ 2 ´ 1) = 1 / 120 Summing these, we get: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + . . . = 2.718…

7 6 5 4 3 2

y = x2

1

–4

–3

–2

–1

1

2

3

4

–1

Figure A.14  Graph of y = x2 (from Wikimedia Commons)

3.  The graph of y = x2 is shown in Figure A.14. It has a parabolic shape. 4. The answer is Option B. After the first year, with Option A you would receive just the $4,000. With Option B you would receive $2,000 after the first six months; but then you would get an increase of $200. So, during the last six months of that year, you would get $2,200 dollars. Adding the two semesters up, you would get $4,200 at the end of the first year. Now, what income do both options generate after year 2? Well, with Option A you would get an increase of $800 for the year. So, you would end up earning $4,800. But with Option B, you would earn $2,400 in the first semester—the $2,200 you would have started off with at the beginning of the year (= salary from the previous semester) and the $200 raise you would have gotten for that

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50 | An sw e r s semester. Then, in the last six months you would get another increase of $200 on top of this new salary: that is, you would earn another $2,600 ($2,400 + $200). Adding the two semesters up, you would get $5,000 at the end of the second year. If you continue calculating the incomes generated by the two options in this way for, say, six years, you would see that Option B actually generates more income, and is therefore the better option: Years

Option A

Option B

1

$4,000

$4,200

2

$4,800

$5,000

3

$5,600

$5,800

4

$6,400

$6,600

5

$7,200

$7,400

6

$8,000

$8,200

5. The value is x = 7: e7 = 1096.633. As can be seen, the rate of change is very big.

Chapter 7 1.  The answers are shown below: (1) i 3 = i ´ i ´ i = Ö- 1 ´ Ö- 1 ´ Ö- 1= - 1 ´ Ö- 1= - Ö- 1 = - i (2) i 6 = i ´ i ´ i ´ i ´ i ´ i = Ö- 1 ´ Ö- 1 ´ Ö- 1 ´ Ö- 1 ´ Ö- 1 ´ Ö- 1 = -1 ´ - 1 ´ -1 = 1 ´ -1 = - 1 (3) i0 = 1 (Recall from chapter 5 that any number to the power of 0 equals 1) 2.  The answer is 3i. Ö ( -9 ) = Ö ( 9 ´ -1) Ö ( -9 ) = Ö ( 9 ) ´ Ö ( -1) Ö ( -9 ) = 3 ´ Ö ( -1) Since √(−1) = i: Ö ( -9 ) = 3i 3.  a2 + b2 Proof: ( a + bi ) ( a - bi ) = a2 + ( ab )i - ( ab )i - b2 i 2

( )

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An sw e r s

| 5

a 2 + (ab)i - (ab)i - (b2 )i 2 = a 2 - (b2 )i 2 Since i2 = −1: -(b2 )(i 2 ) = - (b2 )(-1) = b2 So: a 2 - (b2 )i 2 = a 2 + b2 4. (a) (3 + 2i) ´ (3 - 2i) = 13 Proof: (a + bi) ´ (a - bi) = a 2 + b2 a = 3, b = 2 a 2 + b2 = 32 + 22 = 9 + 4 = 13 (b) (5 + 3i) ´ (5 - 3i) = 34 Proof: (a + bi) ´ (a - bi) = a 2 + b2 a = 5, b = 3 a 2 + b2 = 52 + 32 = 25 + 9 = 34 2 2 5. (a) (4 - 5i) = (4 - 5i) (4 - 5i) = 16 - (40)i + (25)i = 16 - 40i - 25 (since i 2 = - 1) = (-9 - 40i)

(b) (3 - 3i)3 = (3 - 3i) (3 - 3i) (3 - 3i) = (-54i + 54i 2 ) = (-54 - 54i)



Chapter 8 1. How can the sportsmen travel as stipulated and end up back at the camp? On a two-dimensional surface this is, of course, impossible. But the earth’s surface is spherical, not planar. The camp is pitched at the North Pole, and the travel directions described by the puzzle will lead the sportsmen back to the pole, no matter how far east they go. Hence, the bear is a polar bear, which is white.

