Mathematics through English

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Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»

Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»

Министерство образования и науки Российской Федерации Сибирский федеральный университет

Т. Н. Свиридова

MATHEMATICS THROUGH ENGLISH Рекомендовано федеральным государственным бюджетным образовательным учреждением высшего профессионального образования «Санкт-Петербургский государственный университет» в качестве учебного пособия по дисциплине «Английский язык» для студентов высших учебных заведений, обучающихся по направлению подготовки 040400 «Прикладная математика и информатика», № 3085 от 21.07.2015

Красноярск СФУ 2016

Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»

УДК 811.111:51(07) ББК 81.432.1я73+22.1я73 С247

Свиридова, Т. Н. С247 Mathematics through English : учеб. пособие / Т. Н. Свиридова. – Красноярск : Сиб. федер. ун-т, 2016. – 138 с. ISBN 978-5-7638-3415-4 Представлены материалы для совершенствования умений профессионально ориентированного чтения и навыков устной и письменной речи. Приведены тексты аутентичного характера. Предложена серия упражнений, способствующих расширению и систематизации словарного запаса, развитию разговорных навыков и умений, закреплению грамматического материала. Предназначено для студентов высших учебных заведений, обучающихся по направлению подготовки 040400 «Прикладная математика и информатика». Электронный вариант издания см.: http://catalog.sfu-kras.ru

ISBN 978-5-7638-3415-4

УДК 811.111:51(07) ББК 81.432.1я73+22.1я73

© Сибирский федеральный университет, 2016

Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»

CONTENTS Введение........................................................................................................4 To the Student................................................................................................5 Module 1. Mathematics and mathematicians.............................................6 UNIT 1....................................................................................................6 UNIT 2................................................................................................. 11 UNIT 3................................................................................................. 16 Module 2. Algebra...................................................................................... 24 UNIT 1................................................................................................. 24 UNIT 2................................................................................................. 29 UNIT 3................................................................................................. 34 Module 3. Geometry.................................................................................. 42 UNIT 1................................................................................................. 42 UNIT 2................................................................................................. 47 UNIT 3................................................................................................. 54 Module 4. Mechanics................................................................................. 61 UNIT 1................................................................................................. 61 UNIT 2................................................................................................. 67 UNIT 3................................................................................................. 73 Module 5. Informatics............................................................................... 80 UNIT 1................................................................................................. 80 UNIT 2................................................................................................. 88 UNIT 3................................................................................................. 91 Module 6. Unsolved problems................................................................... 99 UNIT 1................................................................................................. 99 UNIT 2............................................................................................... 105 UNIT 3............................................................................................... 111 References................................................................................................. 117 Appendix 1................................................................................................ 119 Appendix 2................................................................................................ 131 Appendix 3................................................................................................ 134

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Введение Основная цель пособия – совершенствование умений профессионально-ориентированного чтения и перевода современной научной литературы, а также развитие навыков устной и письменной речи на основе изученных текстов. Следует подчеркнуть, что конкретные задачи, решаемые при работе с пособием, определяются конечными целями обучения, предусматривающими свободное чтение литературы по специальности, умение вести беседу, дискуссию, выступать с сообщениями, докладами, презентациями и т. п. Пособие состоит из шести модулей, каждый из которых посвящен определенной области математики. Тематика текстов включает сведения из истории математики, описание ее основных проблем, современную оценку развития данной области науки и занимательные факты из жизни ученых. При отборе текстов учитывалась как их познавательная ценность, так и их насыщенность лексико-грамматическими структурами. Каждому тексту предшествует ряд упражнений разминочного характера, в которых студентам предлагается высказать свое мнение по ряду проблем, затрагиваемых в тексте – посредством этого достигается большая вовлеченность учащихся в процесс дальнейшей работы над текстом и основной лексической темой урока. Включение в учебный материал текстов с занимательной информацией и элементами математических игр способствует, по мнению автора, оживлению учебного процесса. Каждый модуль состоит из разделов, предусматривающих работу над закреплением грамматического материала, расширением потенциального словаря, развитием навыков чтения, говорения и письма. В каждом модуле учебного пособия представлены упражнения, отличающиеся по своему содержанию, целевой направленности и форме выполнения. Система речевых упражнений построена в соответствии с методикой проблемного обучения: проблемный вопрос, проблемное высказывание, дискуссия по проблеме. Равномерное чередование различных видов упражнений и использование в них различных моделей помогает более эффективной работе студентов.

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To the Student This book is intended for intermediate students of English. It is designed to improve your professional English. You will be given a chance to: • master communicative competence; • develop thoughts and ideas; • activate personal opinion; • stimulate your imagination; • enrich your vocabulary; • experience various ways of exploring a language and developing your skills: speaking, reading and writing; • prepare for the examination. To get the most out of this book, you have to think. Don’t be a passive learner. With your group-mates work out the solutions to different activities. Help in understanding mathematical terms may be obtained from the mini-dictionary in Appendix. The material is taken from the Internet, math textbooks, authentic sources – thus they are related to your own experience. This book is designed so that it can be used sequentially from Module 1 to 6. However, each Module stands on its own, therefore you can use it selectively. Every Module starts with a list of objectives so you always know exactly what you are going to do. Good luck and enjoy using this book! Author

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Module 1 Mathematics and mathematicians Learning objectives: In this module you will: • understand basic mathematical concepts, and acquire vocabulary connected with it • develop the idea • express agreement, disagreement • do mental math tricks • review sentence-structure • write a composition • check spelling, punctuation and grammar

UNIT 1 Before you read 1. Discuss these questions with your partner. 1. How far do you agree that: Mathematics is not: • just doing things with numbers and letters and other symbols; • just a collection of facts and rote recipes; • just computational and arithmetic skills; Mathematics is: • a way of thinking; • the language of science; • a creative discipline; • a source of pleasure and wonder; • a means of problem solving. 2. What is mathematics? 3. Where does the word mathematics come from? 4. How many branches of mathematics do there exist nowadays? 5. What field of maths is the most interesting (important, essential, significant) to your mind? 6

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Mathematics and mathematicians

Reading 2. Read the text. The six sentences below have been removed. Decide where they should go. A. It is believed that mathematics is the mother of modern science. B. The second reason of the importance of studying mathematics is its everyday use in real life by everyone. C. Mathematics is important; all of it. D. Finally, mathematics is the only mutual language used by all human beings regardless of their origin, gender, religion, or culture. E. Mathematics is the mother science of the abstract world. F. Another good example of the use of the importance of mathematics in daily life is managing how money grows and how to keep it from shrinking. The Importance of Mathematics Mathematics is the tool to keep balance in our life. 1) .......... Mathematics is the confusing course that causes my headaches. These are three different definitions of the word mathematics according to three different people. However mathematics is considered as the most important science, and its development affects the development of science in all of its kind medical, physics, biology, technology, and more. The importance of mathematics is being an essential, creative, and powerful discipline recognized globally. First reason of studying mathematics is its importance for the development of modern science. 2) .......... The basics developed in mathematics such as equations, laws, algorithms, they are used implicitly in other sciences. For instance, the development of algorithms in mathematics will affect the future of computer science since ninety percent of the computer science theories and programming structures are based on abstract algorithms. Another example is related to the medical field where according to The Ourant Institute, New York University, “equations that describe cardiac mechanics including blood, muscle, and valve mechanics and electrophysiology are different, in both cases a realistic treatment 7

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demands the use of methods that account for anisotropy, in homogeneity, and complex geometries” which are fields in abstract mathematics. Mathematics is the science that develops explicitly other kind of science to apply its capacity for speed and precision. 3) .......... Since the early ages the basic arithmetic operations have been used by human regarding his/her origin, culture, and language. These four operations construct the elementary parts of mathematics. Cooking, for instance; is a simple task done by anybody; however, the quality of cooking is based on the amount of ingredient used; people use mathematics to weigh the quantity used through adding and subtracting using balances and units like kilogram and liter. 4) .......... The simple arithmetic operations are used for the calculation of the income from a savings account or the charges accumulated on a credit card; these operations could mean big payoffs over time. For thousands of years mathematics has been used daily if it is in commerce while trading or in kitchens while cooking. 5) .......... One plus one is still equal to two despite of what country is used in or what language is expressed with. Adding the cost of the shopping items is the same whether it is in dollars, euros, or dirham. Few are the people that can speak more than one tongue language; however, all of us possess the ability to be literate in the mutual language of math. The language of math is what connects the world because it is a tool using which we can express the mysteries of the universe, paranormal phenomenon, or the secret of curing extreme diseases. Mathematics connects people with other cultures in order to communicate and transfer the understanding of strange phenomena. Mathematics is more than just the science of numbers taught by teachers in schools to students. Mathematical theories are used everyday either by expert in applied science like physicians or by chefs in cheap restaurants. It needs to be amplified in education to provide students with skills required to achieve higher education, career aspirations, and reaching personal fulfillment. 6) .......... (abridged from http://voices.yahoo.com/the-importance-mathematics-5224583.html)

3. Look through the text and find the word that means the same as. 1) common to two or more people; 2) quality of being of the same kind; 3) an ability to hold, contain, learn things/qualities/ideas etc.; 4) made or become greater in number or quantity; 5) made larger or fuller; 6) able to read and write; cultured.

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4. Decide if the following statements are true or false, or there is no information in the text. a) true b) false c) no information 1. Mathematics is a tool for describing the world. 2. Mathematics connects people with other cultures. 3. Mathematics is just the science of numbers. 4. Mathematicians do not deal with applications of mathematics. 5. Like all other sciences maths arose out of the needs of men. 6. Mathematical theories are used everyday. 7. Mathematics as a science is a collection of branches. 5. Divide the text into logical units. Suggest titles for each unit. Find the sentences expressing the main ideas of each logical unit in the text. Combine the topic sentences using the following connectives. Because, in case of, besides, still, nevertheless, hence, accordingly, therefore, as well as, consequently, either …or. Speaking 6. Agree or disagree with the following statements. Use the introductory phrases. I quite agree to it. I don’t quite agree to it. I think, it is right. On the contrary. Far from it. I accept it fully. Not at all. Quite the reverse. Exactly. Certainly. Just the other way round. 1. Maths is a game, a free creation of the mind divorced from practical problems. 2. Maths is a universal tool for describing the world around. 3. Maths is an activity which has as its goal the formulation and understanding of a complete model of the universe. 4. Scientists can use maths as a shorthand script to codify relationships. 5. Mental reasoning from obvious truths, i.e., axioms and postulates, constitutes applied science. 6. Maths is an inspiration to the artist as well as a tool to scientist. 7. The present role of maths is the same as in previous stages of its development. 8. Maths is the technique of discovering and expressing in the most economical possible way useful rules of reliable reasoning about calculation, shape and measurement. 9. Math language was not developed all of a sudden. 9

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10. A sharp dividing line between “pure” and “applied” maths can, in fact, be drawn. 7. Discussion 1. Which image of maths do you prefer? • “Maths is general learnable knowledge” (The Pythagoreans). • “Maths is great oaktree with deep roots and powerful trunk, and from the trunk’s top numerous branches had issued and subdivided into smaller branches” (H. Eves, historian of maths). • “Maths is the whole universe”. (Experts of modern mathematical logic.) 2. Most mathematicians object to the separation of pure and applied aspects of maths. Why? Do pure and applied maths have common language, methods, applications? 3. The requirement of rigour in reasoning is proverbial in maths. Maths rigour – who needs it? 4. What is the role of a) experience, b) common sense, c) intuition, d) talent, e) genius, f) imagination, g) flashes of insight in maths? 5. What is meant by the phrase “mathematization of science”? What are its advantages/disadvantages and implications? 6. It is convenient to keep the old classification of maths as one of the sciences, but it is more just to call it an art. If maths is an art with cultural bearings it must be a part of the liberal education of a doctor, lawyer or average educated person. Agree or disagree. 8. Your friend can’t decide if he should study mathematics or not. Work out arguments in favour of studying mathematics. Try to sound convincingly. 9. Fill in the gaps with the derivative. It is said that Mathematics is the gate and key of the Science. According to the famous Philosopher Kant, “A Science is exact only in so far as it (1) … Mathematics”. So, all (2) … education which does not commence with Mathematics is said to be defective at its foundation. Neglect of mathematics works injury to all knowledge. One who is (3) … of mathematics cannot know other things of the World. Again, what is worse, who are thus ignorant are unable to perceive their own ignorance and do not seek any remedy. So Kant says, “A natural Science is a Science in so far as it is mathematical”. And Mathematics (4) ... a very important role in building up modern Civilization by perfecting all Science. 10

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In this modern age of Science and Technology, emphasis (5) … on Science such as Physics, Chemistry, Biology, Medicine and Engineering. Mathematics, which is a Science by any criterion, also is an efficient and necessary tool being employed by all (6) … . As a matter of fact, all these Sciences progress only with the aid of Mathematics. So it is aptly remarked, “Mathematics is a Science of all Sciences and art of all arts”. Mathematics is a (7) … of human mind concerned chiefly with ideas, processes and reasoning. It is much more than Arithmetic, more than Algebra more than Geometry. Also it is much more than Trigonometry, Statistics, and Calculus. EMPLOY _____________________ 1 SCIENCE _____________________ 2 IGNORANCE _________________ 3 PLAY ________________________ 4 GIVE ________________________ 5 THIS _ _______________________ 6 CREATE _ ____________________ 7

UNIT 2 Before you read 1. Discuss these questions with your partner. 1. How far do you agree that: • Mathematics is only for math-minded boys? • Women cannot be genuine mathematicians? 2. When you think of a woman mathematician, what image comes to mind? 3. Do you think women mathematicians are unusual in any way? If yes, in what way? 2. Match these words with their definitions. 1) collaborate 2) embody 3) enchantress 4) implication

a) a woman who charms b) what is implied; smth hinted at or suggested, but not expressed c) to work in partnership, esp in literature or art d) to include; comprise 11

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3. Match the words to make phrases. 1) intellectual 2) lengthy 3) punched 4) pure 5) a series of 6) printed

a) tables b) addition c) diagrams d) notes e) card f) pursuits

Reading 4. Read the text below and choose the correct word for each space. Ada Lovelace (1815–1852) is often referred to as the world’s first computer programmer. The daughter of the famous poet Lord Byron, and the admired intellect, Annabella Milbanke, Ada Lovelace represented the meeting of two alternative worlds: the romanticism and art of her father versus the rationality and science of her mother. In her attempt (1) .......... draw together these polar opposites and create a ‘poetical science’ during the Victorian age, Ada collaborated (2) .......... the renowned mathematician and inventor, Charles Babbage. In Victorian Britain printed mathematical tables were used by navigators, architects, engineers, mathematicians and bankers, but these tables (3) .......... by human clerks (literally called ‘calculators’) and they were riddled with errors. Charles Babbage became interested in mechanising the production of these tables and he developed a series of diagrams and prototypes which enabled (4) .......... to explore his ideas. Babbage designed two types of engine, Difference Engines and Analytical Engines. The Difference Engines are calculators that work (5) .......... pure addition. In contrast The Analytical Engines mark a progression towards a machine that can conduct a number of different functions, such as addition, subtraction, multiplication and division – in short a general purpose machine with a design that embodies many of the characteristics to today’s modern computers. Ada Lovelace’s reputation comes from her important work interpreting Charles Babbage’s Analytical Engine. Following a visit to Turin in 1840 by Babbage, and the Italian Engineer, Luigi Menabrea, wrote a paper describing the principles of the machine. Ada then translated this paper from French but 12

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in doing so she added lengthy notes and further level of understanding which perhaps even Babbage himself (6) .......... Babbage was impressed with her work, describing her as ‘the Enchantress of Numbers’ and addressed her in a letter as ‘my dear and much admired interpreter’. Ada had understood the significance of the Analytical Engine and its implications (7) .......... computational method. She saw that through the punched card input device the Analytical Engine opened up a whole new opportunity for designing machines that could manipulate symbols rather than just numbers. Her achievements are even more exceptional given the attitudes of Victorian Britain towards the intellectual pursuits of women. (abridged from http://www.sciencemuseum.org.uk/onlinestuff/stories/ada_lovelace.aspx)

1. A from B in C to 2. A with B to C on 3. A calculated B were calculated C are calculated 4. A he B him C his 5. A using B use C is used 6. A had not achieved B has not achieved C doesn’t achieved 7. A to B for C with 5. These are answers to questions about the text. Write the questions. 1. The famous poet Lord Byron. 2. In 1840. 3. Difference Engines and Analytical Engines. 4. She translated it from French. 5. ‘The Enchantress of Numbers’. 6. It opened up a whole new opportunity for designing machines that could manipulate symbols rather than just numbers. Grammar review 6. Find the word which shouldn’t be in the sentence. 1. There is no reason that women cannot be an outstanding (famous, prominent) mathematicians and the Russian women mathematicians have proved it. a) the b) an c) have 2. Today mathematicians be frequently liken maths and its creations to music and art rather than to science. a) its b) be c) than 13

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3. Mathematicians do not rely on their intuitive judgement – they are seek to give a rigorous proof. a) on b) are c) to 4. The development of a meaningful, adequate and consistent system of notations does in various branches of maths is part of the history of maths. a) does b) is c) in 5. To develop a rigorous and elegant proof for the mathematician builds a structure of logic and form which to his eye is as beautiful as the finest poem. a) of b) is c) for 6. The full significance of maths can be seen and are taught only in terms of its intimate relationships to other fields of knowledge. a) in b) are c) be 7. It was by deductive reasoning that the Greeks has derived all math conclusions. a) has b) derived c) was 7. Fill in the gaps with the correct form of present tense. 1. No mathematician .......... (know) the name of the man who was the first to say 1, 2, 3 ... 2. She .......... (not study) algebra since last term. 3. The computer .......... (not work). It broke down this morning. 4. There are two ways in which maths .......... (become) so effective in our age. 5. For the decades universities .......... (look) for ways of offering courses to students who .......... (not have) access to the university campus, usually because of physical distance. 6. The first actual contact which most people .......... (have) with maths is through arithmetic. 7. Bare math formulas .......... (explain) nothing; they simply .......... (describe) symbols and signs in precise language. 8. Mental Math Trick Are you ready to do tough math calculations in your head? This Mental Math Trick will give you the edge and help you become a human calculator! Squaring a 2-digit number ending in 1: • take a 2-digit number ending in 1; • subtract 1 from the number; • square the difference; • add the difference twice to its square; • add 1. 14

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Example: • if the number is 61, subtract 1: 61 – 1 = 60; • 60 × 60 = 3600 (square the difference); • 3600 + 60 + 60 = 3720 (add the difference twice to its square); • 3720 + 1 = 3721 (add 1); • so 61 × 61 = 3721. Here's another: • for 81 × 81, subtract 1: 81 – 1 = 80; • 80 × 80 = 6400 (square the difference); • 6400 + 80 + 80 = 6560 (add the difference twice to its square); • 6560 + 1 = 6561 (add 1); • so 81 × 81 = 6561. Math Word Search Puzzle

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There are 41 math words hidden in this puzzle, but only 40 listed below. Can you findthem all? The mystery math word may not seem like much to many people, but it really is important! ADD ADDEND COUNT DENOMINATOR DIFFERENCE DIVIDE DIVISOR EIGHT EIGHTY EQUALS FACTOR FIFTY FIVE FORTY FOUR FRACTION HUNDRED MINUS MULTIPLE MULTIPLY NINE NINETY NUMBER NUMERAL NUMERATOR ONE PLUS PRODUCT QUOTIENT SEVEN SEVENTY SIX SIXTY SUBTRACT SUM TEN THIRTY THREE TWENTY TWO