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52 | An sw e r s 2.  (a) 1 2 3 4 5 ¼ n       1 3 5 7 9 ¼ (2n + 1) (b) 1

2



3

 1

10

10

 2

8

10

4 

5 

¼ n

5

 3

10

(c) 1 2 3    1

4

¼ 

 4

10

5

¼ 10n

¼ n 

27 64 125 ¼ n3

3. Alexia is either a truth-teller or a liar. Assume that she is a truth-teller. Then, her statement is true. But the statement then implies that she is a liar. This is a contra­ dic­tion. So, she must be a liar. If she is a liar, then her statement is false. But in this case the statement turns out to be true—she is indeed a liar. So, we are left with another contradiction. It is impossible to tell if she is a truth-teller or a liar. 4.  (a) Deflate the tires. (b) The man had the hiccups. He requested a glass of water to help get rid of them. The bartender took out the gun, instead, to scare the man’s hiccups away. It worked and so the man thanked him and left, no longer needing the water. (c) The weight rises or falls with a constant acceleration, as does the monkey with an equal acceleration. Both the weight and m ­ onkey mirror each other. So, the monkey cannot get away from the weight. 5.  ( a ) À0 + 1 = À0

( b)

À0 + À0 = 2À0 = À0

ℵ0 represents the set of numbers with the same cardinality (the integers). If you add “1” to it, problem (a), you are simply going one number further down the number line. Indeed, no matter how many numbers you add to the number line, you will never go past it or beyond it. You will thus always end up on the number line. Similarly, you can double the line, problem (b), whatever that means in infinite terms, but in so doing you will not go past it or beyond it. The line is infinite and will always have the same cardinality, no matter what arithmetical operation you perform on ℵ0.

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| 53

Chapter 9 1. The statement is: You cannot prove that I am a knight. If the native were a knave, then this statement would, of course, be false. Its opposite would be true—namely, You can prove that I am a knight. But this means, in truth, that he must actually be a knight (because it can be proved). He cannot be both, so we discard the assumption that he is a knave. Thus, the native must be a knight. This means that the statement You cannot prove that I am a knight is true. But if it is true, then, as it says, we cannot prove it. So, even though the native is a knight, we will never be able to prove it. 2. We know: (1) that the boxes contain 10¢ (= two nickels), 15¢ (= three ­nickels), and 20¢ (= four nickels); (2) that each box is mislabeled, for ex­ample, if it says 10¢, then we know for certain that it does not have 10¢ in it, but 15¢ or 20¢; (3) that the contents of box B, labeled 15¢, are two nickels (10¢). From these facts there are two possible scenarios: Scenario 1 contradicts a given fact—C is labeled correctly as containing 20¢, contrary to the fact that all three boxes are labeled incorrectly. So, we can Scenario 1

←

←

three nickels

two nickels

four nickels

←

Possible contents

←

Actual contents

←

Possible contents

←

←

Box C 20c/

←

Box B 15c/

←

Box A 10c/

15c/

10c/

20c/

Scenario 2

←

←

two nickels

three nickels

←

Possible contents

←

Actual contents

←

Possible contents

←

←

Box C 20c/

←

Box B 15c/

←

Box A 10c/

20c/

10c/

15c/

four nickels

Figure A.15  Representations of boxes problem

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54 | An sw e r s reject this scenario. Scenario 2, on the other hand, produces no contra­dic­ tions. Thus, A contains 20¢, B 10¢, and C 15¢. 3. This genre of puzzle is solved typically with the aid of a cell chart, putting the positions—programmer, analyst, accountant—on one axis and the names of the persons—Amy, Sharma, Sarah—on the other. This allows us to keep track of all the deductions made along the way: We are told that: (1) the accountant is an only child, and (2) Amy has a Table A.1  Cell chart Programmer

Analyst

Accountant

Amy Sharma Sarah

brother (to whom, incidentally, Sarah is married). So, we can eliminate Amy as the accountant, who is an only child, while Amy is not. We show this by placing an × in the cell opposite her name under the column headed accountant. This indicates that this possibility is eliminated for Amy:

Table A.2  Cell chart partially completed: part 1 Programmer Amy

Analyst

Accountant ×

Sharma Sarah

We are also told that the accountant earns the least of the three, and that Sarah earns more than the analyst. From these two facts, two obvious things about Sarah can be established: (1) she is not the accountant (who earns the least); (2) she is not the analyst (for she earns more than the analyst does).