UNIT 3 Before you read 1. Discuss these questions with your partner. • Mathematicians – what are they? • What do mathematicians do? • Why does a person make up his mind to become a mathematician? • What motives and directs the activities of mathematicians? • Do you think it is difficult to be a mathematician? • What mathematicians to your mind, are the most distinguished and why? 2. Match these words with their definitions. a) producing much or many 1) mortal b) very large 2) anticipate c) to do, make use of, before the right or 3) immense natural time d) to make or become less 4) insight 5) prolific e) understanding 6) diminish f) which must die; which cannot live forever 3. Match the words to make phrases 1) prolific 2) molten 3) subconscious 4) infinitesimal 5) anticipating 16

a) calculus b) theory c) intellectual d) core e) mind

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Reading 4. Read the text. The five sentences below have been removed. Decide where they should go. Leibniz' thoughts on mathematical physics had some influence. His childhood IQ has been estimated as second-highest in all of history, behind only Goethe's. Mathematical physicists influenced by Leibniz include not only Mach, but perhaps Hamilton and Poincaré themselves. Leibniz pioneered the common discourse of mathematics, including its continuous, discrete, and symbolic aspects. He invented more mathematical terms than anyone, including “function”, “analysis situ”, “variable”, “abscissa”, “parameter”, and “coordinate”. Gottfried Wilhelm von Leibniz (1646–1716) Germany Leibniz was one of the most brilliant and prolific intellectuals ever; and his influence in mathematics (especially his co-invention of the infinitesimal calculus) was immense. 1) .......... Descriptions which have been applied to Leibniz include “one of the two greatest universal geniuses” (da Vinci was the other); “the most important logician between Aristotle and Boole”; and the “Father of Applied Science”. Leibniz described himself as “the most teachable of mortals”. Mathematics was just a self-taught sideline for Leibniz, who was a philosopher, lawyer, historian, diplomat and renowned inventor. Because he “wasted his youth” before learning mathematics, he probably ranked behind the Bernoullis as well as Newton in pure mathematical talent, and thus he may be the only mathematician among the Top Fifteen who was never the greatest living algorist or theorem prover. We won't try to summarize Leibniz' contributions to philosophy and diverse other fields including biology; as just three examples: he predicted the Earth's molten core, introduced the notion of subconscious mind, and built the first calculator that could do multiplication. Leibniz also had political influence: he consulted to both the Holy Roman and Russian Emperors; another of his patrons was 17

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Sophia Wittelsbach, who was only distantly in line for the British throne, but was made Heir Presumptive. (Sophia died before Queen Anne, but her son was crowned King George I of England.) 2) .......... (His ideas on symbolic logic weren't pursued and it was left to Boole to reinvent this almost two centuries later.) Mathematical innovations attributed to Leibniz include the notations ∫f(x)dx, df(x)/dx, and even 3√x; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. 3) .......... His works seem to anticipate cybernetics and information theory; and Mandelbrot acknowledged Leibniz' anticipation of self-similarity. Like Newton, Leibniz discovered The Fundamental Theorem of Calculus; his contribution to calculus was much more influential than Newton's, and his superior notation is used to this day. As Leibniz himself pointed out, since the concept of mathematical analysis was already known to ancient Greeks, the revolutionary invention was notation (“calculus”), because with “symbols [which] express the exact nature of a thing briefly ... the labor of thought is wonderfully diminished”. 4) .......... He developed laws of motion that gave different insights from those of Newton. His cosmology was opposed to that of Newton but, anticipating theories of Mach and Einstein, is more in accord with modern physics. 5) .......... Mathematical physicists influenced by Leibniz include not only Mach, but perhaps Hamilton and Poincaré themselves. Although others found it independently (including perhaps Madhava three centuries earlier), Leibniz discovered and proved a striking identity for π: π/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 –... (abridged from http://knowyourmathematician.blogspot.ru/p/gottfried-wilhelm-vonleibniz-1646-1716.html)

5. Decide if the following statements are true or false, or there is no information in the text. a) true b) false c) no information 1. Leibniz was “one of the two greatest universal geniuses”. 2. Leibniz developed a binary numeration system. 3. Aristotle described Leibniz as “the most teachable of mortals”. 4. Boole invented such mathematical terms as “function”,“analysis situ”. 5. Leibniz made his calculating machine. 6. Leibniz' thoughts had great influence on mathematical physics. 18

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Mathematics and mathematicians

7. It was Leibniz who consulted to both the Holy Roman and Russian Emperors. 8. Poincare was influenced by Leibniz. Speaking 6. Suppose that the statement is incomplete. Repeat the statement and add your own reasoning, developing the idea further. Use the following phrases: There is one more point … I may as well add that ... One more remark seems reasonable, namely… Moreover… More than that… Exactly… Indeed… Model. Maths is often termed the language of science. Exactly. Maths is, indeed, the language of science. But I like to emphasize that maths is not only a language but a science in its own right. 1. Mastery of maths does not demand a “math mind”, peculiar talents or genius. The subject is within anybody’s grasp. 2. We say that while the creative work in maths is done by individuals, the results are the fruition of centuries of thought and development. 3. Mathematicians see beauty where others find only confusion of signs and symbols. 4. Maths is a wide science which is open to anyone who enjoys thinking and precise thinking in particular. 5. Mathematicians are human beings like “you and me”. 6. Like all other sciences maths arose out of the needs of men. 7. Read the quotations given below and agree or disagree with them. Your opinion should be followed by some appropriate comment. Here are some expressions that may be used to show your attitude: Agreement Disagreement I think so, too. I wouldn’t say so. That’s quite true… That’s not right, surely… I agree absolutely with… (I’m afraid) I can’t accept… • “The greatest unsolved theorem in mathematics is why some people are better at it than others”. (Adrian Mathesis) • “Mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination”. (Unknown) 19

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Module 1

• “Mathematics may not teach us how to add love or how to minus hate. But it gives us every reason to hope that every problem has a solution”. (Unknown) • “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true”. (Bertrand Russell) • “There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else – but persistent”. (Raoul Bott) 8. What qualities does a person need to be a good mathematician? Mark the following qualities 1 for the most important and 10 for the least important. Justify your order. • Diligence; • persistence; • being mathematically-minded; • being able to prove theorems; • having a creative mind; • being able to introduce new concepts; • having a higher education; • knowing all the scientific developments; • being able to pass exams at the University; • patience. Grammar review 9. Read the information about C. F. Gauss. Fill in the gaps with a, an, the or – (no article). Carl Friedrich Gauss (1777–1855) His great genius was demonstrated at (1) .......... young age. When he was in (2) .......... school, his teacher gave (3) .......... class (4) .......... task that he thought would take (5) .......... long time to complete. His task was to sum (6) .......... first 100 numbers. Gauss completed (7) .......... task in (8) .......... very short time. Much later, when (9) .......... teacher checked (10) .......... answers, Gauss had (11) .......... correct answer. He summed (12) .......... first hundred numbers not by addition, but by multiplication. He found 20

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Mathematics and mathematicians

that there are 50 pairs of 101. And 50 · 101 = 5050. He would always tell this story later in his life on how he was (13) .......... first to complete (14) .......... task. 10. Make sentences with these words. 1) mathematicians, the, modern, system, use, numeration, Greeks’, classical. 2) is, the, it, to, possible, precisely, of, the, system, development, numeration, Hindu-Arabic, describe. 3) abstract, people, from, and, generalize, numbers, process, counting. 4) number, problems, the, of, maths, in, inexhaustible, is. 5) teaches, continuity, history, the, of, science, of, development, the. 6) knowledge, not, is, math, a, set, alone, concepts, of. 7) among, maths, the, man, of, developments, cultural, highest, ranks. 11. Use the proper tense form (indefinite, continuous, perfect or perfect continuous) in the sentences. 1. The scientists express this number as a terminating decimal. (by the end of the last century, in future, since the birth of modern civilization, usually, nowadays, for a long time, if necessary) 2. Axiomatic inquiry brings forth new concepts in algebra. (still, from the outset, in 1873, eventually, this year, already, next decade) Writing 12. Read the text. Punctuate it as you think appropriate. What is mathematics most people would say it has something to do with numbers but numbers are just one type of mathematical structure saying math is the study of numbers or something similar is like saying that zoology is the study of giraffes math may be better thought of as the study of patterns but this too falls short. The more I study math the more I wonder about what exactly math is actually nobody knows it seems to be a product or our mind, and yet reflects the external universe with uncanny accuracy a mathematician develops a mathematical theory for its aesthetic unworldly beauty and its compelling evolution with no thought of how it might be applied to the world a century later a physicist finds this theory to be perfect to use as a framework to express his physics this sort of thing happens frequently pretty weird how intimately connected our innermost mind and the outermost universe really are this is a profound mystery. 21

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Module 1

Bruce Bennett my advisor in grad school defines mathematics as unified consciousness theory as you come to master a branch of mathematics it's as though you've grown a new abstract organ of perception through which you may then view the world you've grown a new mind's eye that can perceive realities literally inconceivable without this new organ of perception. Rafael Espericueta Professor of Mathematics Bakersfield College (from http://www2.bc.cc.ca.us/math/what_is_math.htm)

13. Essay Writing An essay is a creative literary work dealing with one problem or topic in detail. The University newspaper announced a literary competition. The students were asked to continue the essay on the topic “The study of Mathematics”. Why should anyone study mathematics? There are actually many different reasons. To some people it is an absolute necessity as a part of their professional training. To others maths can provide an introduction to systematic and logical thinking .......... Continue the essay and write 230–250 words. Feedback Did you have any problems with this Module? What are you going to do to help yourself? Activity Problem Solution 1 2 3 Just for Fun A boy was teaching a girl arithmetic, he said, it was his mission. He kissed her once; he kissed her twice and said, “Now that’s addition”. In silent satisfaction, she sweetly gave the kisses back and said, “Now that’s subtraction”. Then he kissed her, she kissed him, without an explanation. And both together smiled and said, “That’s multiplication”. Then her Dad appeared upon the scene and made a quick decision. 22

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Mathematics and mathematicians

He kicked that boy three blocks away and said, “That’s long division!” A group of 4 mathematicians and 4 engineers are traveling to a big conference, and to save money they all take the train. Each engineer buys a train ticket, but only one mathematician buys a ticket. The engineers don't understand this. They all board the train, and one mathematician stands by the window. He then shouts, “the ticket-collector is coming!” and all of the mathematicians run and hide in the bathroom stall. The collector comes by, and each engineer shows him his ticket, and then he knocks on the bathroom door and says “ticket please!”. The mathematicians slide their one ticket under the door, and the collector goes away. On the return trip, the engineers think they're smart, and they buy just one ticket. But this time, the mathematicians don't buy any tickets. The engineers don't understand this. Again, they board the train, and one mathematician watches and yells “the ticket-collector is coming!” The engineers hurry up and run to the bathroom stall. Then one of the mathematicians knock on their door and says “ticket please!”, and the engineers slide their ticket under the door. The mathematicians grab it and hurry to hide in the other bathroom stall…

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Module 2 Algebra Learning objectives: In this module you will: • work with basic algebraic concepts • practice questions and answers • express your personal point of view • write a letter • check spelling, punctuation and grammar

UNIT 1 Before you read 1. Discuss these questions with your partner. 1. When did algebra originate? 2. What types of equations do you know? 3. Are groups the only algebraic structures in modern algebra? 4. Does algebra evolve as a unique and integral subject? 5. What does modern algebra deal with? 6. Is modern algebra the most sophisticated subject in maths? Reading 2. Read the text.

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Algebra Algebra is often referred to as a generalization of arithmetic: problems and operations are expressed in terms of variables as well as constants. A  constant is some number that always has the same value, such as 3 or 14.89. A variable is a number that may have different values. In  algebra, letters such as a, b, c, x, y, and z are often used to represent variables. In any given situation, a variable

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Algebra

such as x may stand for one, two, or any number of values. For example, in the expression x + 5 = 7, the only value that x can have is 2. In the expression x2 = 4, however, x can be either +2 or −2. And in the expression x + y = 9, x can have an unlimited number of values, depending on the value of y. Algebra became popular as a way of expressing mathematical ideas in the early ninth century. Arab mathematician Al-Khwarizmi is credited with writing the first algebra book, Al-jabr wa’l Muqabalah, from which the English word algebra is derived. The title of the book translates as “restoring and balancing”, which refers to the way in which equations are handled in algebra. Al-Khwarizmi's book was influential in its day and remained the most important text in algebra for many years. Al-Khwarizmi did not use variables in the same way they are used today. He concentrated instead on developing procedures and rules for solving many types of problems in arithmetic. The use of letters to stand for variables was first suggested in the sixteenth century by French mathematician Françoise Vièta (1540–1603). Vièta appears to have been the first person to recognize that a single letter (such as x) can be used to represent a set of numbers. The rules of elementary algebra deal with the four familiar operations of addition, subtraction, multiplication, and division of real numbers. A real number can be thought of as any number that can be expressed as a point on a line. Constants and variables can be combined in various ways to produce algebraic expressions. Numbers such as 64x2, 7yt, s/2, and 32xyz are examples. Such numbers combined by multiplication and division only are monomials. The combination of two or more monomials is a polynomial. The expression a + 2b − 3c + 4d + 5e − 7x is a polynomial because it consists of six monomials added to and subtracted from each other. A polynomial containing only two parts (two terms) is a binomial, and one containing three parts (three terms) is a trinomial. Examples of a binomial and trinomial, respectively, are 3x2 + 2y2 and 4a + 2b2 + 8c3. One primary objective in algebra is to determine the conditions under which some statement is true. Such statements are usually made in the form of a comparison. One expression can be said to be greater than (>), less than ( 12. In this case, an unlimited possible number of answers exists. That is, x could be 10 (because 10 + 3 > 12), or 11 (because 11 + 3 > 12), or 12 (because 25

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Module 2

12 + 3 > 12), and so on. The answer to this problem is said to be indeterminate because no single value of x will satisfy the conditions of the algebraic statement. In most instances, equations are the tool by which problems can be solved. One begins with some given equality, such as the fact that 2x + 3 = 15, and is then asked to find the value of the variable x. The rule for dealing with equations such as this one is that the same operation must always be performed on both sides of the equation. In this way, the equality between the two sides of the equation remains true. In the above example, one could subtract the number 3 from both sides of the equation to give: 2x + 3 − 3 = 15 − 3, or 2x = 12. The condition given by the equation has not changed since the same operation (subtracting 3) was done to both sides. Next, both sides of the equation can be divided by the same number, 2, to give: 2x/2 = 12/2, or x = 6. Again, equality between the two sides is maintained by performing the same operation on both sides. Algebra has applications at every level of human life, from the simplest day-to-day mathematical situations to the most complicated problems of space science. Suppose that you want to know the original price of a compact disc for which you paid $13.13, including a 5 percent sales tax. To solve this problem, you can let the letter x stand for the original price of the CD. Then you know that the price of the disc plus the 5 percent tax totaled $13.13. That information can be expressed algebraically as x (the price of the CD) + 0.05x (the tax on the CD) = 13.13. In other words: x + 0.05x = 13.13. Next, it is possible to add both of the x terms on the left side of the equation: 1x + 0.05x = 1.05x. Then you can say that 1.05x = 13.13. Finally, to find the value of x, you can divide both sides of the equation by 1.05: 1.05x/1.05 = = 13.13/1.05, or x = 12.50. The original price of the disc was $12.50. (abridged from http://www.scienceclarified.com/A-Al/Algebra.html)

3. Look through the text and find the word that means the same as: 1) taken as a starting-point, source or origin; 2) to take (a number, quantity) away from (another number, etc.); 3) statements of equality between two expressions by the sign =; 4) not fixed; vague or indefinite; 5) to represent; 6) exactly. 4. Which of the following sentences correspond to the text content? 1. The origin of the word “Algebra” is rather exotic. 2. We owe the word “algebra” to the Arab mathematician AlKhwarizmi. 26

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Algebra

3. Constants and variables cannot be combined to produce algebraic expressions. 4. Algebra is not an end in itself; it has to listen to outside demands issued from various parts of maths. 5. Algebra is changing constantly and rapidly. 6. Equations are the tool by which problems can be solved. 7. Algebra may be used at every level of human life. 5. Find these numbers in the text. What do they describe? 1. 14.89. 2. 2. 3. 1540–1603. 4. 13.13. 5. 5. 6. 12.50. 6. These are answers to questions about the text. Write the questions. 1. A constant is some number that always has the same value. 2. In the early ninth century. 3. Such numbers are monomials. 4. This combination is called a polynomial. 5. The objective is to determine the conditions under which some statement is true. 6. They help to solve the problems. 7. The original price of the disc was $ 12.50. 7. Divide the text into logical units. Suggest titles for each unit. Find the sentences expressing the main ideas of each logical unit in the text. Combine the topic sentences using the following connectives. First of all, besides, nevertheless, hence, accordingly, therefore, as well as, consequently, in short, etc. Grammar review 8. Choose the correct words/phrases. 1. Contemporary algebra is concerned as a mixture of much/many that is very old and still important, e.g., counting and newer concepts such as structures. 2. Scientists do not have enough/several evidence to fix the date when the epochal discovery of cardinal number was made. 27

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Module 2

3. You can’t find this book nowhere/anywhere, it is practically unavailable. 4. I know a few/a little problems of algebra. 5. I always have many/plenty of time to do the research. 6. He got toomany/hardly any sleep last night. 7. Did he find something/anything as a result of his operation? 9. Find grammar mistakes in these sentences and correct them. 1. Vector algebra and matrix algebra are two subjects that have made already significant contributions to both maths and science. 2. What does cause the Greeks to give their algebra geometrical formulation? 3. The complex numbers remained on the purely manipulative level until the nineteen century. 4. Hamilton and Grassmann was forced by physical considerations to invent an algebra in which the commutative law of multiplication does not hold. 5. A polynomial equation is solvable if, and only if, it’s group over the coefficient field is solvable. 6. It should said that a proof constitutes the principal part of the math method. 7. Thanks to the fundamental theorem of algebra the solution of polynomial equations require no new kinds of numbers. 10. Make sentences with these words. 1) algebra, and, linear, spaces, linear, transformations, studies, vector, matrices, including. 2) algebra, by, people, using, are, to, able, calculations, with, perform, quantities, unknown. 3) common, the, to, ideas, all, are, structures, algebraic, algebra, in, studied, universal. 4) of, early, was, the, Greeks, algebra, geometric, the. 5) elementary, in, stand, algebra, symbols, for, number. 6) no, is, general, solution, possible, algebraic, the, for, polynomial, of, than, equation, degree, greater, four. Writing 11. Read the text. Punctuate it as you think appropriate. The history of algebra is very complex and went through many centuries of development to the algebra that we know today algebra is still being developed and will never quite being developed and added on to algebra 28

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Algebra

is a relatively new form of math in the European countries and the Americas algebra helped mathematicians and scientists to develop many tools and theorems that people over the world use on a daily basis although algebra and geometry were considered separate subsets of math the two subjects are now unified who knows what inventions and discoveries will be made with algebra in the future if mathematicians continue to study the discipline of mathematics.

UNIT 2 Before you read 1. Match these words with their definitions. 1) erroneous 2) vague 3) contemplate 4) calculus 5) perceive 6) integrity 7) dominion 8) treatise

a) to have in view as a purpose, intention or possibility b) a branch of mathematics divided into two parts, that deals with variable quantities, used to solve many mathematical problems c) incorrect; mistaken d) a state or condition of being complete e) authority to rule; control over f) not clear or distinct g) to become aware of h) book, etc. that deals systematically with one subject

2. Match the words to make phrases 1) dominion of 2) anomalous 3) commutative 4) unlimited 5) multiplication

a) applicability b) result c) of fractions d) algebra e) law

Reading 3. Read the text below and choose the correct word for each space. ALGEBRA is that branch of the mathematical sciences which has for its object the carrying (1) .......... of operations either in an order different from that 29

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Module 2

which exists in arithmetic, or of a nature not contemplated in fixing the boundaries of that science. The circumstance that algebra has its origin in arithmetic, however widely it may in the end differ (2) .......... that science, led Sir Isaac Newton to designate it “Universal Arithmetic”, a  designation which, vague as it is, indicated its character better than any other by which it has been attempted to express its functions-better certainly, to ordinary minds, than the designation which has been applied to it by Sir William Rowan Hamilton, one of the greatest mathematicians the world (3) .......... since the days of Newton – “the Science of Pure Time”; or even than the title by which De Morgan would paraphrase Hamilton's words – “the calculus of Succession”. To express in few words what it is which effects the transition from the science of arithmetic into a new field is not easy. It will serve, probably, to convey some notion of the position of the boundary line, when it is stated that the operations of arithmetic are all capable of direct interpretable only by comparison with the assumptions on which they (4) .......... For example, multiplication of fractions – which the older writers on arithmetic, Lucas de Burgo in Italy, and Robert Recorde in England, clearly perceived to be a new application of the term multiplication, scarcely at first sight reconcilable with its original definition as the exponent of equal additions, – multiplication of fractions becomes interpretable by the introduction of the idea of multiplication into the definition of the fraction itself. On the other hand, the independent use of the sign minus, on which Diophantus, in the 4th century, laid the foundation of the science of algebra in the West, by placing in the forefront of his treatise, as one of his earliest definitions, the rule of the sign minus, “that minus multiplied (5) .......... minus produced plus” – this independent use of the sign has no originating operation of the same character as itself, and might, if assumed in all its generality as existing side by side with the laws of arithmetic, more especially with the commutative law, have led to erroneous conclusions. As it is, the unlimited applicability of this definition, in connection (6) .......... all the laws of arithmetic standing in their integrity, pushes the dominion of algebra into a field on which the oldest of the Greek arithmeticians, Euclid, in his unbending march, could never have advanced a step without doing violence to his convictions. In asserting that the independent existence of the sign minus, side by side with the laws of arithmetic, might have led to anomalous results, had not the 30

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Algebra

operations been subject to some limitation, we are introducing no imaginary hypothesis, but are referring (7) .......... a fact actually existing. The most recent advance beyond the boundaries of algebra, as it existed fifty years ago, is that beautiful extension to which Sir W. R. Hamilton has given the designation of Quaternion, the very foundation of which requires the removal of one of the ancient axioms of arithmetic, “that operations may be performed in any order”. (abridged from http://www.1902encyclopedia.com/A/ALG/algebra-01.html)

1. A on B in C to 2. A to B from C in 3. A has seen B saw C was seeing 4. A based B are based C base 5. A on B by C with 6. A with B on C from 7. A of B to C on 4. Which of the following sentences correspond to the text content? 1. Algebra is the branch of mathematics which studies the structure of things, the relationship between things and quantity. 2. Like all branches of mathematics, algebra has developed because we need it to solve our problems. 3. Algebra has its origin in arithmetic. 4. It is possible to classify algebra by dividing it into four areas. 5. In the 4th century Diophantus proved that minus multiplied by minus produced plus. 6. Algebra uses symbols, usually letters, and the operators. 5. Divide the given words into four categories: 1. Words which make their opposites with un-. Example: uncertain. 2. Words whichmake their opposites with dis-. Example: disproportional. 3. Words which make their opposites with in-. Example: inexpensive. 4. Words which make their opposites with im-. Example: impatient. productive agreeable active kind permanent similar convenient liked advantageous experienced exact popular used satisfied adequate protected organized employed profitable variable successful dependent limited 31

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Module 2

6. Fill in the missing words in the sentences below. Choose from the following: reduces, quaternion, symbols, be applied, analytical, shift An important development in algebra was the introduction of (1) .......... for the unknown in the sixteenth century. As a result of the introduction of symbols, Book III of La géometrie by René Descartes strongly resembles a modern algebra text. Descartes's most significant contribution to algebra was his development of (2) .......... algebra. Analytical algebra (3) .......... the solution of geometric problems to a series of algebraic ones. In 1799, German mathematician Carl Friedrich Gauss was able to prove Descartes's theory that every polynomial equation has at least one root in the complex plane. Following Gauss's discovery, the focus of algebra began to (4) .......... from polynomial equations to studying the structure of abstract mathematical systems. The study of the (5) .......... became extensive during this period. The study of algebra went on to become more interdisciplinary as people realized that the fundamental principles could (6) .......... to many different disciplines. Today, algebra continues to be a branch of mathematics that people apply to a wide range of topics. Grammar review 7. Fill in the gaps with the correct form of past tense. 1. Early in the 19th century a new view of maths .......... (begin) to emerge. 2. Abel .......... (die) when he .......... (be) only 27 leaving behind a wealth of highly original work which .......... (stimulate) research for many years after. 3. I .......... (prove) the fundamental theorem of algebra for an hour when he ………(come). 4. He .......... (obtain) the difference after he .......... (subtract) the numeral. 5. In 1846 Cayley ………(begin) the work on the theory of algebraic invariants, which .......... (be) in the air for some time and which .......... (receive) some of its motivation from determinants. 6. I .......... (wait) for 20 minutes when I .......... (realize) that I .......... (come) to the wrong place. 7. He .......... (drive) never before. 8. Find the word which shouldn’t be in the sentence. 1. How did unsolved problems has influence the development of maths? 32

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Algebra

a) did b) has c) the 2. Vector algebra and matrix algebra are two subjects that have already made significant contributions to both on maths and science. a) on b) have c) to 3. The discriminant is a combination of constants which vanishes if at least two factors of a function are do the same. a) are b) do c) which 4. Linear combinations of matrices with scalar coefficients obey to the rules of ordinary algebra. a) with b) to c) the 5. The fact that the foundations of some math concept or theory are not be secure do not prevent mathematicians from using them. a) are b) from c) be 6. Despite the crisis in logic and theory of sets, there was been a feeling of confidence that the new foundations for analysis, geometry and abstract algebra could be safely used to build new theories. a) was b) been c) be 7. What mathematicians are contributed to the creation of secure and rigorous foundations of maths? a) to b) the c) are Speaking 9. Discussion 1. “The four rules of arithmetic may be regarded as the complete equipment of the mathematician” (J. C. Maxwell). Agree or disagree. 2. Maths is forever and again a study of different functions. Prove it. 3. The complexity of a civilization is mirrored in the complexity of its numbers. Illustrate the statement. 4. The irrational numbers have a long history. The world of maths is far richer in irrational numbers than it is in rational ones. Prove it. 10. Answer the questions. 1. In how many ways can 5 students be seatedin a row of 5 seats? 2. How many different numbers of 3 different digits each can be made from the digits 1, 3, 5, 7, 9? 3. How many different symbols each consisting of 4 letters in succession can be formed from letters a, b, c, d, e (repetitions are permitted)? 4. In how many ways can we select: • A committee of 3 from a group of 10 people? 33

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Module 2

• A set of 3 books from a set of 7 different books? • A set of 3 maths books and 5 physics books, all are different? 11. Do the test Three Minute Timed Test 1. Read everything before doing anything. 2. Put your name in the upper left-hand corner of this paper. 3. Circle the word “Name” in sentence two. 4. Draw five small squares in the upper left-hand corner of this paper. 5. Put an X in each square. 6. Sign your name under the title of this paper. 7. After the title, write “Yes, yes, yes”. 8. Put a circle around sentence seven. 9. Put an X in the lower left-hand corner of this paper. 10. Draw a triangle around the X you just put down. 11. On the back of this paper, multiply 703 by 66. 12. Draw a rectangle around the word “paper” in sentence four. 13. Call out your first name when you get to this point in the test. 14. If you think you have followed directions carefully to this point, call out “I have”. 15. On the reverse side of this paper, add 8950 and 9850. 16. Put a circle around your answer and put a square around the circle. 17. Count out in your normal speaking voice, from ten to one backward. 18. Punch three small holes in the top of this paper with your pencil. 19. If you are the first person to get this far, call out loudly, “I am the first person to this point, and I am the leader in following directions”. 20. Underline all even numbers on the side of this page. 21. Put a square around every number written out on this test. 22. Say out loud, “I have nearly finished, I have followed directions”. 23. Now that you have finished reading carefully, do only number 2.

UNIT 3 Reading 1. Read the text. Entitle it and the parts of the text. 1. The general theory of algebraic equations, the elementary parts of which are studied in high school, has a long and distinguished history in mathematics. The proof by Niels Henrik Abel in 1824 that solutions of an 34

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Algebra

algebraic equation of degree five or greater, where the degree is the highest exponent of any term in the equation, cannot be expressed in terms of radicals (that is, expressions definable in terms of square roots) was one of the most important mathematical results of the first half of the nineteenth century. Another result of basic importance is the fundamental theorem of algebra, which was first proved in the eighteenth century but which was proved rigorously only in the last half of the nineteenth century. This theorem asserts that every algebraic equation always has at least one root that is a real or a complex number. Also of great significance were the proofs that not all numbers are roots of algebraic equations; numbers that are not such roots are called transcendental numbers. The most famous proofs of this sort are Charles Hermite’s (in 1873) that e is transcendental and F. Lindemann’s (in 1882) that π is transcendental. 2. Much of the work in algebra during the present century has been devoted to generalized mathematical systems that are characterized not in terms of the four fundamental arithmetical operations but in terms of generalizations of these operations and of the familiar ordering relations of “less than” and “greater than”. In a number of the social sciences the theory of binary relations has received extensive application. From an algebraic standpoint a binary relation structure may be characterized as consisting of a set A and a set R of ordered pairs (x, y), where x and y are both elements of A. Such an R is called a binary relation on A. A relation R is said to be a partial ordering of A when it is reflexive, antisymmetric, and transitive – that is, when it satisfies the following three properties: reflexive; for every x in A, xRx; antisymmetric: for every x and y in A, if xRy and yRx, then x – y; transitive: for every x, y, and z in A, if xRy and yRz, then xRz. If R is also connected in A (that is, if for any two elements x and y in A with x ≠ y, either xRy or yRx) then R is said to be a complete or simple ordering or, sometimes, a linear ordering of A. The concept of a complete ordering is a direct abstraction of the order properties of “≤” with respect to the real numbers. A familiar use of the concept of an ordering relation is in utility theory, particularly in the classical theory of demand in economics, in which it is assumed that each individual has an ordering relation over the set of commodity bundles or, more generally, over the set of alternatives with which he is presented. The general concept of ordering relations also has far35

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Module 2

ranging applications in the theory of measurement within psychology and sociology, and more general binary relations have been extensively applied in anthropology in the study of kinship systems. Partial orderings can be extended in another direction by imposing additional conditions to obtain lattices, which have also been used in the social sciences. In a different direction, but still within the framework of binary relations, is the theory of graphs, in which no restrictions are placed on the binary relation, R. Applications of graph theory have been made to socialpsychological and sociological problems, especially to provide a mathematical method for representing various kinds of relationships between persons. 3. Another direction of generalization of classical algebra has been to what are called groups, rings, and fields. A group is a set A together with a  binary operation, o, satisfying the following axioms. First, the operation o is associative, that is, for x, y, and z in A, x o (y o z) = (x o y) o z. Second, there is an element e, called the identity, of the set A such that for every x in A, x o e = e o x = x. And, finally, for each element x of A there is an inverse element x – 1 such that x o x – 1 = e. It is obvious that if A is taken as the set of integers, o as the operation of addition, e as the number 0, and the inverse of x as the negative of x, then the set of integers is a group under the binary operation of addition. The theory of groups has had profound ramifications in other parts of mathematics and in the sciences, ranging from the theory of algebraic equations to geometry and physics. The reason for the fundamental importance of group theory is perhaps best summarized by stating that a group is the appropriate way to formulate the very important concept of symmetry. In the range of applications of group theory just mentioned, the underlying thread is the concept of symmetry, whether it is in the symmetry of the roots of an equation or the symmetry properties of the fundamental particles of physics. As a simple example, consider the finite group of rotations 90°, 180°, 270°, and 360°. A square does not change its apparent orientation under such a rotation about its center, but an equilateral triangle does. This group of rotations is the symmetry group of rotations for a square but not, of course, for an equilateral triangle. Although the methods and results of group theory have not yet had special applications of depth in the social sciences, they are important to many of the general mathematical results that have been applied. 4. Algebraic aspects of the theory of sets have been studied under the heading of Boolean algebras. The concept of an algebra of sets, that is, a collection of sets closed under union and complementation, is fundamental in the modern theory of probability, where events are interpreted as sets of possible outcomes and numerical probabilities are assigned to events. 36

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Algebra

5. Linear algebra is one of the most important generalizations of classical elementary algebra. The objects to which the operations of addition and multiplication are applied are now matrices, vectors of an n-dimensional space, and linear transformations (an n × n matrix is a particular representation of a linear transformation in n-dimensional space). More particularly, linear algebra arises as a generalization of the linear equations so familiar in elementary algebra, and historically one of the most important tasks of linear algebra has been to find solutions of systems of linear equations. As much ofresearch can be an extremely laborious and difficult affair when the number of equations is large. The set of coefficients of a system of linear equations gives rise to the concept of a rectangular array of numbers, which is precisely what a matrix is. An algebra of matrices in terms of addition and multiplication may be constructed; the distinguishing feature of this algebra, as compared with the algebra of the real numbers, is that multiplication is not commutative – that is, AB is not usually equal to BA, and the product of two nonzero matrices can be zero. (abridged from http://www.encyclopedia.com/topic/mathematics.aspx)

2. Look through the text and find the word that means the same as: 1) strictly, severely; 2) quality of being useful; 3) effects or results of events, or of circumstances; 4) having all sides equal; 5) put forward as a time, place, reason, etc.; 6) clearly seen or understood. 3. Match the words to make phrases. 1) apparent 2) numerical 3) profound 4) square 5) partial

a) root b) ordering c) ramification d) orientation e) probability

4. Decide if the following statements are true or false, or there is no information in the text. a) true b) false c) no information 1. The general theory of algebraic equations is not studied in high school. 2. In 1824 Abel proved solutions of an algebraic equation of degree five. 3. A polynomial equation is solvable if, and only if, its group over the coefficient field is solvable. 37

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Module 2

4. Much of the work in algebra during the 19th century had been devoted to generalized mathematical systems. 5. The general concept of ordering relations has far-ranging applications in the theory of measurement within psychology and sociology. 6. Boolean algebra studies algebraic aspects of the theory of sets. 7. Algebra is changing constantly and rapidly. 8. Matrix algebra is not commutative but associative. 5. These are answers to questions about the text. Write the questions. 1. The general theory of algebraic equations. 2. It was Abel. 3. In a number of the social sciences. 4. Groups, rings and fields. 5. Boolean algebra. 6. Linear algebra. Speaking 6. Agree with the statements below and try to give some proof or justification. Use the introductory phrases. There is no point in denying that… I see no point at all to disagree that… I hold a similar view… This is the case. 1. It took centuries of development and sophistication in maths to state explicitly the existence of algebraic structures. 2. The term “group” stands for a special kind of math system and it has nothing to do with the colloquial meaning ordinary attached to the word “group”. 3. There is another law of common algebra, besides the commutative law of multiplication that is broken in matrix algebra and this is the cancellation law of multiplication. 4. Modern algebra is being made to apply to situations, which at first sight in no way related to algebra. 5. Algebra is not only a part of maths; it also plays within maths the role which maths itself had been playing for a long time with respect to physics. 6. Maths is changing constantly, and algebra must reflect these changes if it wants to stay alive. 7. Try to solve the problem There was a certain number of crows sitting on trees in the garden. If there had been just one crow sitting on each tree, then one crow would not 38

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Algebra

have had a tree to sit on. However, if two crows had been sitting on each tree, then there would not be any crows on one tree. How many trees were there in the garden? (3, 4, 5, 6) Grammar review 8. Restate the following sentences using Participle instead of the subordinate clause. 1. When these data were collected they were of great use to us. 2. As the data were of high accuracy, the results could be calculated more precisely. 3. When the experimental data were corrected in this way they were plotted on the diagram. 4. We are familiar with the method of calculating which is being described here. 5. After he had attended a course of lectures on the subject he got a better understanding of it. 6. After they had considered that hypothetical result they could tell the difference between these two models. 7. Several factors which are understood poorly are involved in that system. 9. Make sentences inserting the participial phrase of the verb given in brackets. 1. (to make) his report, he was thinking hard. 2. (to buy) magazines, he looked back from time to time, hoping to see his colleagues. 3. (to read) the report she closed the file and switched off the computer. 4. (to call) his students, he went to the University. 5. (to look) through some magazines, I came across an article about unsolved problems of algebra. Writing 10. Choose one of the words given below and write a paragraph, illustrating the concept. Model. Equations Equations are expressions of equality between two quantities connected by the sign =. To be more exact, an equality which is not true for 39

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Module 2

all the values of the letters in it is referred to as an equation, e.g., x – 7 = 0 is true only if x = 7. Much of maths is concerned with equations. To solve an equation means to find the values of the unknowns that satisfy the equation, i.e., to reduce it to an identity. Groups fields, rings, functions. 11. Write a letter to your friend telling him or her all about your university course in algebra. Use these notes, your own ideas and information from texts and exercises to help you. Hi (friend’s name) PARAGRAPH 1 Opening remarks: Ask how your friend is. Apologize for taking so long to write back. Say how you are and what you have been doing. PARAGRAPH 2 AND 3 Main body: Tell your friend about your university and your studies. Explain what algebra is. PARAGRAPH 4 Closing remarks: Invite your friend to visit you for a few days. Say you are looking forward to seeing him/her. Sign off (Bye for now, All the best, Love, etc.) (your first name) Write 100–150 words Feedback Have you used vocabulary connected with algebra? checked your spelling, punctuation and grammar? reviewed sentence structure? written 100–150 words? Just for Fun A math student is pestered by a classmate who wants to copy his homework assignment. The student hesitates, not only because he thinks it's wrong, but also because he doesn't want to be sanctioned for aiding and abetting. His classmate calms him down: “Nobody will be able to trace 40

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Algebra

my homework to you: I'll be changing the names of all the constants and variables: a to b, x to y, and so on”. Not quite convinced, but eager to be left alone, the student hands his completed assignment to the classmate for copying. After the deadline, the student asks: “Did you really change the names of all the variables?” “Sure!” the classmate replies. “When you called a function f, I called it g; when you called a variable x, I renamed it to y; and when you were writing about the log of x + 1, I called it the timber of x + 1”... New York (CNN). At John F. Kennedy International Airport today, a Caucasian male (later discovered to be a high school mathematics teacher) was arrested trying to board a flight while in possession of a compass, a protractor and a graphical calculator. According to law enforcement officials, he is believed to have ties to the Algebra network. He will be charged with carrying weapons of math instruction.