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| 55

To keep track of these two conclusions, we enter two ×’s in their appropriate cells, eliminating accountant and analyst as possibilities for Sarah: Table A.3  Cell chart partially completed: part 2 Programmer

Analyst

Amy

Accountant ×

Sharma Sarah

×

×

Now, the only cell left vacant under accountant is opposite Sharma. Therefore, by the process of elimination, Sharma is the accountant. We show this by putting a dot opposite her name in the cell, and eliminating all other possibilities for her with ×’s, because Sharma can hold only one of the stated positions—if she is the accountant, then, logically, she is neither the programmer nor the analyst: Table A.4  Cell chart partially completed: part 3 Programmer

Analyst

Amy Sharma

Accountant ×

×

Sarah

×

•

×

×

The chart above now shows that Sarah is the programmer, since the only cell left vacant for her is under programmer. We show this again with a dot. This eliminates the programmer possibility for Amy as well—since there is only one programmer. We show this with ×: Table A.5  Cell chart partially completed: part 4 Programmer

Analyst

Accountant

Amy

×

×

Sharma

×

×

•

Sarah

•

×

×

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56 | An sw e r s Now, as can be seen the only cell left vacant for Amy is under analyst. That is what she is, therefore: Table A.6  Cell chart completed Programmer

Analyst

Accountant

Amy

×

•

×

Sharma

×

×

•

Sarah

•

×

×

4.  The coin is in B. Let’s assume that the inscription on A is true: Scenario 1

←

True A The coin is in here.

B The coin is not in here.

C The coin is not in A.

Figure A.16  Coin scenario: part 1

Now, we can quickly ascertain that B’s inscription is also true—if the coin is in A, then, as B’s inscription proclaims, it is certainly not in B. But this is contrary to the condition that at most one inscription is true. Here we have two true inscriptions, instead. So, we can reject scenario 1. In the process, however, we have discovered that A’s inscription is necessarily false—the coin is not in A. That makes C’s inscription true, since it merely confirms that the coin is not in A: Scenario 2 True

A The coin is in here.

←

←

False B The coin is not in here.

C The coin is not in A.

Figure A.17  Coin scenario: part 2

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An sw e r s

| 57

Since at most only one of the inscriptions is true, then B’s inscription has to be false. This completes scenario 2:

B The coin is not in here.

←

A The coin is in here.

True

←

Scenario 2 False

←

False

C The coin is not in A.

Figure A.18  Coin scenario: part 3

B’s inscription reads: “The coin is not in here.” According to scenario 2 this is a false statement. Thus the opposite is true—the coin is in B, contrary to what B’s inscription says. 5. In this puzzle, the key is to realize that each sister does not have a different brother. So, the farmer had seven daughters and one son, who was the brother of each of the seven daughters. In total, he had eight children.

Chapter 10 1.  Here is the solution, step-by-step: 1. Fill the 5-pint jar from the cask.

3-pint

5-pint

Figure A.19  Pouring problem: part 1

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58 | An sw e r s 2.  Fill the 3-pint jar from the 5-pint one, leaving 2 pints in the 5-pint jar.

3-pint

5-pint

Figure A.20  Pouring problem: part 2

3.  Empty the 3-pint jar back into the cask.

3-pint

5-pint

Figure A.21  Pouring problem: part 3

4.  Pour the 2 pints that are in the 5-pint jar into the 3-pint jar.

3-pint

5-pint

Figure A.22  Pouring problem: part 4

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An sw e r s

| 59

5.  Fill the 5-pint jar from the cask.

3-pint

5-pint

Figure A.23  Pouring problem: part 5

6. Pour liquid into the 3-pint jar from the 5-pint jar. This will add a pint to the 3-pint jar and leave 4 pints in the 5-pint jar, which is the required solution.

3-pint

5-pint

Figure A.24  Pouring problem: part 6

2.  The sequence of movements is shown below: W = White; B = Black: 0. 1. 2. 3. 4. 5. 6. 7. 8.

WWW_BBB WW_WBBB WWBW_BB WWBWB_B WWB_BWB W_BWBWB _WBWBWB BW_WBWB BWBW_WB

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60 | An sw e r s 9. B W B W B W _ 10. B W B W B _ W 11. B W B _ B W W 12. B _ B W B W W 13. B B _ W B W W 14. B B B W _ W W 15. B B B _ W W W 3. The traveler cannot start with the cabbage, since the wolf would eat the goat if the two were left alone; nor the wolf, since the goat would eat the cabbage. So, his only choice is to start with the goat. Once this critical decision is made, the rest of the puzzle is solved easily. He goes across, drops off the goat, and comes back alone. When he gets back to the original side, he could pick up either the wolf or the cabbage. Let’s go with the cabbage. He goes across with the cabbage to the other side, drops it off, but goes back to the original side with the goat (to avoid disaster). Back on the original side, he drops off the goat and goes over to the other side with the wolf. When there, he drops off the wolf safely with the cabbage. He travels back alone to pick up the goat. He then travels to the other side with the goat and, together with the wolf and cabbage, continues on his journey. 4.  It takes 12 crossings, organized as follows: 0. Both boys are at the opposite bank. 1. One boy brings the boat to the soldiers and gets off. 2. One of the soldiers crosses the river and gets off on the opposite bank. 3. The boy who is there returns with the boat. 4. When he gets to the original side, both boys get on and cross the river. 5. One boy gets off and the other returns with the boat. 6. That boy gets off and a second soldier crosses the river. 7. He gets off on the opposite bank and the second boy returns with the boat. 8. When he gets there, both boys get on and cross the river. 9. One boy gets off and the other one returns with the boat. 10. He gets off and the third soldier crosses the river over to the opposite bank. 11. The soldier gets off and the second boy returns with the boat. 12. The two boys get on and cross over.

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| 6

5. It is not possible to cover the altered checkerboard in this way, for the simple reason that the two squares that are to be removed are of the same color. A  domino placed on the checkerboard always covers a white and a black square. With two opposite corners removed, of the same color, the board does not have an equal number of black and white squares for dominoes to cover.

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REFERENCES AND BIBLIOGRAPHY

This section contains the references used in this book as well as additional bibliography for further consultation. Al-Khalili, Jim. 2012. Paradox: The Nine Greatest Enigmas in Physics. New York: Broadway. Archimedes. 1897. The Sand Reckoner, Thomas L. Heath (trans.). Cambridge: Cambridge University Press. Banks, Robert S. 1999. Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics. Princeton: Princeton University Press. Barrow, John D. 2014. Maths & the Arts. London: Bodley Head. Beckmann, Petr. 1971. A History of π. New York: St. Martin’s. Beiler, Albert H. 1966. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover. Bellos, Alex. 2014. The Grapes of Math. Toronto: Doubleday. Benjamin, Arthur. 2015. The Magic of Math. New York: Basic Books. Berggren, Len, Borwein, Jonathan, and Borwein, Peter. 1997. Pi: A Source Book. New York: Springer. Berlinski, David. 2000. The Advent of the Algorithm. New York: Harcourt. Berlinski, David. 2013. The King of Infinite Space: Euclid and His Elements. New York: Basic Books. Blatner, David. 1997. The Joy of Pi. Harmondsworth: Penguin. Boole, George. 1854. An Investigation of the Laws of Thought. New York: Dover. Borel, Émil. 1909. “Le continu mathématique et le continu physique.” Rivista di Scienza 6: 21–35. Bronowski, Jacob. 1973. The Ascent of Man. Boston: Little, Brown, and Co.