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Module 3

Geometry Learning objectives: In this module you will: • understand basic geometric concepts, and acquire vocabulary connected with it • confirm or deny the statements • give definitions • review grammar • write a Cinquain Poetry

UNIT 1 Before you read 1. Match these words with their definitions. 1) quadrilateral 2) compass 3) circumference 4) cone 5) available 6) sphere 7) permit 8) proposition

a) a solid figure that is entirely round b) a solid body which narrows to a point from a round, flat base c) a problem or a question (with or without the answer or solution) d) V-shaped instrument with two arms joined by a hinge, used for drawing circles, measuring distances on a map or chart, etc. e) to allow f) four-sided (plane figure) g) able to be used h) a line that marks out a circle or other curved figure

Reading 2. Read the text and entitle it. The six sentences below have been removed. Decide where they should go. a) A fundamental part of geometric proofs involves constructions. 42

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Geometry

b) A plane extends forever in both directions. c) In general, lines can have one of three relationships to each other. d) The figure produced is a cone. e) An axiom is a statement that mathematicians accept as being true without demanding proof. f) In its most basic sense, then, geometry was a branch of mathematics originally developed and used to measure common features of Earth. The term geometry is derived from the Greek word geometria, meaning “to measure the Earth”. 1) .......... In its most basic sense, then, geometry was a branch of mathematics originally developed and used to measure common features of Earth. Most people today know what those features are: lines, circles, angles, triangles, squares, trapezoids, spheres, cones, cylinders, and the like. Humans have probably used concepts from geometry as long as civilization has existed. But the subject did not become a real science until about the sixth century B.C. At that point, Greek philosophers began to express the principles of geometry in formal terms. The one person whose name is most closely associated with the development of geometry is Euclid (c. 325–270 B.C.), who wrote a book called Elements. This work was the standard textbook in the field for more than 2,000 years, and the basic ideas of geometry are still referred to as Euclidean geometry. Elements of geometry Statements. Statements in geometry take one of two forms: axioms and propositions. 2) .......... An axiom is also called a postulate. Actually, mathematicians prefer not to accept any statement without proof. But one has to start somewhere, and Euclid began by listing certain statements as axioms because they seemed so obvious to him that he couldn't see how anyone would disagree. One axiom is that a single straight line, and only one, can be drawn through two points. Another axiom is that two parallel lines (lines running next to each other like train tracks) will never meet, no matter how far they are extended into space. Indeed, mathematicians accepted these statements as 43

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Module 3

true without trying to prove them for 2,000 years. Statements such as these form the basis of Euclidean geometry. However, the vast majority of statements in geometry are not axioms but propositions. A proposition is a statement that can be proved or disproved. In fact, it is not too much of a stretch to say that geometry is a branch of mathematics committed to proving propositions. Proofs. A proof in geometry requires a series of steps. That series may consist of only one step, or it may contain hundreds or thousands of steps. In every case, the proof begins with an axiom or with some proposition that has already been proved. The mathematician then proceeds from the known fact by a series of logical steps to show that the given proposition is true (or not true). Constructions. 3) .......... A construction in geometry is a drawing that can be made with the simplest of tools. Euclid permitted the use of a straight edge and a compass only. An example of a straight edge would be a meter stick that contained no markings on it. A compass is permitted in order to determine the size of angles used in a construction. Many propositions in geometry can be proved by making certain kinds of constructions. For example, Euclid's first proposition was to show that, given a line segment AB, one can construct an equilateral triangle ABC. (An equilateral triangle is one with three equal angles.) A plane. A plane is a geometric figure with only two dimensions: width and length. It has no thickness. The flatness of a plane can be expressed mathematically by thinking about a straight line drawn on the plane's surface. Such a line will lie entirely within the plane with none of its points outside of the plane. 4) .......... Planes encountered in everyday life (such as a flat piece of paper with certain definite dimensions) and in mathematics often have a specific size. But such planes are only certain segments of the infinite plane itself. Plane and solid geometry. Euclidean geometry dealt originally with two general kinds of figures: those that can be represented in two dimensions (plane geometry) and those that can be represented in three dimensions (solid geometry). The simplest geometric figure of all is the point. A point is a figure with no dimensions at all. The points we draw on a piece of paper while studying geometry do have a dimension, of course, but that condition is due to the fact that the point must be made with a pencil, whose tip has real dimensions. From a mathematical standpoint, however, the point has no measurable size. Perhaps the next simplest geometric figure is a line. A line is a series of points. It has dimensions in one direction (length) but in no other. A line can also be defined as the shortest distance between two points. Lines are used to construct all other figures in plane geometry, including angles, triangles, 44

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Geometry

squares, trapezoids, circles, and so on. Since a line has no beginning or end, most of the “lines” one deals with in geometry are actually line segments – portions of a line that do have a limited length. 5) .......... They can be parallel, perpendicular, or at an angle to each other. According to Euclidean geometry, two lines are parallel to each other if they never meet, no matter how far they are extended. Perpendicular lines are lines that form an angle of 90 degrees (a right angle) to each other. And two lines that cross each other at any angle other than 90 degrees are simply said to form an angle with each other. Closed figures. Lines also form closed figures, such as circles, triangles, and quadrilaterals. A circle is a closed figure in which every part of the figure is equidistant (at an equal distance) from some given point called the center of the circle. A triangle is a closed figure consisting of three lines. Triangles are classified according to the sizes of the angles formed by the three lines. A quadrilateral is a figure with four sides. Some common quadrilaterals are the square (in which all four sides are equal), the trapezoid (which has two parallel sides), the parallelogram (which has two pairs of parallel sides), the rhombus (a parallelogram with four equal sides), and the rectangle (a parallelogram with four right- or 90-degree angles). Solid figures. The basic figures in solid geometry can be visualized as plane figures being rotated through space. Imagine that a circle is caused to rotate around its center. The figure produced is a sphere. Or imagine that a right triangle is rotated around its right angle. 6) .......... Area and volume. The fundamental principles of geometry involve statements about the properties of points, lines, and other figures. But one can go beyond those fundamental principles to express certain measurements about such figures. The most common measurements are the length of a line, the area of a plane figure, or the volume of a solid figure. In the real world, length can be determined using a meter stick or yard stick. However, the field of analytic geometry provides a way to determine the length of a line by using principles adapted from geometry. Mathematical formulas are available for determining the area of any figures in geometry, such as rectangles, squares, various kinds of triangles, and circles. For example, the area of a rectangle is given by the formula A = l · h, where l is the length of the rectangle and h is its height. One can find the areas of portions of solid figures as well. For example, the base of a cone is a circle. The area of the base, then, is A = π · r2, where π is a constant whose value is approximately 3.1416 and r is the radius of the base. (Pi [π] is the ratio of the circumference of a circle to its diameter, and it is always the 45

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Module 3

same, no matter the size of the circle. The circumference of a circle is its total length around; its diameter is the length of a line segment that passes through the center of the circle from one side to the other. A radius is a line from the center to any point on the circle.) (adapted from http://www.scienceclarified.com/Ga-He/Geometry.html)

3. Which of the following sentences correspond to the text content? 1. The necessity for geometric drawings and models is as old as geometry itself. 2. Visual method is especially important in geometry. 3. Figures of unequal size but of the same shape have many geometric properties in common. 4. Plane geometry requires drawings, but solid geometry – models. 5. Many propositions in geometry can be proved by making certain kinds of constructions. 6. From a mathematical standpoint, however, the point has no measurable size. 7. The field of analytic geometry provides a way to determine the length of a line by using principles adapted from geometry. 4. These are answers to questions about the text. Write the questions. 1. It is derived from the Greek word geometria. 2. It was Euclid. 3. Axioms and propositions. 4. It is a point. 5. They can be parallel, perpendicular, or at an angle to each other. 6. A triangle is a closed figure consisting of three lines. 7. Mathematical formulas are available for determining the area of any figures in geometry. 5. Make the right choice and complete the sentences: 1. The set of all points in geometry is a … (volume, plane, line, space, model, surface). 2. Sets of points which all lie in one plane are … (circles, rays, angles, solids, ellipses, plane figures, squares). 3. The regular polyhedra are a part of geometric study in antiquity. How many different types are there? (rhombus, trapezoid, square, cube, tetrahedron octahedron, dodecahedron, icosahedron). 4. A solid with opposite faces equal and parallel is a … (cube, cylinder, prism, pyramid, sphere). 46

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Geometry

5. The set of 3 points not all on one line and all the points between them on the segments is a … (parallelepiped, triangle, rhombus, rectangle, cone). 6. When we cut a cone at different angles we obtain a set of curves such as … (circle, cycloid, catenary, ellipse, parabola, hyperbola, conchoid, quadratrix, spiral, circumference). 7. It is convenient to have labels for angles and we usually classify them according to … (the measures of their angles, the measures of their sides). 8. The angle of 90° … (straight, right, acute, obtuse, adjacent, complementary). 9. A triangle with its sides equal is … (right, acute, isosceles, equilateral). 10. The distance around the circle is a (an) … (perimeter, parabola, hyperbola, circumference, ellipse). 11. The area of a rectangle is the product of two dimensions: … (the side and the base, the side and the altitude, the base and the height). 6. Determine whether the following statements are true or false. Draw figures to help with your decision. 1. Every square is a rhombus. 2. Every trapezoid is a parallelogram. 3. The “opposite sides” of a parallelogram are congruent to each other. 4. A rectangle that is inscribed in a circle is a square. 5. No parallelogram is a trapezoid. 6. Every rhombus with one right angle is a square. 7. No trapezoid has two right angles. 8. If a rectangle has a pair of congruent sides, then it is square. 9. If a trapezoid has one right angle, then it has two right angles. 10. If two diameters of a circle are perpendicular to each other, then their end points determine the vertices of a square. 11. No trapezoid has a pair of congruent sides.

UNIT 2 1. Try to define the given geometric objects, using the list of verbs and paying attention to the predicate of a definition. A definition is a phrase that signifies a thing's essence. A scientific definition is both a description of a scientific concept such as “force”, “distance”, “velocity”, etc., and the way to measure it. 47

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Module 3

Model: A drawing is (stands for) a visual picture of a geometric object. An angle a ray a square а circle a cone a triangle a quadrilateral a prism a polygon Predicates of definition: defines, signifies, means, implies, symbolizes, assigns, marks, represents, illustrates, classifies, denotes, describes, formulates, gives a name of, refers to, equals, suggests. 2. Read the text Triangle Inequality Theorem Let's explore the question: Can a triangle be formed by given any three segment lengths? Your first response is probably “yes”. If you have a piece of uncooked spaghetti at your house, go to get two pieces. Break the first piece into three relatively the same length pieces and see if you can form a triangle. Remember to join the pieces end-to-end. It should have worked. Now take the second piece of spaghetti and break two very small pieces off one end of it. (About 1 inch each.) The long piece should still be 4–5 inches long. Now try making a triangle as you just did. Any problems this time? Did your pieces of spaghetti look something like this? Tape the two sets of spaghetti pieces on a sheet of paper to be turned into your teacher. Now, you are going to read about the Triangle Inequality Theorem. Triangle Inequality Theorem: The sum of the measures of any two sides of any triangle is greater than the measure of the third side. In other words, you can pick any two sides measure and when they are added together the sum will be greater than the measure of the third side. Example: Can the following lengths form a triangle? 1. 8, 6, 2. 8 + 6 = 14 > 2 ok. 8 + 2 = 10 > 6 ok. 6 + 2 = 8 not > 8 fails. So, no, can’t form a triangle. 2. 3, 4, 5. 3 + 4 = 7 > 5 ok. 3 + 5 = 8 > 5 ok. 4 + 5 = 9 > 3 ok. So, yes, can form a triangle. 48

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Geometry

3. Now practice the Triangle Inequality Theorem. Can a triangle be formed with the following lengths? Explain each answer. 1. 2", 3", 5". 2. 4 cm, 1 cm, 2 cm. 3. 6', 8', 1'. 4. 5 m, 5 m, 5 m. 4. The blanks in the story below can be filled with the words below except for number words such as two or three. All words are not used. Acute ....... Area ....... Base ....... Circle ....... Cone Congruent ....... Cube ....... Cylinder ....... Edge ....... Face Hexagon ....... Intersect ....... Line of symmetry ....... Obtuse Octagon ...... Parallel ...... Pentagon ...... Perimeter ...... Perpendicular Quadrilateral ....... Ray ....... Rectangle ....... Rectangular prism Reflection ....... Right angle ....... Rotation ....... Segment ....... Sphere Square ....... Translation ....... Triangle ....... Vertex ....... Volume Once upon a time, Joyce decided to build a home with an ________ – sided front. She called it an octagon house. A big party was planned to celebrate. Joyce went to the ________ store with a five sided entry to buy party hats with one vertex and a circle base cut out. They looked like colorful ________, and they were all the same shape and size. In other words, they were ________. She also bought ________ – sided hexagon ice cubes for the punch. Once she sent out the invitations it was time to decorate the game room. She wanted to hang streamers along the border of the room. Thus, she need to measure to find out the ________ of the room. The thought to paint the room would require her to know the ________ of the walls and more time, so Joyce dismissed the idea. However, while shopping for the perfect streamers, Joyce stopped by a furniture store and found a couple of perfect pieces for the guests to sit on. The first was the shape of a rectangular prism. It was covered in soft dark green fabric with ________ edges, ________ vertices, and ________ faces. The other piece had two faces with a curved surface covered with an animal print nicknamed the safari ________ seat. It was possible to lift the top base in order to store things inside. Joyce wondered how much it could store or the ________ of such seat storage. Two days before the party, guests began to call Joyce because directions to her home were not on the invitations. “OMG!” she exclaimed. “Let me just tell you over the phone as if you were coming from the library. The street in front of the library, Brooks, will dead end into Jacks Road. In 49

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Module 3

other words, Brook Street is ________ to Jacks Rd. Take a right. Jacks Rd runs ________ to Perry Road and will never meet. So in order to get to Perry Rd, Jacks Road will cross or ________ at mills road. Make a left. Continue on Mills Rd until you pass a statue shaped like a basketball or a ________ Then, slow down, and the next stop sign is at Perry Road. Make a right and my house is the third house on the left. She emailed everyone else the directions. The party was awesome. Jamie was telling jokes. Joyce laughed so hard she was leaning so far back in her chair that her body formed an ________  angle. However, Brittany leaned forward laughing and holding her stomach. She obviously formed an ________ angle. Since Louis came to the party straight from work, he was napping on the sofa. The loud laughter startled Louis so much that he sat straight up forming a ________ angle. Once Louis was fully awake, he moved to the far left of the sofa to join the fun. Louis’ move on the couch would be called a ________ in the Math world. Jamie stayed in his seat and made a ________ or turned to the right to acknowledge Louis’ presence, and the jokes continued. In no time, the punch was gone, the food was gone; thus, the party was over. (from http://www.bellaonline.com/articles/art61807.asp)

Grammar review 5. Choose the correct form of the verb, singular or plural. 1. The regular polyhedra is/are a part of geometric study chiefly in antiquity. 2. The abscissa denote/denotes the distance of the point from a fixed vertical reference line, called the Y-axis. 3. For the general cubic equations such formulas was/were obtained in the 16th century. 4. The following loci leads/lead to particular type of second degree equations, in two variables. 5. The hyperbola is/are the locus of a point which moves so that the difference of its distances from two fixed points is a constant 2a. 6. What other radii is/are shown? 6. Make the sentences more emphatic. Turn active into passive, if necessary. Model 1. The only numbers accepted by the Greeks were the natural numbers. It was the natural numbers that were the only numbers accepted by the Greeks. 50

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Geometry

1. The Greeks first appreciated the power of math reasoning. 2. Math theory emerged and evolved first in the maths of the early Greeks. 3. Ancient Greece created the intellectual miracle of a logical system. 4. In constructing methods of proof, mathematicians employ a high order of intuition, imagination and ingenuity. 5. Euclidean geometry was the only geometry for more than two thousand years. 6. Euclid's Elements were the first successful attempt to build all geometry based on postulation thinking. Model 2. Modern maths ignores the distinction between postulates and axioms. Modern maths does ignore this distinction. 1. Mathematicians tried to construct a geometry in which the negation (the converse) of the parallel postulate holds. 2. N. Lobachevsky published the first account dealing with nonEuclidean geometry and created the subject concerned. 3. The discoverers of non-Euclidean geometry wanted to show that the 5th postulate is, in fact, deducible. 4. Priority arguments are very important in science and that is why we honour N. Lobachevsky as the creator of non-Euclidean geometry. 5. The futile and fruitless efforts to produce a proof of the parallel postulate led to the idea of a geometry with more than one parallel. 6. Many mathematicians after Euclid attempted to prove the parallel postulate by an indirect method (i.e., reductio ad absurdum). 7. Find grammar mistakes in these sentences and correct them. 1. A circle can easily drawn with the help of a compass. 2. Two lines originating from the same point forms an angle. 3. If you hold the sharp end of a compass fixed on a sheet of paper and then turn the compass completely around you will drew a curved line enclosing part of a plane. 4. We should like to show that the Pythagorean Property is true for all right triangles, there being several proofs of these property. 5. Such a mistake is unlikely have remained unnoticed. 6. We have drawn a triangle, the measure of it altitude being three times the measure of it base. 7. If you wished to abbreviate the expression “geometric progression”, you might wrote G.P.

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Module 3

8. Ask questions as in the model using the question words suggested in brackets. Model. The word “geometry” was derived from the Greek words for “earth measure”. (Where … from?) Where was the word “geometry” derived from? 1. The ancients believed that the Earth was flat. (What?) 2. The early geometers dealt with measurements of line segments, angles, and other figures in a plane. (What …with?) 3. Greek mathematicians considered geometry as a logical system. (Who?) 4. They assumed certain properties and try to deduce other properties from these assumptions. (How?) 5. During the last century geometry was still further extended to include the study of abstract spaces. (Why?) 6. Any modern geometric discourse starts with a list of undefined terms and relations. (What … with?) 9. Fill in the gaps with the derivative. It is interesting to note that the (1) … of the special quadrilaterals is based upon the so-called parallel postulate of Euclidean geometry. This postulate is now usually stated as follows: Through a point not on line L, there is no more than one line parallel to L. Without (2) … that there exists at least one parallel to a given line through a point not on the given line, we could not state the definition of the special quadrilaterals which have given pairs of parallel sides. Without the assumption that there exists no more than one parallel to a given line through a point not on the given line, we could not deduce the conclusion we (3) … for the special quadrilaterals. An important aspect of geometry (or any other area of mathematics) as a deductive system is that the (4) … which may be drawn are consequences of the assumptions which have been made. The assumptions made for the geometry we have been considering so far are essentially (5) … made by Euclid in Elements. In the nineteenth century, the famous mathematicians Lobachevsky, Bolyai and Riemann (6) … non-Euclidean geometries. EXIST _ ________________ 1 ASSUME _______________ 2 STATE _________________ 3 CONCLUDE ____________ 4 THAT __________________ 5 DEVELOP_ _____________ 6 52

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Speaking 10. Agree or disagree with the following statements. Use the introductory phrases and develop the idea further. Agreement Disagreement I quite agree to it. This is not the case. Exactly. Quite so. Far from that. I doubt it. It's hardly likely that... It’s too much to say that … Not quite so. All geometric models are inaccurate and misleading. The points of geometry have no size and no dimensions. Models of geometry are idealizations abstracted from physical objects. The figure is to the geometer what the numerical example is to the algebraist. Geometry as a science never exhausts itself. Geometry is the only field of maths in which two-thousand-year-old traditions and theories are still valid and there is always a flood of fresh ideas. The necessity for geometric drawings and models is as old as geometry itself. Writing Cinquain is a type of poetry. In many ways it is similar to a Japanese Haiku. It was invented by a US poet born in 1878. She enjoyed the Haiku style and adapted it to her own techniques. She named her new construction cinquain, based on the French word for five. A cinquain is a five-line poem that describes a person, place, or thing. dessert a one-word title, a noun cold, creamy two adjectives eating, giggling, licking three -ing participles cone with three scoops a phrase ice cream a synonym for your title, another noun 11. Use this organizer to write your own cinquain about any geometric concept. 1 __________________________________________________ a one word title, a noun that tells what your poem is about

2 __________________________________________________ two adjectives that describe what you’re writing about

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3 __________________________________________________ three -ing participles that describe what your poem about

4 __________________________________________________ a phrase that tells more about what you’re writing about

5 __________________________________________________ a synonym for your title, another noun that tells what your poem is about

UNIT 3 Reading 1. Read the titles of the text and say: 1) what information you can get from it; 2) if the subject of the text is familiar to you. Non-Euclidean Geometry Euclid and his legacy Working circa 300 BC, the Greek mathematician, Euclid, made geometry into an exact discipline by defining five axioms and then building the edifice of geometry on top of these. This structure endured undisturbed for some two thousand years. His geometry was deeply entrenched in the mind-set of all subsequent mathematicians. What we now call “Euclidean geometry” was simply called “geometry”. It was regarded as being not just the only geometry but also as being synonymous with truth. Yet there was also a counter-current. Certain mathematicians had their suspicions about the 5th Euclidean postulate, the one about parallel lines. Being more complex and less obvious than the others, it seemed more like a theorem than an axiom. So people set about trying to prove that it followed from the first four axioms. Others tried to prove the postulate by assuming its negation in the expectation that they would produce a logical contradiction. As it turned out in this case, failure can be as productive as success. In modern terms, Euclid's 5th postulate is normally stated as: Given a line L and a point P not on the line, there is precisely one line through P in the plane determined by L and P that does not intersect L. The unsuccessful efforts to prove the 5th postulate of Euclid showed that many results are equivalent to it. Three such results are: The sum of the angles in a triangle is 180 degrees. 54

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Geometry

The ratio of the circumference to the diameter of a circle is the same for all circles, regardless of size. The Pythagorean Theorem. Non-Euclidean geometry By 1830 it was realized that the 5th postulate was a strange beast, neither fish nor foal. For it is neither true nor false. Asked whether the 5th postulate is true or not, a modern mathematician will answer, “It depends...” It depends on the surface one is talking about. If one is doing geometry on a plane or cylindrical surface then the postulate is true. If one is performing geometrical activities on a sphere or a saddle shape then the axiom is false. Riemann took the bold step of treating the 5th postulate not as a theorem, or axiom, nor as a true or false statement, but as a definition. This led to the birth of non-Euclidean geometry. Geometry now split into three broad classes: the geometry of positive, zero and negative curvature. The term curvature is neither abstruse nor vague. It refers to the sum of the angles of a triangle on the surface in question. If that sum is exactly 180 degrees then one is dealing with a “flat” surface, and this is the domain of application of Euclidean geometry. If the angle sum is greater than 180 degrees then one is dealing in spherical geometry. If the sum is less than 180 degrees then this is hyperbolic geometry. Equivalently, this relates to parallels. In hyperbolic geometry there are at least two distinct lines through P which do not intersect L. In spherical geometry there are no parallel lines at all. Flat or Euclidean geometry applies to the plane and also to the surface of a cylinder. Spherical geometry applies to the surface of a sphere. Hyperbolic geometry applies to a saddle-shaped surface. Spherical geometry Is spherical geometry useful? It sure is, being the right one to use for the surface of the Earth. We define straight lines (geodesics) as those that travel the shortest distance between two points. On a sphere such as our Earth, geodesics are called great circles (since larger circles are not possible), each having the same diameter as the sphere. The Equator and the lines of longitude are great circles, parallels of latitude are not. Beijing and Philadelphia lie on the same latitude, and if we travelled from one to the other along this line of latitude, we would go 10,130 miles. By contrast, a geodesic between these two cities, which passes close to the North Pole, is only 6,878 miles. Yet on a flat map of the Earth the path along the line of latitude appears straight. This is because maps of the world on flat paper necessarily distort distance. This is most obvious near the North Pole, where the distance scale is distorted by a factor of about 60 %. 55