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164 | R e f e r e n c e s an d B i b li o g r aph y Cantor, Georg. 1874. “Über eine Eigneschaft des Inbegriffes aller reelen algebraischen Zahlen.” Journal für die Reine und Angewandte Mathematik 77: 258–62. Cayley, Arthur. 1854. “On the Theory of Groups, as Depending on the Symbolic Equation θn = 1.” Philosophical Magazine 7: 40–7. Chaitin, Gregory J. 2006. Meta Math. New York: Vintage. Chamberlain, Marc. 2015. Single Digits. Princeton: Princeton University Press. Changeux, Pierre. 2013. The Good, the True, and the Beautiful: A Neuronal Approach. New Haven: Yale University Press. Clark, Michael. 2007. Paradoxes from A to Z. London: Routledge. Clegg, Brian and Pugh, Oliver. 2012. Infinity: A Graphic Guide. London: Icon Books. Crilly, Tony. 2011. Mathematics. London: Quercus. Davis, Phillip  J. and Hersh, Reuben. 1986. Descartes’ Dream: The World According to Mathematics. Boston: Houghton Mifflin. Dehaene, Stanislas. 1997. The Number Sense: How the Mind Creates Mathematics. Oxford: Oxford University Press. Derbyshire, John. 2004. Prime Obsession: Bernhard Riemann and His Greatest Unsolved Problem in Mathematics. Washington: Joseph Henry Press. Devlin, Keith. 2005. The Math Instinct. New York: Thunder’s Mouth Press. Devlin, Keith. 2011. The Man of Numbers: Fibonacci’s Arithmetic Revolution. New York: Walker and Company. Dormehl, Luke. 2014. The Formula. New York: Perigee. Doxiadis, Apostolous. 2000. Uncle Petros and Goldbach’s Conjecture. London: Faber and Faber. Du Sautoy, Marcus. 2004. The Music of the Primes: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: HarperCollins. Dudeney, Henry  E. 1958. The Canterbury Puzzles and Other Curious Problems. New York: Dover. Dunlap, Richard A. 1997. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific. Eco, Umberto. 1998. Serendipities: Language and Lunacy, trans. by William Weaver. New York: Columbia University Press. Elwes, Richard. 2014. Mathematics 1001. Buffalo: Firefly. Eymard, Pierre and Lafon, Jean-Pierre. 2004. The Number Pi. New York: American Mathematical Society. Flood, Raymond and Wilson, Robin J. 2011. The Great Mathematicians: Unravelling the Mysteries of the Universe. London: Arcturus. Fortnow, Lance. 2013. The Golden Ticket: P, NP, and the Search for the Impossible. Princeton: Princeton University Press.

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Frege, Gottlob. 1879. Begiffsschrift eine der aritmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Nebert. Freiberger, Marianne and Thomas, Rachel. 2014. Numericon: Journey Through the Hidden Lives of Numbers. New York: Quercus. Gaines, Helen Fouché. 1989. Cryptanalysis: A Study of Ciphers and Their Solutions. New York: Dover. Gamow, George. 1947. One, Two, Three . . . Infinity. New York: Dover. Gessen, Masha. 2009. Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century. Boston: Houghton Mifflin Harcourt. Gödel, Kurt. 1931. “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Teil I.” Monatshefte für Mathematik und Physik 38: 173–89. Hardy, G. H. 1967. A Mathematician’s Apology. Cambridge: Cambridge University Press. Hardy, Michael and Woodgold, Catherine. 2009. Prime Simplicity. Mathematical Intelligencer 31: 44–52. Harel, Guershon and Sowder, Larry. 2007. “Toward Comprehensive Perspectives on the Learning and Teaching of Proof,” in F. K. Lester (ed.), Second Handbook of Research on Mathematics Teaching and Learning, 805–42. Charlotte, NC: Information Age Publishing. Havil, Julian. 2008. Impossible? Princeton: Princeton University Press. Heath, Thomas L. 1949. Mathematics in Aristotle. Oxford: Oxford University Press. Henshaw, John. M. 2014. An Equation for Every Occasion. Baltimore: Johns Hopkins University Press. Hersh, Reuben. 1999. What Is Mathematics Really? Oxford: Oxford University Press. Hilbert, David. 1931. “Die Grundlagen der elementaren Zahlentheorie.” Mathematische Annalen 104: 485–94. Hofstadter, Douglas. 1979. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books. Hofstadter, Douglas and Sander, Emmanuel. 2013. Surfaces and Essences: Analogy as the Fuel and Fire of Thinking. New York: Basic. Isacoff, Stuart. 2003. Temperament: How Music Became a Battleground for the Great Minds of Western Civilization. New York: Knopf. Jung, Carl G. 1972. Synchronicity: An Acausal Connecting Principle. Routledge and Kegan Paul. Jung, Carl G. 1983. The Essential Jung. Princeton: Princeton University Press. Kasner, James and Newman, John. 1940. Mathematics and the Imagination. New York: Simon and Schuster. Li, Yan and Shiran, Du. 1987. Chinese Mathematics: A Concise History. New York: Oxford University Press.

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166 | R e f e r e n c e s an d B i b li o g r aph y Livio, Mario. 2002. The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. New York: Broadway Books. Loomis, Elisha Scott. 1968. The Pythagorean Proposition. The National Council of Teachers of Mathematics. Mackenzie, Dana. 2012. The Universe in Zero Words. London: Elwin Street Publications. Maor, Eli. 1994. e: The Story of a Number. Princeton: Princeton University Press. Maor, Eli. 2007. The Pythagorean Theorem: A 4,000-Year History. Princeton: Princeton University Press. Mazur, Joseph. 2008. Zeno’s Paradox: Unravelling the Ancient Mystery behind Space and Time. New York: Plume. Musser, Gary  L., Burger, William  F., and Peterson, Blake  E. 2006. Mathematics for Elementary Teachers: A Contemporary Approach. Hoboken: John Wiley. Neugebauer, Otto and Sachs, Abraham. 1945. Mathematical Cuneiform Texts. New Haven: American Oriental Society. O’Shea, Donal. 2007. The Poincaré Conjecture. New York: Walker. O’Shea, Owen. 2016. The Call of the Primes. New York: Prometheus. Parker, Matt. 2014. Things to Make and Do in the Fourth Dimension. Toronto: Doubleday. Peano, Giuseppe. 1973. Selected Works of Giuseppe Peano, H.  Kennedy, ed. and trans. London: Allen and Unwin. Peirce, Charles S. 1923. Chance, Love, and Logic. New York: Harcourt, Brace. Peirce, Charles  S. 1931–1958. Collected Papers of Charles Sanders Peirce, Vols. 1–8, C. Hartshorne and P. Weiss (eds.). Cambridge, Mass.: Harvard University Press. Petkovic, Miodrag  S. 2009. Famous Puzzles of Great Mathematicians. Providence, RI: American Mathematical Society. Posamentier, Alfred S. 2004. Pi: A Biography of the World’s Most Mysterious Number. New York: Prometheus. Posamentier, Alfred S. 2017. The Joy of Mathematics. Amherst: Prometheus Books. Posamentier, Alfred S. and Lehmann, Ingmar. 2007. The (Fabulous) Fibonacci Numbers. New York: Prometheus. Rockmore, Dan. 2005. Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers. New York: Vintage. Rudman, Peter  S. 2010. The Babylonian Theorem: The Mathematical Journey to Pythagoras and Euclid. Amherst: Prometheus. Russell, Bertrand and Whitehead, Alfred N. 1913. Principia Mathematica. Cambridge: Cambridge University Press. Sabbagh, Karl. 2004. The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics. New York: Farrar, Strauss & Giroux.

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INDEX

A abacus principle  39 absolute value  40, 41, 42, 43 Achilles and the Tortoise Paradox 106 Ahmes Papyrus  58, 59 al-Khwarizmi, Muhammad ibn Musa  39, 127 algebra  53, 60, 74, 99, 119, 127 algorithm  27, 29, 59, 96, 116, 127, 128, 129, 130, 131 amicable number  15, 16 analytic geometry  43, 44, 45, 46 Archimedes  59, 60, 62, 80, 113 Argand, Jean-Robert  98 Argand plane  98, 100 Aristotle  105, 106, 116, 121 arithmetic  38, 41, 42, 43, 46, 50, 51, 52, 53, 54, 55, 70, 71, 74, 78, 99, 101, 118, 119 Aronofsky, Darren  57, 63, 66 Arrow Paradox  106 Aryabhata 95 Augustine, Saint  17 axiom  5, 116, 117, 118, 119, 120, 121