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Imagine a triangle consisting of two lines of longitude coming from the North Pole plus the segment of the Equator between them. Since the two base angles are both right angles, the sum of the angles in this triangle is more than 180 degrees. Hence geometry on a sphere has positive curvature. There are no parallel lines on a sphere, as all great circles cross one another. To see this, imagine one great circle, which cuts the sphere into two equal parts. Any other great circle cannot remain within only one of these two halves. If it did, its diameter would be less than that of the sphere. A rectangle of cloth can be used to perfectly wrap a cylinder, which is mathematically flat. On the other hand, a sphere is not flat, so that a sheet cannot fully cover it without wrinkling. One can imagine a clever tailor fashioning a piece of cloth that will fit perfectly over the top half of a beachball. The cloth would have to have less area inside a circle of fixed radius than on a bedsheet. To do so the tailor would need to cut out darts from the fabric and then sew it together again. One would have to crease a piece of positively curved cloth to lay it on a dresser. Hyperbolic geometry On a hyperbolic surface there is more than one parallel line through each point P and the angles of a triangle add up to less than 180 degrees. In contrast to the sphere, whose triangles seem to bulge out, on a saddle the geodesic sides of a triangle seem sucked in. A tiny being living on either surface would view the lines as perfectly straight, and would only be able to tell whether the triangle was bulging or sucked in by measuring its angle sum. In contrast to a sphere, the saddle-shaped area on a woman's side above her hip has negative curvature. To drape this part of her body exactly a cloth with negative curvature is required. The region inside a circle of a given radius contains more material than the same circle in a plane. To make suitable cloth a tailor might start with a flat piece of fabric, make a cut as if to make a dart, but instead of stitching the cut edges together, insert an extra piece of fabric. Negatively curved cloth has lots of folds when laid flat on a dresser. Three dimensions, and up The discussion so far has been limited to two dimensional geometry, i.e. to the study of surfaces, where there are only two independent directions. The notion of curvature extends naturally to three dimensions and beyond. The problem with discussing three dimensional geometries that are spherical or hyperbolic is that, unlike the curved surfaces mentioned above, we cannot picture or see these spaces. 56

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For each positive integer n, a n-space is the set of all ordered n-tuples of real numbers. Euclidean space of dimension n is n-dimensional space with distance defined by the Pythagorean Theorem. The crochet coral reef Since hyperbolic space is not something that we can perceive directly, various models have been constructed that give us a feel for its properties. Going down one dimension, we can show what a hyperbolic surface looks like, e.g. a saddle shape. The crocheted coral reef is an attempt to give a feel for hyperbolic 3-dimensional space within our familiar framework of Euclidean space. The crocheted reef is a hyperbolic surface made of negatively curved cloth. In 1997 mathematician Daina Taimina worked out how to make a physical model of hyperbolic space that allows us to feel, and to tactilely explore, the properties of this unusual geometry using crochet. The essence of the hyperbolic construction can be implemented with crochet simply by increasing the number of stitches in each row. As you increase, the surface naturally begins to ruffle and crenellate. Many of the properties of hyperbolic space become visible to the eye and can be directly experienced by playing with the models. Geodesics on the hyperbolic surface can be sewn onto the crochet surface for easy examination. You can also see multiple parallel lines that happily violate Euclid's 5th postulate. A coral reef is not only one of nature's visual marvels, it also shows an amazing adaptation to efficiently filter feed water by using the expanded surface area that negative curvature provides. Is the universe Euclidean? Most cosmologists believe that if we left the Earth in a spaceship headed out in a fixed direction, then after a very long time, we would come back close to where we started. This is because the universe is thought to be finite but unbounded (like the surface of a sphere). However, this leaves God the choice of geometry. Although the non-Euclidean geometries are logically consistent and are as mathematically significant as the Euclidean, one could ask whether Euclidean geometry is more “true” than the others in the sense that it applies to the space we live in. Is the universe an Euclidean space (i.e. flat), or does it have negative or positive curvature? People are so used to thinking in terms of Euclidean geometry that it is hard to step outside this mind-set. Since it took mathematicians thousands of years to do this, don't feel inadequate if non-Euclidean three-space seems abstruse to you – it is! To all intents and purposes the space we live in seems 57

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to be Euclidean, i.e. angles of a triangle are always measured as adding up to 180 degrees and the Pythagorean theorem has yet to fail in practice. However, when we speak about the overall shape of the universe itself, things become problematic. It may be that although the universe appears flat on the human scale it has a very slight positive or negative curvature. If we could measure the angles of a sufficiently large triangle in space we might find the sum to be different from 180 degrees. Note that it is not an easy matter to perform the equivalent operation on the surface of the Earth, and to thus demonstrate the positive curvature of the world we live on. We would need to draw a large triangle flat on the Earth's surface and to measure its angles sufficiently accurately to note that they add up to more than 180 degrees. It can be done, but it is not a simple matter because of the large scale involved and the fact that the Earth is not a perfect sphere. Physicists tried to see whether they could detect a cosmic deviation from the Euclidean sum by measuring the angles between distant stars. If the universe weren’t flat then the angles between three stars wouldn’t total to 180 degrees. All measurements, however, revealed the standard 180 degrees and for most of the past century the evidence has pointed to a Euclidean cosmos. At present (2007), the combination of cosmic microwave background, type 1a supernova observations and mass estimates leads astronomers to suggest that the average curvature of our universe is very close to zero. Astrophysicists favour a flat universe but do not rule out a slight positive curvature. An implication of Einstein's theory of general relativity is that Euclidean geometry is a good approximation to the properties of physical space only where the gravitational field is not too strong. It has also been suggested that the shape of the universe is akin to a twoholed hyperbolic torus. The WMAP satellite currently taking pictures of the early universe will hopefully provide evidence one way or other in the next few years, so that humanity may at last know the shape of existence itself. Tad Boniecki August 2009 (abridged from http://soler7.com/IFAQ/NonEuclideanGeometry.html)

2. Find in the text the synonyms to the following words or phrases. 1) lack of success; 2) later, following; 3) about; 4) the state of being curved; 58

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5) to disturb the peace, calm or smoothness of; 6) that which makes a thing what it is; the inner nature or most important quality of a thing; 7) to discover; 8) smooth and level; even; having an unbroken surface; 9) absence of agreement; 10) of similar character; like; 11) to break; to act contrary to. 3. Which of the following sentences correspond to the text content? 1. Euclid’s masterpiece serves as a model for all pure math theories. 2. Non-Euclidean geometry can be presented in various ways. 3. The discoverers of non-Euclidean geometry wanted to show that the 5th postulate is, in fact, deducible. 4. Spherical geometry is useful. 5. Hyperbolic geometry is one of many non-Euclidean geometries. It is particularly important. 6. The universe is Euclidean. 7. Euclidean and non-Euclidean geometries are alternative ways geometrically to describe physical space, both the former and the latter are consistent. 4. These are answers to questions about the text. Write the questions. 1. This structure endured undisturbed for some two thousand years. 2. It was simply called “geometry”. 3. This led to the birth of non-Euclidean geometry. 4. It applies to the plane and also to the surface of a cylinder. 5. There are no parallel lines on a sphere, as all great circles cross one another. 6. We would need to draw a large triangle flat on the Earth's surface. 7. Euclidean geometry is a good approximation to the properties of physical space only where the gravitational field is not too strong. 5. Verbs and prepositions often go together. Find these verbs in the text. Then fill in the gaps with the correct prepositions. 1) to depend ....... the surface; 2) to refer ....... the sum; 3) to apply ....... the plane; 4) to cut ....... darts from; 5) to work ....... how to make; 6) to rule ....... a slight positive curvature. 59

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Writing 6. Imagine you have started an advanced geometry course at university. Write a letter to a friend telling him/her what you have learnt about it so far. Dear (first name) PARAGRAPH 1 Hi! You asked me to tell you all about my studies. Last week we learnt about the history of geometry ... PARAGRAPH 2 Now I’ll briefly explain exactly what Non-Euclidean geometry is … PARAGRAPH 3 Finally you may wonder ... Best wishes, (your first name) Write 150–200 words Feedback Answer the following questions: How did you feel about the activities in this Module? Can you try out some of the activities in Russian? Are there any activities here that you would like to do again? Just for Fun How to divide a garden A man had a house with a beautiful garden. The house occupied a quarter of the ground. The man had four children who liked the garden. One day they asked their father to divide the garden among them. “All right”, said the man, “I'll divide the garden into four equal parts and give everybody one part”. “We don't understand how you can do it”, said the children. But their father did it and each child received an equal part of the garden. Can you explain how the father divided the garden into four parts of equal size and shape? Answer: We know that the house occupies one quarter of the ground. So we can divide the garden into three equal quarters. Then we must divide each of these squares into four equal squares. If we take three small squares for each child we can divide the garden into four parts of equal size and shape. Each child will receive one part.

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Module 4 Mechanics Learning objectives: In this module you will: • work with vocabulary connected with mechanics • characterize problems • develop the idea • express agreement, disagreement • review sentence-structure • write a summary • prepare a short presentation • check spelling, punctuation and grammar

UNIT 1 Before you read 1. Answer the questions 1. What does classical mechanics deal with? 2. What problems can be solved in classical mechanics? 3. What is the main subject of theoretical mechanics? 2. Read the text and check your answers. What is classical mechanics? Classical mechanics is a part of physics that deals with the motion of point masses (very small things) and rigid bodies (large things that can rotate as a whole but cannot change their shape). This is very useful in practice, since many objects in real life can be approximately considered to be either point masses or rigid bodies in most situations. 61

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Typical problems solved in classical mechanics are: To find the trajectory of a stone thrown into the air with known initial velocity. (The stone is considered to be a point mass.) To predict the motion of a spacecraft approaching some planet, if its initial position and velocity far from the planet are known. (The spacecraft is considered to be a point mass.) To find out how many revolutions per minute a disc will be executing if we know the strength of the engine driving the rotation. (The disc is considered to be a rigid body rotating as a whole.) To find out how much energy and how much time is needed to accelerate a small object to a given speed. (Point mass.) To find the frequency of oscillations in a system of point masses connected by springs. Of course, one can consider also much more complicated problems than these. For example: A light spinning top stands at an angle on the surface of a heavy cylinder that can roll along a horizontal plane without sliding. Determine the conditions that would allow the top to avoid falling off the cylinder. (Both the cylinder and the top are considered rigid bodies.) A spacecraft launched from the Earth needs to reach the surface of Mars within a certain time. Predict the most appropriate time of year for this mission and determine the least amount of rocket fuel needed. (The spacecraft is considered a point mass moving in the gravitational field of the Sun, the Earth, and Mars.) Classical mechanics uses ordinary differential equations (ODEs) to describe the properties of bodies mathematically. Thus, coordinates, angles, etc. are numbers that depend on time, e.g., and satisfy certain (systems of) ODEs. One can use several mathematical methods to solve such equations. In some cases, solutions can be found exactly, for instance: has the general solution. In  other cases, solutions are found only in terms of integrals that one cannot evaluate in closed form. Sometimes, one can solve the equation approximately using some method such as perturbation theory. Finally, any ODE can be solved numerically (using a computer program) up to a certain precision. Students of mechanics are expected to learn methods of solving certain standard differential equations that are exactly solvable, for instance: multidimensional harmonic oscillators, motion in 1-dimensional force field, motion in 3-dimensional central force field. Numerical methods for solving ODEs are important in practice but are usually not studied as part of classical mechanics because these methods are not specific to mechanics but are 62

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Mechanics

equally applicable to every differential equation. Numerical methods for solving various equations are best studied in a dedicated course that involves hands-on computer programming. Newtonian mechanics The first successful theory of classical mechanics is contained in Newton's three laws of mechanics that govern the motion of point masses: There exist reference frames where a point mass not interacting with other bodies will move with constant speed in the same direction. (If this is not true in some reference frame, then that reference frame is not inertial. Further laws are formulated in inertial frames.) A point mass interacting with other bodies moves with the acceleration found from, where is the sum of all forces acting on the body, is the mass of the body, and is the acceleration, i.e. the second derivative of the position vector with respect to the time. All forces are caused by other point masses, and whenever a point mass 1 exerts a forceon a point mass 2, the point mass 2 also exerts the force on the point mass 1. The motion of all point masses is described by differential equations which can be solved directly as long as all relevant forces could be predicted or measured. I assume that you are already familiar with these laws and with typical situations where they apply (e.g. motion of bodies thrown at an angle near Earth) before you start studying theoretical mechanics. In Newtonian mechanics, a rigid body is simply a collection of point masses connected by “rigid sticks”. These sticks are “rigid” because they always produce exactly such forces as to keep constant distances between all points, regardless of any other forces or motions. Thus the motion of rigid bodies can be described without introducing any other special rules. One derives the concept of angular momentum, torque, etc., from Newton's laws without any additional postulates. From Newtonian mechanics to “theoretical” mechanics The necessity to consider point masses is certainly inconvenient if one needs to describe liquids and gases, so a special branch of mechanics with its own formalism was developed for that purpose, namely continuum mechanics (mechanics of continuous media). The formalism of continuum mechanics is generalized to field theory where the basic object is not a point mass but a field, i.e. some abstract “substance” that is present at all points in space and shows its influence at every point at once. (Examples are: gravitational field, electric field, and magnetic field.) Such substances may be described by a function of space and time, for example the vector field describes the electric field. 63

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The behavior of fields is usually governed by partial differential equations; for example, the electric field and the magnetic field satisfy Maxwell's equations. As more and more complicated problems needed to be solved, various mathematical tools were developed to simplify and to generalize the mathematical description of mechanics. Finally, the Lagrangian and the Hamiltonian formulations of mechanics were discovered. These two formulations still remain the cornerstones of classical mechanics and field theory, as well as Einstein's theory of relativity, and thus indirectly of all modern theoretical physics. These formulations of mechanics are not based on the assumption of “forces” and are equally applicable to point masses, rigid bodies, fields, and continuous media. The main subject of theoretical mechanics (sometimes also called “analytical mechanics”) is the study of these more refined and more general mathematical formulations of classical mechanics. (abridged from http://en.wikibooks.org/wiki/Classical_Mechanics/Introduction)

3. Answer the questions. 1. Who do you think made the most important contribution to mechanics? 2. What was that contribution? 4. Look through the text and find the word that means the same as: 1) very near correct; about right; 2) a curved path of a projectile; 3) strength; power of body or mind; physical power; 4) rate of increase; 5) firm, strict; not changing; not to be changed; 6) substances like water or oil that flow freely and are neither solids nor gases; 7) puts forth; 8) something supposed but not proved. 5. Which of the following sentences correspond to the text content? 1. Mechanics is closely identified with physics and engineering. 2. All the fields and subjects of mechanics apply the methods and equations of theoretical mechanics. 3. Most mechanical laws were derived inductively from experiments. 4. The formalism of continuum mechanics is generalized to field theory where the basic object is not a point mass but a field. 5. The behavior of fields is usually governed by partial differential equations. 6. The fundamental entity in mechanics is the material particle and bodies are considered as aggregates of such particles. 64

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6. Verbs and prepositions often go together. Find these verbs in the text. Then fill in the gaps with the correct prepositions. a) to deal ....... the motion; b) to throw ....... the air; c) to find ....... how; d) to interact ....... other bodies; e) to act ....... the body; f) to base ....... the assumption. Writing 7. Write a summary of the text What is classical mechanics. It should include a statement of the main idea, a note on the introduction and conclusion and an outline of the main topics. Add to the summary your comments concerning the contents. Speaking 8. Comment on and characterize classical, Newtonian and theoretical mechanics using the characteristics mentioned in the text. 9. Starting from general assumptions, Newton tied together in a  single scheme many diverse and disconnected things. Explain what enabled him to make his great guess, concerning each and all taken together. 1. Moon's circular motion. 2. Disturbances of Moon's simple motion. 3. Planetary motions (Kepler's laws). 4. Planetary perturbations. 5. Motion of comets. Tides. 6. Bulge of the Earth. 7. Differences of gravity. 8. Precision of equinoxes. 9. The motion of a gyroscope. All related by inverse-square-law of gravitation and a spinning Earth. 10. What law can be deduced from the following statements? 1. A spinning top has the same weight as a still one. 2. Mass is constant, independent of speed. 3. Mass increases with velocity, but appreciable increases require velocities near that of light. 65

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4. If an object moves with a speed of less than one hundred miles a second, the mass is constant to within one part of a million. 11. Characterize the problems the fundamental fields of mechanics deal with. If you are the mechanics student, what field are you going to specialize in? Give your reasons. Model. It is easy to show that forces are vectors, i.e., that they obey geometrical addition. The vector treatment of balanced forces is called statics. The geometrical properties of the motion of rigid bodies constitute the subject matter of kinematics. The study of motion subject to external forces is dynamics. There are two basic problems in dynamics: to find the forces exerted that cause the motion and determine the motion of the body when the forces applied are specified. Theoretical Mechanics. It is as old as mechanics itself. It makes research in various trends. The theories of stable motion, of automatic and optimal operations, the dynamics of flight, etc. are treated. The problems concerning the behavior of mass-points, motion, density, liquids, gases, plasma and their states are dealt with. Ideal physical and mechanical models are introduced to make formulation of mechanics possible with the aid of differential and integral calculus. Analytical Mechanics formulated at least 200 years ago studies systems of mass-points of rigid bodies, finite in number. Analytic methods and the methods of differential equations are used. Applied Mechanics. The theory of machines and mechanical devices, the theories of vibrations, regulations, gyroscope, automatic control, etc. are developed. Most problems are of engineering and manufacturing aspect. Electromechanics commonly implies the interaction of currents with fluids and the construction of practical electromechanical energy-converting devices. The theory covers topics regarding the nature of the mechanical and electrical properties of the interacting medium. It makes a great difference whether the fluid is a gas, a liquid, or a plasma to say nothing of the diversity of properties associated with each of these media. Hydrodynamics. Aerodynamics. Magnetohydrodynamics. Electrohydrodynamics. They are all based on classical Newton's mechanics and the model of material continuum. The theories of perfect fluid and viscous fluid, the perfectly flexible line, the membrane, gas and wave dynamics, the perfectly elastic solid, the infinitesimally visco-elastic material and plastic material, the phenomena in the Earth's interior and in its atmosphere are created. Celestial Mechanics. Stellar Astronomy. Astrophysics. The investigations of gravitational fields, celestial bodies with various configurations, the 66

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Mechanics

evolution of planetary and satellite systems and their stability, the motion of cosmic dust, topological peculiarities of the rotation of celestial bodies, stellar atmosphere, cosmic gas dynamics, the structure and evolution of stars, etc. are being performed in all these fields. Continuum Mechanics. The theories of elasticity, plasticity, creep and relaxation in solids, the theory of motion of plasma constitute the subject matter of the field. The problems of the elastic limit, the origin of a residual plastic deformation, the motion of highly compressed liquids and gases, or conversely, rarefied gases, mechanical models for polymeric plastic materials, etc. are dealt with.