B Babylonian mathematics  2, 13, 39, 56, 95, 96, 97 Bacon, Francis  51, 52 Barber Paradox  108, 120 Berkeley, George  112 Bernoulli, Jacob  84, 86, 87, 91 Bhāskara  4, 20, 97, 137, 138

Bi-Literary Alphabet  52 binary digit  51, 52, 53, 54, 118, 119, 129, 142 Bombelli, Rafael  94, 98, 99 Boole, George  53, 118, 119, 120 Brahamagupta 39 Briggs, Henry  78 Brun, Viggo  34 Buffon, Georges-Louis Leclerc, Comte de  62 Buffon’s Needle Problem  62, 67

C calculus  13, 46, 49, 50, 83, 85, 106, 107, 112 Cantor, Georg  109, 113, 123 Cantor’s diagonal proof  110 Cantor’s infinity proofs  109, 111, 117 Cardano, Gerolamo  40, 94, 98 cardinality  109, 110, 111, 152 Carroll, Lewis  114 Cartesian plane  43, 44, 46, 48, 49, 75, 98 Chebyshev, Lvovich  24 Chinese mathematics  3, 39, 52, 56, 60, 72, 77 circle  45, 46, 56, 57, 58, 59, 60, 63 complex number  88, 93, 94, 97, 98 composite number  23, 24, 25, 27, 29, 39 compound interest  75, 83, 84, 89, 90 computability  129, 130, 131 computer science  116, 127, 131 conjecture 9

conjugate  102, 103 connectivity  84, 85 consistency  116, 117, 119 contradiction  8, 9, 24, 108, 116, 121 coordinate geometry  43, 44 cosine 88

D D’Alembert, Jean  99 da Vinci, Leonardo  5 De Bono, Edward  114 decidability  116, 127, 130 decimal number  11, 38, 39, 40, 53 deduction  5, 116 defective number  16 Descartes, René  43, 71, 93, 99, 122 Dichotomy Paradox  105 differentiation 85 digit  38, 39, 51, 57, 69, 116, 126, 127, 129 Diophantus  17, 18, 95 Dirichlet, Johann  17 dissection proof  3, 4 division by zero  46, 47, 48 Dudeney, Henry E.  34, 35, 108, 124, 141

E e  82, 83, 84, 85 Egyptian mathematics  2, 56, 58 Einstein Albert  64, 80, 113 engineering  1, 13, 14, 60, 94, 118, 119 Epimenides  107, 108

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170 | I N D E X Eratosthenes  25, 26 Eubulides of Miletus  107 Euclid  4, 9, 11, 12, 16, 17, 23, 24, 25, 26, 27, 28, 30, 113, 116, 117, 119 Euclid’s algorithm  127, 128 Euclidean geometry  117, 118 Euler, Leonhard  32, 56, 60, 62, 82, 83, 85, 88, 93, 99 Euler’s identity  88, 89, 90 excessive number  16 exponent  50, 69, 70, 71, 72 exponential arithmetic  73, 74 exponential function  75, 84, 85, 86, 87 exponential notation  71, 72, 73, 74, 77, 80

F factorial 83 Fermat, Pierre de  17, 18, 31 Fermat’s Last Theorem  17, 18, 33, 79 Fibonacci, Leonardo  39, 40, 143 figurate number  14 Frege, Gottlob  120 function  45, 46, 50, 85, 86 Fundamental Theorem of Algebra  99, 100 Fundamental Theorem of Arithmetic  26, 27, 129

G Galilei, Galileo  108, 109, 113, 133 Galois, Evariste  100 Gamow, George  112 Gardner, Martin  21, 124, 140 Garfield, James  15 Gauss, Karl Friedrich  17, 27, 98, 99 geometry  9, 14, 46, 118 Germain, Sophie  33 Germain prime  33 Girard, Albert  99 Gödel, Kurt  116, 121, 123, 131 Goldbach, Christian  9, 10, 82 Goldbach’s conjectures  9, 10 googol 70

googolplex 70 Greek mathematics  5, 39, 59, 72 group theory  100

Lobachevski, Nikolai  117 logarithm  71, 77, 78, 80, 83, 87 logarithmic spiral  86