UNIT 2 1. Make the right choice and complete the sentences: 1. Mechanics is a classical subject which deals with the motion of bodies and construction of machines and various mechanical .......... (equations, devices, particles) 2. The general principles in mechanics are illustrated by constitutive equations which abstract the differences among .......... (bodies, devices, motions) 3. Most mechanical laws were derived inductively from .......... (principles, experiments, schemes) 4. Mechanics is seeking the simplest possible .......... (law, idea, description) of how bodies actually move, it makes no pretence of explaining why bodies move. 5. Einstein presented a new concept of .......... (gravitation, mass, force) There is, he claimed, no absolute force of gravity pulling objects down. 6. In the past few years several versions of Einstein’s “thought experiments” were carried .......... (in, to, out) with real apparatus to verify some of the new hypotheses. 2. Fill in the gaps with the derivative. The general theory of relativity, which is being brilliantly confirmed by experiments, the equations of which are used by astrophysicists to compute the gravitational fields of space objects, (1) .......... fundamental difficulties which are not clarified to this day. The chief one is the problem of determining the energy of the gravitational field. In the framework of Einstein’s theory this question remains a veritable “headache” for scientists. Opinions on how exactly to (2) .......... this energy invariably differ. 67

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Recently Academician A. Logunov and Professor M. Mestvirishvili advanced a new theory of gravitation, in which the energy of the gravitational field can always (3) .......... Unlike Einstein, Logunov and Mestvirishvili maintain that (4) .......... world is homogeneous, while gravitational attractions in it are conditioned not by the curvature of space but by some physical force field like the electromagnetic field. They draw on the methods (5) .......... in the field theory of elementary particles. POSSESS _____________________ 1 COMPUTATION _______________ 2 DETERMINE _ ________________ 3 WE __________________________ 4 USE _________________________ 5 Reading 3. Read the text and say, what type of information this text represents: description, comparison, classification, disproof. What details are the most important? 4. The five sentences below have been removed. Decide where they should go. A. Hooke’s law may also be expressed in terms of stress and strain. B. Mathematically, Hooke’s law states that the applied force F equals a constant k times the displacement or change in length x, or F = kx. C. The deforming force may be applied to a solid by stretching, compressing, squeezing, bending, or twisting. D. Sometimes Hooke’s law is formulated as F = −kx. E. Under these conditions the object returns to its original shape and size upon removal of the load.

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Hooke's Law Hooke’s law, law of elasticity discovered by the English scientist Robert Hooke in 1660, which states that, for relatively small deformations of an object, the displacement or size of the deformation is directly proportional to the deforming force or load. 1) .......... Elastic behaviour of solids according to Hooke’s law can be explained by the fact that small displacements of their constituent molecules, atoms, or ions from normal positions is also proportional to the force that causes the displacement. 2) .......... Thus, a metal wire exhibits elastic behaviour according to Hooke’s law because the small increase in its length when stretched by an applied force doubles each time the force is doubled. 3) .......... The value of k depends not only on the kind of elastic material under consideration but also on its dimensions and shape. At relatively large values of applied force, the deformation of the elastic material is often larger than expected on the basis of Hooke’s law, even though the material remains elastic and returns to its original shape and size after removal of the force. Hooke’s law describes the elastic properties of materials only in the range in which the force and displacement are proportional. 4) .......... In this expression F no longer means the applied force but rather means the equal and oppositely directed restoring force that causes elastic materials to return to their original dimensions. 5) .......... Stress is the force on unit areas within a material that develops as a result of the externally applied force. Strain is the relative deformation produced by stress. For relatively small stresses, stress is proportional to strain. (adapted from http://global.britannica.com/EBchecked/topic/271336/Hookes-law)

5. Decide if the following statements are true or false, or there is no information in the text. a) true b) false c) no information 1. We meet similar Hooke's law – behaviour in many cases of stretching, compression, twisting, bending – all varieties of elastic deformations. 2. Hooke discovered that when a spring is stretched by an increasing force, the stretch varies directly as the force. 3. Hooke's law is remarkable not just for its simplicity but for its wide range. 4. One cannot express Hooke's law in terms of stress and strain. 5. Elastic behaviour of solids according to Hooke’s law can be explained by the fact that small displacements of their constituent molecules, atoms, or ions from normal positions is also proportional to the force that causes the displacement. 69

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6. Match the words to make phrases. 1) behaviour 2) deforming 3) deformation 4) elastic 5) original 6) relative 7) law

a) dimension b) deformation c) of material d) of elasticity e) of solids f) force g) property

Grammar review 7. Find the word which shouldn’t be in the sentence. 1. Rules and knowledge together with a techniques for gaining more knowledge made the beginning of the genuine science. a) with b) a c) for 2. Historical records reveal the origin of some laws; others are concealed, because the man who was first proposed them did not let us know how they had occurred to him first. 3. We take it as for granted that there are simple laws to be found and they are true descriptions of nature when we do find them. a) as b) for c) do 4. All the Greek scientists were been trying to build a picture and a the ory of the universe founded on observed facts and speculation. a) were b) been c) to 5. Though our modern tradition of experimenting and our modern wealth of scientific tools made changes, we still hold about the Greek delight in a theory that can account for the natural phenomena. a) about b) for c) that 6. The changes from Newton’s predictions on to Einstein’s, though fundamental in nature, are usually too small in effect to make any difference in laboratory experiments or even in most astronomical measurements. a) even b) on c) from 7. The formula of the dependence of the spin and mass found by Krechet and Ivanenko for the universe coincides with the dependence have known for elementary particles. a) found b) have c) known 8. Complete the following sentences with to if necessary. 1. George said we had better .......... go to the annual conference of mathematicians. 70

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shine.

2. Nuclear reactions must be going on in the stars .......... make them

3. The rules of quantum mechanics can be employed .......... calculate the results of the experiments. 4. We are .......... consider the following condition so as to imagine the consequences. 5. I saw him .......... enter the University building. 6. They demand that an appropriate solution .......... be found. 7. He asked me not … go. 9. Change active into passive. Model. Scientists are seeking for laws describing nature. Laws describing nature are being sought for. 1. He has observed direct proportionality between a stretch of a spring and the force applied. 2. Scientists claim constancy as the most essential characteristic of all scientific laws. 3. While experimenting the scientists are looking for constancy. 4. Scientists considered conservation laws as a basis of mechanics. 5. Hooke was experimenting with springs and loads. 6. Natural scientists are using deduction to extract common behaviour from a few laws. 7. Ancient scientists were discovering laws by collecting facts and speculating. 10. Read the text below and choose the correct word for each space. The third law (1) .......... that all forces exist in pairs: if one object A exerts a force FA on a second object B, then B (2) .......... exerts a force FB on A, and the two forces are equal and opposite: FA = −FB. The third law means that all forces are interactions between different bodies, and thus that there is no such thing as a unidirectional force or a force that acts on only one body. This law is sometimes referred (3) .......... as the action-reaction law, with FA called the “action” and FB the “reaction”. The action and the reaction are simultaneous, and it does not matter which is called the action and which is 71

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called reaction; both forces are part of a single interaction, and (4) .......... force exists without the other. The two forces in Newton's third law (5) .......... of the same type (e.g., if the road exerts a forward frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third law predicts for the tires pushing backward on the road). From a conceptual standpoint, Newton's third law is (6) .......... hen a person walks: they push against the floor, and the floor pushes against the person. Similarly, the tires of a car push against the road while the road pushes back on the tires – the tires and road simultaneously push against each other. In swimming, a person (7) .......... with the water, pushing the water backward, while the water simultaneously pushes the person forward – both the person and the water push against each other. The reaction forces account (8) .......... the motion in these examples. (9) .......... forces depend on friction; a person or car on ice, for example, may be unable to exert the action force to produce the needed reaction force. 1. A states B speaks C develops 2. A usually B simultaneously C often 3. A to B at C on 4. A either B neither C another 5. A be B is C are 6. A saw B see C seen 7. A works B interacts C feels 8. A on B to C for 9. A this B these C that 11. Solve the problem The duty-free shop At a duty-free shop at the airport, a toy teddy bear is hanging at the end of a spring. The spring is 51.0 cm long when hanging vertically. When the teddy bear of mass 400 g is hung from the end of the spring, the length of spring becomes 72.0 cm. a) Calculate the spring constant. Write a unit with your answer. b) Calculate the energy stored in the spring when a second toy of mass 300 g is also hung along with the teddy bear on the spring. c) The 400 g teddy bear is now hung on a stiffer spring which has double the spring constant. Discuss how this affects the extension and the elastic energy stored in the spring.

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Speaking 12. Discuss the significance of the great discoveries in mechanics and physics. (1807) J. Fourier presented to the French Academy a theorem of unprecedented importance for the progress of science, which advanced the math mastery of the motion of waves: Any wave, whatever its form, can be treated as a sum of a set of simple harmonic waves. (1864) J. Maxwell published a paper synthesizing notions about electricity, magnetism and light. According to Maxwell's theory of electromagnetic radiation visible light, ultraviolet light and any possible radiations of still higher frequencies must be emitted by oscillating electric charges within atoms. Accelerated charges produce the electromagnetic waves that we observe as light when they strike our eyes. (1911) A. Einstein. General relativity. The light is curved in the gravitational field. (1915) Accelerated mass should radiate energy in the form of gravitational waves. (1924) De Broglie Duality. Any moving particle (electron, atom, neutron, a quantum of light) is an extensive wave in some of its behaviour, and a compact particle in some of its behaviour. All objects should have with them a wavelength related to their momentum, e.g., an electron must have a wavelength associated with it. (1927) Heisenberg's Uncertainty Principle. There is some uncertainty in the specification of position and velocity of a quantum particle. We can say, at least, that there is a certain probability that any particle will have a position near some coordinate x. The most precise description of nature must be in terms of probabilities.

UNIT 3 Reading 1. Put the paragraphs of the article in order and read the article. More Than One Brain Behind Einstein's Famous Equation: E = mc2 Jan. 25, 2013 – A new study reveals the contribution of a little known Austrian physicist, Friedrich Hasenöhrl, to uncovering a precursor to Einstein famous equation.

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a) Before Hasenöhrl focused on cavity radiation, other physicists, including French mathematician Henri Poincaré and German physicist Max Abraham, showed the existence of an inertial mass associated with electromagnetic energy. In 1905, Einstein gave the correct relationship between inertial mass and electromagnetic energy, E = mc2. Nevertheless, it was not until 1911 that German physicist Max von Laue generalized it to include all forms of energy. b) Given the lack of recognition for Hasenöhrl's contribution, the authors examined the Austrian physicist's original work on blackbody radiation in a cavity with perfectly reflective walls. This study seeks to identify the blackbody's mass changes when the cavity is moving relative to the observer. c) According to science philosopher Thomas Kuhn, the nature of scientific progress occurs through paradigm shifts, which depend on the cultural and historical circumstances of groups of scientists. Concurring with this idea, the authors believe the notion that mass and energy should be related did not originate solely with Hasenöhrl. Nor did it suddenly emerge in 1905, when Einstein published his paper, as popular mythology would have it. d) They then explored the reason why the Austrian physicist arrived at an energy/mass correlation with the wrong factor, namely at the equation: E = (3/8) mc2. Hasenöhrl's error, they believe, stems from failing to account for the mass lost by the blackbody while radiating. e) Friedrich Hasenöhrl in establishing the proportionality between the energy (E) of a quantity of matter with its mass (m) in a cavity filled with radiation. In a paper about to be published in the European Physical Journal H, Stephen Boughn from Haverford College in Pensylvannia and Tony Rothman from Princeton University in New Jersey argue how Hasenöhrl's work, for which he now receives little credit, may have contributed to the famous equation E = mc2. (adapted from http://www.sciencedaily.com/releases/2013/01/130125103931.htm)

2. Which of the following sentences correspond to the text content? 1. Two American physicists established the proportionality between the energy (E) of a quantity of matter with its mass (m) in a cavity filled with radiation. 2. French mathematician Henri Poincaré and German physicist Max Abraham, didn’t show the existence of an inertial mass associated with electromagnetic energy. 74

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3. Hasenöhrl made a serious error. 4. Thomas Kuhn believes that the nature of scientific progress occurs through paradigm shifts, which depend on the cultural and historical circumstances of groups of scientists. 5. Hasenöhrl was the first to give the correct relationship between inertial mass and electromagnetic energy. 6. Einstein published his paper in 1911. 3. Here are the answers to some questions about the article. What are the questions? 1. Austrian physicist Friedrich Hasenöhrl. 2. Two. 3. Thomas Kuhn. 4. In 1905. 5. In 1911. Grammar review 4. Make phrases following the order of adjectives and nouns. determiner or article – opinion adjective – size – shape – age – colour – nationality – religion – material – noun used as an adjective – the noun that the adjectives are describing 1) desk office big ugly an wooden brown; 2) hair long black straight my sister’s; 3) photograph black white and oval a family historic; 4) vase jade Ming beautiful a antique little green; 5) computer laptop high-tech brand-new deep university’s blue my. 5. Choose the correct phrase. 1. a) an annual international scientific conference b) an international scientific annual conference c) an annual scientific international conference 2. a) an old boring American opinion b) a boring old American opinion c) an American old boring opinion 3. a) a TV long boring interview b) a long boring TV interview c) a boring long TV interview 4. a) a slender strange German magazine b) a strange slender German magazine c) a German slender strange magazine 75

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6. Find grammar mistakes in these sentences and correct them. 1. To solve the problem of gravitation scientists is considering the time-space geometry in a new way. 2. Mechanics is a classical subject which deal with the motion of bodies and construction of machines and various mechanical devices. 3. Most laws in science state the relationships between measurements two (three) quantities. 4. The natural philosophers of ancient Greece liked do experiments in their heads. 5. In specifying gravitation on the new geometrical view Einstein not proved “Newton's law of gravitation” wrong but offered a refining modification – though this involved a radical change in viewpoint. 6. Galileo created a structure of bodies in motion very many like Euclid developed a structure of relations of objects in space. 7. The work did in going around any path in a gravitational field is never zero. 7. Make sentences with these words. 1) The, apply, general, all, to, laws, of, and, mechanics, bodies, motions, all. 2) gravity, all, with, Under, the, acceleration, influence, fall, the, of, objects, the. 3) The, of, force, acts, inertia, proportional, any, on, vertically, object, and, is, downward, to, it, mass, or, its, gravity. 4) law, The, is, claiming, universal, more, from, more, and, scientists, attention, gravitation, of. 5) taken, granted, is, are, for, It, three, of, dimensions, a, single, time, dimension, usually, that, there, space, of, and. 6) knowledge, nature, of, The, in, was, being, laws, codified, simple. 7) basic, is, find, to, behind, The, that, can, phenomena, theoretical, laws, amalgamate, different, problem, experiment. Speaking 8. Express your views on the following statements trying to prove your point. The given phrases may come in handy. What is missing (lacking) in the statement is that ... In view of the idea … I have reason to believe that … 1. It is characteristic of the scientific laws that they have abstract character. 76

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2. If law does not work even in one place where it ought to, it must be wrong. 3. One can claim for scientific laws a universal validity. 4. Any great discovery of a new law is useful only if we can take more out than we put in. 5. The fundamental hypothesis of science is: the sole test of the validity of any idea is experiment. 6. There are theoreticians, who imagine, deduce and guess at new laws, but do not experiment. 7. There are experimentators, who experiment, imagine, deduce and guess. 8. Observations, reasoning and experiment make up what we call the scientific method. 9. The basic theoretical problem is to find laws behind experiment that can amalgamate different phenomena. 10. There are no absolute laws in the sense of laws independent of observers. 9. Read the quotations given below and agree or disagree with them. Your opinion should be followed by some appropriate comment. Here are some expressions that may be used to show your attitude: Agreement Disagreement I think so, too. I wouldn't say so. That's a very good point. May be, that's true, but... I agree absolutely with... (I'm afraid) I can't accept... Exactly. Certainly. I don't think this is just the case. 1. “Newton was not only the greatest but the most fortunate among scientists, because the science of the world can be created only once, and it was Newton who created it”. (Lagrange) 2. “The reality of the world consists of its math relations. Math laws are the true cause of phenomena”. (J. Kepler) 3. “The scientist must order. One makes science with facts as a house with stones; but an accumulation of facts is no more science than a pile of stones is a house”. (Poincare) 4. “What goes up must come down”. (a classical saying) 10. Prepare a short presentation (5 minutes) to answer the question: What is mechanics? Talk about: • what mechanics deals with; • branches of mechanics; • how it is used. 77

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Writing 11. Punctuate the story as you think appropriate. Newton and Hooke Do you know anybody who always has to be first maybe it’s your brother or sister or a friend when you say you learned something they say I already knew that when you're going to do something they say I already did that There was another famous scientist named Robert Hooke who was very jealous of Newton whenever Newton announced he had discovered something Hooke would say I already discovered that first I just didn't tell anybody yet this made Newton very mad Everybody makes mistakes sometimes even very smart people like Newton one time when Newton made a mistake Hooke was the first to discover the mistake and tell everybody about it Newton was mad and embarrassed he didn't like to make mistakes but he really hated that it was Hooke who figured out the mistake Newton said that he would never tell anybody about his discoveries again he didn't want to ever have Hooke catch him making a mistake again in a while though Newton realized how silly this was and started telling about his discoveries again Newton wrote a big book called Principia this book told all about how things push and pull and gave lots of examples of how machines work and how things like planets and comets move it was a very important book and scientists still like to read it even though it is more than 300 years old but guess what happened when he was writing it Hooke found out about what he was writing and said I already discovered that Newton was so angry that he decided not to write the book it was a good thing that he again realized how silly it is to let somebody else bother you that much he did finish writing it and even mentioned some things that Hooke had done Feedback How useful has this Module been for you? Grade all the activities from the first (useful) to the last (useless). 1……… 2……… 3……… 4……… 5……… Discuss your opinions with other students.

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Just for Fun From the life of well-known scientists When did Galileo Galilei begin his scientific work? We all imagine Galileo Galilei as an old man with long grey hair and fanatic eyes. That idea of him comes from his well-known portrait. But people are always surprised when they learn that Galileo Galilei made his first scientific discovery at the age of nineteen. This great Italian scientist became Professor of Mathematics at the University of Pisa when he was only twenty-one. Fun facts about Isaac Newton • Newton's discoveries about light and movement of planets were used to make the first flights to the moon possible. • Newton at only age 26 became a professor of math. • Newton believed God was invisible but influenced every part of people's lives. • Newton practiced Alchemy. Alchemy is an ancient practice banned in England in 1404. • Newton was elected as a member of parliament. His membership lasted only a year. • Newton earned the title of Warden of the Royal Mint. • Newton was knighted because of his political activities. • Isaac was named after his father who died three months before Isaac was born. • Isaac was born early. He was so small he could have put him in a quart jug. • Isaac's father could hardly write his name.

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Module 5 Informatics Learning objectives: In this module you will: • understand basic concepts in informatics, and acquire vocabulary connected with it • recognize abbreviations associated with programming • express your personal point of view • confirm or deny the statements • identify some programming languages • make a poster • check spelling, punctuation and grammar

UNIT 1 1. Match the word combinations with their definitions. 1) program 2) programming language 3) machine code 4) assembly language 5) high-level language 6) low-level language 7) artificial intelligence

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a) the code actually executed by computer, not easily readable by the programmer b) the discipline concerned with the building of computer programs that perform tasks requiring intelligence when done by humans c) a list of instructions which are used by the computer to perform the user's requirements d) a language such as assembly language in which each instruction has one corresponding instruction in machine code e) a notation for the precise description of computer programs f) a human-readable representation of machinecode programs g) a language in which each instruction represents several machine code instructions, making the notation more easily readable by the programme

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Informatics

2. Read the text. Unfortunately, computers cannot understand ordinary spoken English or any other natural language. The only language they can understand directly is called machine code. This consists of the 1s and 0s (binary codes) that are processed by the CPU. Writing a machine code program takes a very long time and is best left to experts. A solution to the problem of writing programs is to use a computer language that is more easily understood by humans. Then, using a special program, the computer can translate them into machine code. For example, computer languages called assembly languages use mnemonics (abbreviations that are easy to remember) to represent instructions e.g. JMP (JuMP) JSR (Jump to SubRoutine) Programs written in assembly languages are translated into machine code by a special program called an assembler. Since each line of an assembly language program represents one machine code instruction, assembly languages and machine codes are known as low-level languages. Low-level languages allow programs to operate at high speed using the minimum of memory. Although assembly languages simplify the writing of programs, they are still quite complex. Computer languages which resemble English to some extent, are therefore often used. These languages are known as highlevel languages because each instruction translates into many instructions of machine code. High-level languages make programs easier to write, modify and understand although the programs are executed more slowly and use up more memory than programs written in low-level languages. 3. Complete the table. Advantages

Disadvantages

high-level languages low-level languages

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4. Try to identify these programming languages: a) /*this program finds the minimum of two integers*/#include ma i n() { i n t j. k, m ; pn'ntf( "Input two integers: "); scan ("%d%d".&j,&k); m = min( j , k); printf("\n%d is the minimum of %d and %d\n\n". m.j,k); } b) 1740 REM ***************************** 1750 REM Capture alarm data 1760 REM ***************************** 1770 CLS 1780 SCREEN 1 1790 COLOR BACK.PALL 1800 LOCATE 1.1 1810 INPUT "Alarm time: ".ATS 1820 ACS - MID$CAT$,3.1) 1830 AH$ - MI0$(AT$.1.2) 1840 AMS - MID$(ATS.4.2) 1850 IF ACS ":" GOTO 1770 1860 IF LEN (ATS) 5 GOTO 1770 1870 AH%-VAL(AH$) 1880 AM%-VAL(AMS) 1890 IF AH% > 23 GOTO 1770 1900 IF AM% > 59 GOTO 1770 1910 LOCATE 2.1 1920 INPUT "Alarm text: ",ATEXT$ 1930 IF LEN (ATEXT$) > 30 THEN GOTO 1910 1940 ASW% – 1 1950 LSET FAHS – MKIS(AH%) I960 LSET FAMS – MKI$(AM%) 1970 LSET FATEXTS – ATEXTS 1980 LSET FASWS – MKI$ (ASW%) 1990 PUT £2.1 2000 CLS 2010 GOTO 2720 82

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Informatics

c) 092900 B700-CONVERT-R80-DATE SECTION. 093000* 093100* IF CREDIT HAS NOT BEEN GIVEN ON A TRANSACTION. THE 093200* DATE USED IS THE TRANSACTION DATE. THIS SECTION 093300* CONVERTS THE TRANSACTION DATE TO A NUMBER OF DAYS 093400* SINCE 1900, PUTTING THE RESULT IN WS-CALCDATE. 093500* 093600 B700-010. 093700* 093800 MOVE R80-0TTRAN TO WS-DATE-N. 093900 MOVE WS-OATE-X TO FLO-AREA. 094000 MOVE NORM-MDY-INPUT TO N01 - USER-FI ELDS. 094100 MOVE ZEROS TO NORMALIZER-BUFFER 094200 N01-NBR-INTS. 094300 CALL 'X2XEFTB' USING TWAB 094400 N01-ARG-LIST 094500 FLD-AREA 094600 NORMALIZER-BUFFER. 094700 IF N01-RTN-CODE NOT EQUAL TO ZEROS 094800 DISPLAY 'B200: DATE CONVERSION FAIL. CODE. 094900 N01-RTN-C00E. 095000 MOVE DAYS-SINCE-1900 TO WS-CALC-DATE. 095100* 095200 B700-090. 095300* 095400 EXIT. 5. Read these facts about BASIC. BASIC (Beginners All-purpose Symbolic Instruction Code) This language was written in 1964 as a teaching language. It is a general purpose language which is interactive i.e. data can be input while the program is running. It is very easy for beginners to learn and is very user-friendly e.g. it displays helpful error messages to tell the user when he has made a mistake. Although it is found on most microcomputers, there are many different dialects of BASIC. This makes it difficult to use the same program on two different types of computer. 83

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Study these notes about BASIC. What has been missed out? Why do you think this has been done? language

BASIC

full title

Beginners All-purpose Symbolic Instruction Code

date

1964

use

general purpose

special features

easy to learn, user-friendly, interactive

other points

used on most microcomputers but many dialects exist

6. Read the text to find out the information about these computer languages: FORTRAN, ALGOL, LISP, COBOL, PASCAL, LOGO, PL/I, C. FORTRAN (FORmula TRANslator) FORTRAN was the first programming language that significantly rose above the level of assembly language. It was developed in the USA in 1956, and was intended to provide an abstract way of specifying scientific computations. The opposition to FORTRAN was strong for reasons similar to those advanced against all subsequent proposals for higher-level abstractions, namely, most programmers believed that a compiler could not produce optimal code relative to hand-coded assembly language. Like most first efforts in programming languages, FORTRAN was seriously flawed, both in details of the language design and more importantly in the lack of support for modern concepts of data and module structuring. Nevertheless, the advantages of the abstraction quickly won over most programmers: quicker and more reliable development, and less machine dependence since register and machine instructions are abstracted away. Because most early computing was on scientific problems, FORTRAN became the standard language in science and engineering, and is only now being replaced by other languages. FORTRAN has undergone extensive modernization (1966, 1977, 1990) to adapt it to the requirements of modern software development. COBOL (COmmon Business Oriented Language) and PL/1 (Programming Language 1) The COBOL language was developed in 1958 for business data processing. The language was designed by a committee consisting of representatives of the US Department of Defense, computer manufacturers 84

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and commercial organizations such as insurance companies. COBOL was intended to be only a short-range solution until a better design could be created; instead, the language as defined quickly became the most widespread language in its field (as FORTRAN has in science), and for a similar reason: the language provides a natural means of expressing computations that are typical in its field. Business data processing is characterized by the need to do relatively simple calculations on vast numbers of complex data records, and COBOL’S data structuring capabilities far surpass those of algorithmic languages like FORTRAN or C. The language PL/1 was created in 1964 as a universal language having all the features of FORTRAN, COBOL and ALGOL. PL/1 has replaced FORTRAN and COBOL on many IBM computers, but this very large language was never widely supported outside of IBM, especially on the mini- and microcomputers that are increasingly used in data processing organizations. ALGOL (ALGOrithmic Language) Of the early programming languages, ALGOL has influenced language design more than any other. Originally designed by an international team for general and scientific applications, it never achieved widespread popularity compared to FORTRAN because of the support that FORTRAN received from most computer manufacturers. The first version of ALGOL was published in 1958; the revised version ALGOL 60 was extensively used in computer science research and implemented on many computers, especially in Europe. A third version of the language, ALGOL 68, has been influential among language theorists, though it was never widely implemented. PASCAL The most famous descendent of ALGOL is PASCAL, developed in 1968 by Niklaus Wirth, The motivation for Pascal was to create a language that could be used to demonstrate ideas about type declarations and type checking. As a practical language, PASCAL has one big advantage and one big disadvantage. The original PASCAL compiler was itself written in PASCAL, and thus could easily be ported to any computer. The language spread quickly, especially to the mini- and micro-computers that were then being constructed. Unfortunately, the PASCAL language is too small. The standard language has no facilities whatsoever for dividing a program into modules on separate files, and thus cannot be used for programs larger than several thousand lines. Practical compilers for PASCAL support decomposition into modules, but there is no standard method so large programs are not portable. 85

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LISP (LISt Processing) This language first made its appearance in 1958. It was designed for Artificial Intelligence (AI) research. The most obvious difference between this language and other languages is that the basic and only type of data is the list, denoted by a sequence of items enclosed by parentheses. LISP programs themselves are written as a set of lists, so that LISP has the unique ability to modify itself, and hence grow on its own. LOGO This language derived from LISP appeared in 1969. It is a very simple language to learn and was designed to be used by very young children. It encourages a logical, structured use of language and has most often been used to control a special robot called a turtle and to teach mathematical concepts such as angles. C C was developed by Dennis Ritchie in 1972 as an implementation language for the UNIX operating system. Operating systems were traditionally written in assembly language because high-level languages were considered inefficient. C abstracts away the details of assembly language programming by offering structured control statements and data structures (arrays and records), while at the same time it retains all the flexibility of low-level programming in assembly language (pointers and bit-level operations). Since UNIX was readily available to universities, and since it is written in a portable language rather than in raw assembly language, it quickly became the system of choice in academic and research institutions. When new computers and applications moved from these institutions to the commercial marketplace, they took UNIX and C with them. C is designed to be close to assembly language so it is extremely flexible; the problem is that this flexibility makes it extremely easy to write programs with obscure bugs because unsafe constructs are not checked by the compiler as they would be in Pascal. C is a sharp tool when used expertly on small programs, but can cause serious trouble when used on large software systems developed by teams of varying ability. In the late 1970’s and early 1980’s, a new programming method was being developed. It was known as Object Oriented Programming, or OOP. Objects are pieces of data that can be packaged and manipulated by the programmer. Bjarne Stroustroup liked this method and developed extensions to C known as “C With Classes”. This set of extensions developed into the full-featured language C++, which was released in 1983. 86

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C++ was designed to organize the raw power of C using OOP, but maintain the speed of C and be able to run on many different types of computers. Writing 7. Summarize the information on different high-level computer languages by completing the table below. LANGUAGE

DEVELOPED

FUNCTION

CHARACTERISTIC

FORTRAN 1968 To support Unix operating system Combines features of FORTRAN, COBOL, ALGOL ALGOL 1958 Is used for artificial intelligence research Derived from LISP

8. Try to find information in the Internet about two programming languages that were not mentioned in the text and add boxes to the table. 9. Using the completed table, decide which language would best suit these needs. Then compare the answers. 1. A language for students learning to program for the first time. 2. A language for engineering students who have access to a university mainframe computer. 3. A language for software for an insurance company. The programs allow them to perform a wide range of mathematical calculations as well as keep files on policy holders. 4. A language which helps students with little computing experience to identify any mistakes they have made when keying in programs. 5. A language suitable for university level students and which has a wide range of applications. 6. A simple language for use on microcomputers. 7. A language used for writing software for a large commercial concern. 87

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8. A language which allows data to be input while the program is running. 9. A language for scientists who have to use formulas frequently. 10. A general purpose language suitable for small businesses. 11. A language for controlling a simple robot. 12. A language for research in artificial intelligence. 10. Read the text. Punctuate it as you think appropriate. Informatics is the science of information engineering of information systems and information processing it is the multidisciplinary field which deals with the information analysis its collection transmission storage utilization and dissemination the field is closely related to computer sciences as well as communications mathematics electronics philosophy management and library science informatics and information sciences became popular since the first digital information devices were introduced modern organizations exchange the information using information transmission devices and their programming systems this results the high demand of IT specialists who understand the latest information and communication technology issues develop the software and solve practical engineering problem the study field also deals with the ways how the information is used and affects human welfare there are many contributing disciplines such as computer science communication studies complex systems information theory information technology didactics of informatics and etc.

UNIT 2 Before you read 1. The term “Java” refers to three things: • an island in Indonesia; • a cup of coffee, in American slang; • a language for Internet applications. But what exactly is Java? Try to answer these questions. 1. What is Java, in the world of computers? 2. Have you seen the effects of Java programs on web pages? 2. Choose the proper term according to the given definition. a) send programs or data from a central computer to a remote terminal or PC. (compute, download, execute) 88

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b) a program which converts source programs into machine code. (compiler, instruction, device) c) describing a computer architecture in which all processes, files, I/O operations, etc., are represented as objects (i.e. data structures in memory that may be manipulated by hardware and software). (pen-based, object-oriented, network-compatible) d) an application of computer technology that allows the capture, manipulation and presentation of different types of data e.g. text, graphics, video, animation, sound etc. (machine code, memory, multimedia) 3. Decide if these statements about Java are true or false. Correct the false ones. 1. Java was invented by Microsoft. 2. Small applications written in Java are called “applets”. 3. With the interpreter, a program is first converted into Java bytecodes. 4. Java is not compatible with most computing platforms. 5. Java doesn’t let you watch animated characters on your web pages. Reading 4. Read the text and check your answers. What is Java? Java is a programming language developed by Sun Microsystems which is specially designed to run on the web. When you see a web page that uses Java, a small program called “applet” is executed automatically. Java applets let you watch animated characters and moving, text, play music and interact with information on the screen.

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Java is an object-oriented language similar to C++, but it is more dynamic and simplified to eliminate possible programming errors. A Java program is both compiled and interpreted. First the source code (file with a Java extension) is compiled and converted into a format called bytecode (file with a class extension), which can then be executed by a Java interpreter. Compiled Java code can run on most computers because there are Java interpreters, known as Java Virtual Machines, for most operating systems, including MacOS, Windows, UNIX. Java is multi-threaded. A Java program can have multiple threads (parts), i.e. many different things processing independently and continuously. People are excited about Java because it lets you create moving images and animated drawings. You can also create graphical objects (e.g. bar charts, graphs, diagrams) and new “controls” (e.g. buttons, check boxes, pushbuttons with special properties). A web page that uses Java can have inline sounds that play in real-time, music that plays in the background, cartoon style animations, real-time video and interactive games. One alternative technology is ActiveX, the Microsoft product for including multimedia effects on web pages. Another competitor is Macromedia’s Shockwave, a plug-in that lets you animate pictures, add sound and even make interactive pages so that people can play games on websites. (adapted from http://java.sun.com/)

Grammar review 5. Arrange the following words into sentences to find out some new facts about Java. 1. C++, a, simple, Java, is, object-oriented, based, language, on. 2. Java, the, for, idea, in, started, 1990. 3. On, later, language, this, adapted, was, the, Internet, to. 4. 1.0, the, of, version, was, Java, introduced, officially, 1995, May, in. 5. Application, first, the, with, major, created, was, Java, browser, HotJava, the. 6. Java, hot, a, is, today, computer, on, that, technology, any, runs. 6. Find grammar mistakes in these sentences and correct them. 1. Originally informatics was ones of the numerous fields of cybernetics. 2. At present informatics is a independent multifield science. 3. A computer program is the statement of an algorithm in some welldefined language, although the algorithm itself is a mental concept that exist independently of any representation. 90

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4. Before studying computer systems it necessary to distinguish between computers and calculators. 5. The application of computers scientific problems has become later than the original business applications. 6. The results of the calculations or other processes perform by the computer are obtained from the output devices. 7. When the information in an image is expressed in digital form, it can be manipulated mathematically rather than optically. 7. In the following sentences some errors with the use of articles appear. Underline errors; write what you think the correct form should be as in the example. The / A computer carries out the instructions given in “absolute” machine languages. But people-programmers don’t write programs in machine languages, but in programming languages – i.e., the tools, programmers use to create programs, just as the English and other spoken languages are the tools the writers use to create books. Perhaps first thing we need to know the distinction between assembly languages and all other programming languages, collectively called high-level languages. Writing program in the assembly language is a exceedingly long and tedious process – a medium-sized program has about 20,000 machinelanguage instructions in it. A large and complex program can consist of the hundreds of the thousands of separate machine-language instructions, highlevel languages are designed to eliminate the tedium and error-prone nature of assembly language by letting the computer do itself much of a work of generating the detailed machine-language instructions.

UNIT 3 Before you read 1. Discuss these questions with your partner. 1. How many programming languages do you know? 2. Why are some programming languages more popular than the others? 2. Read the text. The six sentences below have been removed. Decide where they should go. A. In the thirty-five years since C was popularized, there have been enormous leaps in the design of software and operating systems, he says. 91

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B. Google wants to change the way the world writes software. C. At that point, there's little motivation to learn and implement new languages. D. According to Meyerovich's data, programmers see ActionScript as easy to use. E. Most programmers learn three to four languages, the researchers say, but then stop. Part of the problem, he says, is that language designers don't always have practical objectives. F. Part of the problem, he says, is that language designers don't always have practical objectives. Why Do Some Programming Languages Live and Others Die?

1) .......... In recent years, the search giant has unveiled two new programming languages that seek to improve on some of the most widely used languages on the planet. With a language called Go, it seeks to give the world a replacement for the venerable languages C and C++, providing a more nimble means of building really big software platforms inside data centers. And with Dart, it hopes to replace JavaScript, improving the way we build software that's run in our web browsers. But no matter how impressive these new languages are, you have to wonder how long it will take them to really catch on – if they do at all. After all, new programming languages arrive all the time. But few ever reach a wide audience. At Princeton and the University of California at Berkeley, two researchers are trying to shed some light on why some programming languages hit the big time but most others don't. In what they call a “side 92

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project”, Leo Meyerovich and Ari Rabkin have polled tens of thousands of programmers, and they're combing through over 300,000 computing projects at the popular code repository SourceForge – all in an effort to determine why old languages still reign supreme. “Why have we not been reliably able to improve on C?” Rabkin asks. 2) .......... But although C has been beefed up and other new languages have been very successful during that time, C is still a mainstay. 3) .......... “There's a tendency in academics of trying to solve a problem when no one actually ever had that problem”, said Rabkin, who recently received his computer science PhD at Berkeley and is now at Princeton working on a post-doc. He says that academics are so often determined to build a language that stands out from the crowd, without thinking about what's needed to actually make it useful. In some cases, he says, they fail with the simplest of things, like documentation for their language. In other cases, he says, designers will keep adding new features to a language and effectively overload the engineers who are trying to use it. “Maybe the solution isn't entirely technical”, Meyerovich says. “We need to start building more ‘socially aware’ languages”. Yang Zhang, co-founder of analytics outfit Slice-Data and an MIT PhD dropout, is among that many who have jumped behind a new-age programming language called Scala, but he acknowledges that the language was originally hampered by poor documentation and support from the designers. “I was a much bigger masochist back then”, he says of fighting to learn the language in 2006. Meyerovich also says the data he and Rabkin are collecting also indicate that programmers aren't always taking the time to really learn a language when they start using it – and that this trips them up down the road. An example, he says, is ActionScript, an object-oriented language developed by Adobe. 4) .......... But, he says, when they start doing something new with it – moving from, say, media development to game development – they run into problems. Another issue is complacency. 5) .......... “Over time, you'd expect that as developers get older, they'd get more wisdom; they'd learn more languages”, Meyerovich says. “We've found that's not true. They plateau”. Part of the problem is that by the time they hit 35 to 40 years old, they're often moving from hands-on coding to managing other programmers. 6) .......... Meyerovich believes that the language is an issue that the development community as a whole is still struggling to acknowledge. As he and Rabkin 93

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plow through the data – much of which is sortable online – they hope to develop new insights into not only the causes over the problem, but also how to solve it. “This is a hot-button issue I didn't even think we'd be looking at when we went through this data”, Meyerovich says. (abridged from http://www.wired.com/wiredenterprise/2012/06/berkeley-programminglanguages/)

3. Look through the text and find the word that means the same as: 1) has disclosed, revealed; showed publicity for the first time; 2) deserving respect because of age, character, associations; 3) have received votes; 4) to prevail; 5) to put too great a load on; 6) causes to stumble or make a false step; 7) admits the truth, existence or reality of. 4. Match the words to make phrases. 1) venerable 2) computing 3) media 4) sortable 5) hot-button 6) practical

a) issue b) online c) objectives d) projects e) languages f) developments

5. Which of the following sentences correspond to the text content? 1. C++ is the most important language. 2. C++ is favored by the programmers who are looking for the utmost efficiency in this language, for writing programs that need to be tight and efficient. 3. Some programming languages hit the big time but most others don't. 4. Programmers need to start building more ‘socially aware’ languages. 5. Personal taste and convenience play a major part in the selection of a programming language. 6. Most programmers learn three to four languages then stop. 6. Complete the text with the given expressions: a) interprets; b) microprocessor; c) functional unit; 94

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d) control unit; e) execution units; f) architecture; g) chip; h) central processing unit. Central processing unit The 1) ________ (CPU) is the part of a computer that 2) ________ and carries out the instructions contained in the software. In most CPUs, this task is divided between a 3) ________ that directs program flow and one or more 4) ________ that perform operations on data. Almost always, a collection of registers is included to hold operands and intermediate results. When every part of a CPU is on a single physical integrated circuit, it is called a 5) ________ . Practically all CPUs manufactured today are microprocessors. The term CPU is often used vaguely to include other centrally important parts of a computer such as caches and input/output controllers, especially when those functions are on the same microprocessor 6) ________ as the CPU. Manufacturers of desktop computers often erroneously describe the entire personal computer (the system unit or sometimes white box including the computer case and the computer hardware it contains) as the CPU. Rather, the CPU, as a 7) ________ , is that part of the computer which actually executes the instructions (add, subtract, shift, fetch, etc.). A family of CPU designs is often referred to as a CPU 8) ________ . Grammar review 7. Find the word which shouldn’t be in the sentence. 1. Prolog has been designed to exploit the communicational of possibilities associated with the sequential execution of programs. a) of b) with c) to 2. Software developers need speed and flexibility to create applications for a wide array of client that needs. a) for b) of c) that 3. Delphi’s two-way tools help you optimize your programming time by switching on effortlessly between visual design and the underlying source code. a) on b) between c) by 4. Modern computers there are much faster, more complex, multifunctional and useful than most people dreamed of 50 years ago. a) are b) there c) than 95