H

M

halting problem  141 Hamilton, William Rowan  100 Hilbert, David  112, 119 Hindu mathematics  40, 95, 97 Hindu-Arabic system  39

Madhava 62 magic square  34, 35, 141 Mersenne, Marin  23, 28, 29, 31 Mersenne prime (number)  23, 28, 29, 30 music of the spheres  ix, 133

I i (imaginary number)  72, 88, 93, 94, 98, 100 I Ching 52 incompleteness 121 Indian mathematics  4, 20, 24, 39, 40, 62 induction  6, 116 infinite hotel paradox  112 infinity  24, 48, 84, 104, 105, 106, 109, 112, 113, 122 irrational number  11, 13, 43, 60, 100 iterative algorithm  59

N Napier, John  78, 83 natural logarithm  83, 84 negative exponent  32, 74, 86 negative number  39, 40, 41, 42, 43, 44, 54, 94, 97 Newton, Sir Isaac  49, 76 non-Euclidean geometry  117, 118 notation  69, 70 number line  32, 39, 40, 41, 42, 43, 44, 53, 61, 107 numeral  37, 38, 39, 40, 126, 127, 129, 130

J

P

Jones, William  56 Jung, Carl  61

paradox 104 Parallel Postulate  117 Pascal, Blaise  75, 76, 77 Pascal’s triangle  75, 76, 77, 79 pattern  14, 15, 16, 123 Peano, Giuseppe  40, 118, 119, 120 Peirce, Charles S.  66, 122 perfect number  15, 16 Persian mathematics  20, 39, 127 pi (π)  56, 57, 58, 60, 61, 62, 63, 64 Pi Day  64, 65 piem 66 Plato  18, 19, 131, 132 Platonic form  19 polygon  6, 7, 8, 59, 113 postulate  116, 117, 121 prime number  3, 10, 16, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34

K Khallikan Ibn  29, 30 Khallikan’s chessboard problem  29, 30 Kummer, Ernst E.  17

L lateral thinking  114 Legendre, Adrien-Marie  17 Leibniz, Gottfried Wilhelm  45, 49, 51, 52, 60, 62, 82, 122 Liar Paradox  107, 108, 119 limits  48, 49, 106 Lindemann, Ferdinand von 61 Liouville, Joseph  61 Liu Hui  60

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INDEX proof  5, 6, 7, 8, 9, 10 Pythagoras  ix, x, 132, 133 Pythagorean theorem  ix, 3, 4, 5

Q QED 9 quadratic equation  93, 95, 96, 97, 99, 100 quaternion 101

R Ramanujan, Srinivasa  24 rational number  11, 12, 109, 110 real number  61, 63, 98 reductio ad absurdum  8, 12, 24 Rhind Papyrus 58 Richter scale  78, 79 Riemann, Bernhard  32, 117 Riemann Hypothesis  32, 33 right triangle  1, 2, 4, 60 Roman numeral  37, 38, 39 Russell, Bertrand  108

S Sagan, Carl  57, 70 science  13, 31, 60, 61, 87, 94, 133 serendipity  31, 101, 102 sets  109, 113 Shannon, Claude  119 Shaw, Larry  65 Sieve of Eratosthenes  25, 26 sine 88 Smullyan, Raymond  123 spira mirabilis 86 square root of  2 (√2) 11, 12, 13 Stadium Paradox  106 syllogism 116 synchronicity 61

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Turing, Alan  131 twin prime  23, 34

U Ulam, Stanislaw  31 undecidability  119, 120, 130

W Wallis, John  53, 62 Walpole, Horace  101, 102 Whitehead, Alfred North  120, 121 Wiles, Andrew  18 Wiles-Taylor proof  18 Wittgenstein, Ludwig  80, 120

T

Z

Thales of Miletus  5 Theano  ix, 18 transcendental number  60, 61, 81 transfinite number  100, 110, 111 triangular number  15 trigonometry 88

Zeno of Elea  104, 105 Zeno’s paradoxes  104, 105, 107 zero  32, 39, 40, 41, 42, 43, 44, 48, 49, 50, 53, 54, 118 zero exponent  50, 51 zeta function  32