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5. The most smallest point of computer data is called a bit – a contraction of the expression binary digit. a) is b) most c) a 6. For you to understand your computer’s processor you must to remember that the computer’s memory is just a temporary space, a scratch pad, a workbench, a chalkboard where the computer scribbles while the work is being done. a) that b) to c) being 7. Disk storage is the computer’s library where it is keeps its instructions programs, its raw material and any other information. a) is b) keeps c) it 8. Condense the following sentences according to the model using the Participle, the Gerund and the Infinitive instead of subordinate clauses. Model: 1. ALGOL 60 which was published then has many advantages. ALGOL 60 published then has many advantages. 2. The system which provides the user with necessary tools will be very convenient. The system providing the user with necessary tools will be very convenient. 3. When you answer this question you must think. On answering this question you must think. 4. I expect that every identifier is declared to this block. I expect every identifier to be declared to this block. 5. It is known that every identifier is declared to this block. Every identifier is known to be declared to this block. 1. We have a system which enables us to use it for the solution of problems. 2. Perhaps the language which is patterned after the ALGOL 60 description would be desirable. 3. They are also teaching the language to the undergraduates at Stanford who take the courses on digital computers. 4. When you are translating from ALGOL to BALGOL you should determine which variables are global. 5. Identifiers which occur within a block and which are not declared to this block will be nonlocal to it. 6. It was reported that scientists could easily get results. 7. He thought that they would pattern the language after ALGOL. 96

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8. It is known that you should take considerable care in translating from one language into another. 9. They expect that these variables assume the values true or false. 10. You know that every block in ALGOL automatically produces a new level of nomenclature. 9. Read these quotations. Change them into indirect statements. • “How many hardware engineers does it take to change a light bulb?” • None: “We’ll fix it in software”. • “How many software engineers does it take to change a light bulb?” • None: “We’ll document it in the manual”. • “How many Microsoft engineers does it take to change a light bulb?” • None: “Let’s define darkness as the industry standard”. • “How many tech writers does it take to change a light bulb?” • None: “The user can work it out”. 10. Tick the correct sentences. Change the incorrect ones. 1. He asked me whether was popular that scientist. 2. He did not know if they have covered all the problems. 3. I want to know whether we have another alternative. 4. Nick asked me count the people in the room. 5. He asked that when he would come. 6. The teacher asked me do not square the number. 7. The professor asked him whom that device had been invented by. 8. I was not sure if she would needed my help. 9. The student asked in what case the room in the hostel will be given to that postgraduate. 10. He is going to find out whether the library will be open on Sunday. Speaking 11. Read the statements below. Which do you agree with more? Why? Learning a programming language is like learning any natural language. The only difference is that you are communicating with a machine instead of another person. I get annoyed when I hear people comparing programming languages with natural languages. They have almost nothing in common. 12. Make up small groups. You are going to do poster presentations on the programming languages discussed (choose one or two languages). You will have 10 minutes to make your posters. 97

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Feedback Have you used vocabulary connected with programming? checked your spelling, punctuation and grammar? reviewed sentence structure? Just for Fun A language instructor was explaining to her class that French nouns, unlike their English counterparts, are grammatically designated as masculine or feminine. Things like 'chalk' or 'pencil', she described, would have a gender association although in English these words were neutral. Puzzled, one student raised his hand and asked, “What gender is a computer?” The teacher wasn't certain which it was, and so divided the class into two groups and asked them to decide if a computer should be masculine or feminine. One group was composed of the women in the class, and the other, of men. Both groups were asked to give four reasons for their recommendation. The men, decided that computers should definitely be referred to in the feminine gender because: 1) no one but their creator understands their internal logic; 2) the native language they use to communicate with other computers is incomprehensible to everyone else; 3) even your smallest mistakes are stored in long-term memory for later retrieval; 4) as soon as you make a commitment to one, you find yourself spending half your paycheck on accessories for it. The group of women, on the other hand, concluded that computers should be referred to in the masculine gender because: 1) in order to get their attention, you have to turn them on; 2) they have a lot of data but are still clueless; 3) they are supposed to help you solve your problems, but half the time they ARE the problem; 4) as soon as you commit to one, you realize that, if you had waited a little longer, you might have had a better model.

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Unsolved problems Learning objectives: In this module you will: • work with vocabulary • practice questions and answers • dispute the statements • write a dialogue • prepare a short presentation • check spelling, punctuation and grammar

UNIT 1 Before you read 1. Discuss these questions with your partner. 1. What do you know about unsolved problems of mathematics formulated by D. Hilbert in 1900? 2. Which of them are solved? 3. How is it possible to prove that certain problem cannot be solved? 2. Match these words with their definitions. 1) substantiate 2) devastate 3) rigour 4) veil 5) hesitate 6) edifice

a) something that hides or disguises b) to show signs of uncertainty or unwillingness in speech or action c) something built up in the mind d) to give facts to support (a claim, charge, etc.) e) to ruin; to make desolate f) sternness; strictness

Reading 3. Read the text. The five sentences below have been removed. Decide where they should go. A. Nevertheless it was not over. 99

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B. The paradox is connected with the idea of a set of all the sets that are not members of themselves. C. Hilbert’s lecture included a long preamble in which he discussed the nature of mathematics and its role in the progress of the other sciences. D. Hilbert claimed that there was nothing unknowable, no “ignorabimus” in mathematics. E. Another mathematical genius, Kurt Gödel (1906–1970) proved that any consistent axiomatic system for arithmetic was necessarily incomplete, that is to say, there would be true properties which could never be demonstrated. David Hilbert’s challenge Just over a century ago, the mathematician David Hilbert (1862–1942) gave a veritable tour de force at the International Congress of Mathematicians in Paris. At 9 o’clock on the morning of 8 August 1900, in the main lecture theatre the Faculty of Sciences at the Sorbonne, David Hilbert addressed the following words to his expectant audience: Who would not be happy if he could lift the veil that hides the future to take a look at the progress of our science and the secrets of its further developments in future centuries? In so rich and vast field as mathematics, what will be the objectives and what will be the guide for mathematicians’ thought in future times? What will be the new developments and new methods in the new century? 1) .......... Although his list contained 23 problems, he only had time to present 10 of them. At that time, Hilbert, a professor at the University of Göttingen, was considered Germany’s leading mathematician. In the invitation he received to give the inaugural lecture in Paris, his colleague Herman Kosky (1864– 1909), asked him to take a look at the future. But Hilbert finished preparing his lecture at the last minute, and hesitated over the title, which did not arrive on time for the programme, with the result that his lecture was finally held on the third day of the Congress. Hilbert’s effort was in line with the view held by 19th century mathematicians that it was necessary to bring rigour and certainty to the edifice of mathematics. Indeed, in his lecture Hilbert made an explicit reference to the nature of mathematical problems and the role of mathematics in science. 100

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2) .......... Everything could be substantiated and explained in a logical way. If there is a problem, mathematicians will be able to find its solution. In Paris, Hilbert faced a different vision, that of the heroes of the age, the French mathematician Henri Poincaré, who favoured a more intuitive approach. (Poincaré referred to Cantor’s set theory as “a disease that mathematics will eventually recover from with time”.) In any event, nobody wanted to be cast out of the paradise of set theory that George Cantor (1845–1918) had created, and his wonderful conception of the different infinities. Many years later, on 8 September 1930 in Königsberg, at the Congress of the Association of German Scientists and Physicians, David Hilbert gave a lecture of which four minutes was broadcasted by radio. Gilbert ended with his claim: “in opposition to the ignorabimus, we offer our slogan: we must know, we will know”. One landmark in this story is the work of the Italian mathematician Giuseppe Peano (1858–1932), who laid the foundations of arithmetic with a collection of axioms and rules. But the question that arose after the Paris Congress was simple but devastating: is this system complete and consistent? Bertrand Russell’s (1872–1970) Barber paradox cast it into doubt: in a town there is a barber who only shaves the towns’ male residents who do not shave themselves: who shaves the barber? 3) .......... Such a set, if it exists, will be a member of itself, if and only if it is not a member of itself. A breach had opened up in the apparently solid mathematical edifice, as no demonstration could be reliable if it is based on this logic. One of the most exciting periods in the history of the discipline had begun. The so-called Hilbert programme, which aimed to create a formal system for the mathematics which contained a demonstration of the consistency (i.e. not leading to contradictions), completeness (i.e. all truth can be demonstrated) and decidability (i.e. it must be possible to deduce a formula from the axioms by applying the right algorithms), was cut short almost at the same time as he was delivering his war cry against the ignorabimus in 1930. 4) .......... John von Neumann (1903–1957) said after Gödel’s presentation of his findings: “It’s over”. 5) .......... Nevertheless it was not over. Rather, after those dramatic events other heroic mathematicians such as von Neumann himself and Alan Turing (1912–1954) based themselves on this new approach to develop computers and scientific computing as we know them today. Hilbert's 23 Mathematical Problems Problem 1. Cantor's problem of the cardinal number of the continuum. 101

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Problem 2. The compatibility of the arithmetic axioms. Problem 3. The equality of two volumes of two tetrahedra of equal bases and equal altitudes. Problem 4. Problem of the straight line as the shortest distance between two points. Problem 5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group. (i.e., are continuous groups automatically differential groups?) Problem 6. Mathematical treatment of the axioms of physics. Problem 7. Irrationality and transcendence of certain numbers. Problem 8. Problems (with the distribution) of prime numbers. Problem 9. Proof of the most general law of reciprocity in any number field. Problem 10. Determination of the solvability of a Diophantine equation. Problem 11. Quadratic forms with any algebraic numerical coefficients. Problem 12. Extension of Kronecker's theorem on Abelian fields. Problem 13. Impossibility of the solution of the general equation of the 7th degree. Problem 14. Proof of the finiteness of certain complete systems of functions. Problem 15. Rigorous foundation of Schubert's calculus. Problem 16. Problem of the topology of algebraic curves and surfaces. Problem 17. Expression of definite forms by squares. Problem 18. Building space from congruent polyhedra. Problem 19. Are the solutions of regular problems in the calculus of variations always necessarily analytic? Problem 20. The general problem of boundary curves. Problem 21. Proof of the existence of linear differential equations having a prescribed monodromic group. Problem 22. Uniformization of analytic relations by means of automorphic functions. Problem 23. Further development of the methods of the calculus of variations. (abridged from http://www.fgcsic.es/lychnos/en_EN/articles/the_millennium_problems)

4. Decide if the following statements are true or false, or there is no information in the text. a) true b) false c) no information 1. D. Hilbert’s report was clear and vivid, notwithstanding scientific. 102

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2. D. Hilbert spoke about the requirements for the solution of the problems. 3. D. Hilbert thought that the International Congress of Mathematicians was the appropriate and right place for the review and appraisal of the development and achievements of maths. 4. D. Hilbert presented 12 problems. 5. The Italian mathematician Giuseppe Peano laid the foundations of arithmetic with a collection of axioms and rules. 6. Henri Poincaré proved that any consistent axiomatic system for arithmetic was necessarily incomplete. 5. Here are the answers to some questions about the article. What are the questions? 1. It was David Hilbert. 2. He was a professor at the University of Göttingen. 3. Herman Kosky. 4. In 1930 in Königsberg, at the Congress of the Association of German Scientists and Physicians. 5. John von Neumann. 6. Find these numbers in the text. What do they describe? 1. 23. 2. 10. 3. 19. 4. 4. 5. 1900. Speaking 7. Answer the questions. 1. What do the following groups of Hilbert’s problems deal with? • 1–6; • 7–15; • 16–23. 2. Did D. Hilbert reveal great insight in selecting the problems? 3. Is there any problem in the list having anything to do with the unsolved problems of antiquity, with algebra, with number theory? 4. Which of them consists, in fact, of several problems? 5. Do modern mathematicians give a short “yes-no” answer to the (3–7) problems? 6. Which problems are solved? 103

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7. For which of them the impossibility of the solution is proved? 8. What is the contribution of the Russian and Soviet mathematicians to the solution of the problems under study? 9. Are the unsolved problems the very essence of mathematics? 8. Prepare a short presentation “Unsolved Math Problems of D. Hilbert”. Try to give convincing arguments of the significance of both the Second Congress of Mathematicians in 1900 and the report of D. Hilbert for the further development of maths. Remember to: • use simple examples that everyone can understand; • speak slowly and clearly. Speaking tips: • speak to your audience, not your notes; • pause between sections; • speak calmly, but not too softly. Grammar review 9. Fill in the gaps with the correct form of the verbs in brackets. 1. If they .......... (try) to use this method, they would come across a lot of difficulties. 2. He might have made that great discovery earlier if he .......... (had) better conditions for work. 3. Had they obtained the information required, they .......... (turn) it over to us. 4. If the distance between the Sun and the Earth .......... (be) less, the temperature on the surface of the Earth would be much higher. 5. I wish they .......... (finish) writing the program next week. 6. Whatever .......... (be) the error we must detect it. 7. The Earth behaves as though it .......... (be) a large magnet. 8. He suggested that during the experiment the direction of light .......... (be) slightly altered. 9. It is essential that the behaviour of these particles .......... (be) carefully studied. 10. They will discuss this information provided they .......... (know) the result of our work. 10. Tick the correct sentences. Change the incorrect ones. 1. It is important that the information be properly used. 104

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2. It looks as if the computer were out of order. 3. If your teacher introduces the law of exponents at the previous lecture, you could have made use of it for evaluating the result. 4. He could readily evaluate the formula, provided he applied his knowledge of logarithms. 5. But for the teacher’s help, the students would not have solved the problem. 6. I'll give him all the details of the experiment as soon as he will ask me about it. 7. Whatever be the nature of the phenomenon described, it ought be discovered. 8. He insisted that every possible chance be taken into consideration. 9. We explained everything lest they should spoke against our suggestion. 10. The Earth behave as though it were a large magnet.

UNIT 2 1. Match these words with their definitions. 1) defy 2) distort 3) conjecture 4) conceive 5) grapple 6) elevate

a) to form in the mind b) to struggle with something at close quarters c) to raise; to make higher and better d) guess; guessing e) to give a false account of; twist out of the truth f) to offer difficulties that cannot be overcome

2. Match the words to make phrases 1) consciousness of 2) three-dimensional 3) spectacular 4) differential 5) formation of

a) singularities b) equation c) the general public d) space e) result

Reading 3. Read the text. First Clay Mathematics Institute Millennium Prize Announced Today 105

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Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman March 18, 2010 The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture. The citation for the award reads: The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. The Poincaré conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude. James Carlson, President of CMI, said today, “resolution of the Poincaré conjecture by Grigoriy Perelman brings to a close the century-long quest for the solution. It is a major advance in the history of mathematics that will long be remembered”. Poincaré's conjecture and Perelman's proof Formulated in 1904 by the French mathematician Henri Poincaré, the conjecture is fundamental to achieving an understanding of threedimensional shapes (compact manifolds). The simplest of these shapes is the three-dimensional sphere. It is contained in four-dimensional space, and is defined as the set of points at a fixed distance from a given point, just as the two-dimensional sphere (skin of an orange or surface of the earth) is defined as the set of points in three-dimensional space at a fixed distance from a given point (the center). Since we cannot directly visualize objects in n-dimensional space, Poincaré asked whether there is a test for recognizing when a shape is the three-sphere by performing measurements and other operations inside the shape. The goal was to recognize all three-spheres even though they may 106

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be highly distorted. Poincaré found the right test (simple connectivity, see below). However, no one before Perelman was able to show that the test guaranteed that the given shape was in fact a three-sphere. In the last century, there were many attempts to prove, and also to disprove, the Poincaré conjecture using the methods of topology. Around 1982, however, a new line of attack was opened. This was the Ricci flow method pioneered and developed by Richard Hamilton. It was based on a differential equation related to the one introduced by Joseph Fourier 160 years earlier to study the conduction of heat. With the Ricci flow equation, Hamilton obtained a series of spectacular results in geometry. However, progress in applying it to the conjecture eventually came to a standstill, largely because formation of singularities, akin to formation of black holes in the evolution of the cosmos, defied mathematical understanding. Perelman's breakthrough proof of the Poincaré conjecture was made possible by a number of new elements. He achieved a complete understanding of singularity formation in Ricci flow, as well as the way parts of the shape collapse onto lower-dimensional spaces. He introduced a new quantity, the entropy, which instead of measuring disorder at the atomic level, as in the classical theory of heat exchange, measures disorder in the global geometry of the space. This new entropy, like the thermodynamic quantity, increases as time passes. Perelman also introduced a related local quantity, the L-functional, and he used the theories originated by Cheeger and Aleksandrov to understand limits of spaces changing under Ricci flow. He showed that the time between formation of singularities could not become smaller and smaller, with singularities becoming spaced so closely – infinitesimally close – that the Ricci flow method would no longer apply. Perelman deployed his new ideas and methods with great technical mastery and described the results he obtained with elegant brevity. Mathematics has been deeply enriched. (abridged from http://www.claymath.org/sites/default/files/millenniumprizefull)

4. Which of the following sentences correspond to the text content? 1. The Clay Mathematics Institute formulated seven mathematical problems in 1982. 2. We can directly visualize objects in n-dimensional space. 3. No one before Perelman was able to show that the test guaranteed that the given shape was in fact a three-sphere. 4. Perelman described the results he obtained with elegant brevity. 5. Poincaré achieved a complete understanding of singularity formation in Ricci flow. 107

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5. Here are the answers to some questions about the article. What are the questions? 1. In 1904. 2. It is one of the seven Millennium Prize Problems established by CMI in 2000. 3. There were many attempts to prove, and also to disprove, the Poincaré conjecture. 4. This was the Ricci flow method. 5. Joseph Fourier introduced it 160 years earlier. 6. He achieved a complete understanding of singularity formation in Ricci flow. Writing Dialogue Writing A dialogue is a conversation or talk. 6. Reconstruct the article into a dialogue. Write questions and correct answers. 7. Read the text. Punctuate it as you think appropriate. The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000 as of March 2013 six of the problems remain unsolved a correct solution to any of the problems results in a US$ 1,000,000 prize sometimes called a Millennium Prizebeing awarded by the institute the Poincaré conjecture the only Millennium Prize Problem to be solved so far was solved by Grigoriy Perelman but he declined the award in 2010 8. Fill in the missing words in the sentences below. Choose from the following: curves, challenge, conchoid, counterparts, ingenuity, straightedge, insoluble. 1. With a .......... we may draw a line determined by any two points. 2. Every generation of mathematicians ever since the Greek times has to seek a proof that certain problems are solvable or .......... in principle. 3. Math discoveries and the use of higher geometric .......... to effect the solution of the problem show the great .......... of the Greek geometers. 4. Nicomedes invented a special curve, the .......... , with which he could trisect any angle. 5. A ruler and a compass are the physical .......... suggesting the concepts of a straight line and a circle. 108

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6. The conviction of the solvability of every math problem is a powerful  .......... and stimulus to the researcher. Grammar review 9. Choose the proper equivalents of the modal verbs. 1. She (ей придется) to give a reason and possible justification for the restriction. 2. We (нам предстоит) to find a good approximation to the given value. 3. He (ему следует) specify the conditions of the experiment. 4. I (в состоянии) to solve this difficult problem myself. 5. They (им нужно) to check all the calculations again. 6. The students (должны) to appreciate the ancient math in a proper way. 10. Find grammar mistakes in these sentences and correct them. 1. Every math problem must be settled either in the form of a direct answer to the question posed, or by the proof of the impossibility of it’s solution. 2. Every generation of mathematicians ever since the Greek times on has seek a proof that certain problems are solvable or insolvable in principle. 3. Math offer an abundance of unsolved problems. 4. The mathematicians of past centuries will be accustomed to devoting themselves to the solution of difficult particular problems with passionate zeal. 5. What new methods in the wide and rich field of math thought the new centuries can disclose? 6. There are important problems, problems in transonic flow, for example, which possess solutions only for exceptional values of the data; thus the solutions does not depend continuously on the data even when they exist. 7. It should be said that a proofs constitutes the principal part of the math method. 11. Make sentences with these words. 1) right, all, be, can, not, math, away, problems, solved. 2) math, the, solutions, for, the, search, the, led, developments, Greeks, to, in, novel, the. 3) problems, an, math, offers, of, unsolved, abundance. 4) significant, is, for, it, raised, and, not, answer, did, Greek, math, the, questions. 5) teaches, of, history, the, of, science, development, continuity, the. 6) math, of, is, foundation, the, all, of, exact, phenomena, natural, knowledge. 109

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Speaking 12. Agree or disagree with the following statements. Use the opening phrases suggested. Repeat the statement and develop it further. Agreement Disagreement I fully agree to it. Not quite so, I am afraid. This is the case. I don’t think this is just the case. Quite so. Absolutely correct. On the contrary. Far from it. 1. The ancient Greeks are given credit for posing famous unsolved construction problems that challenge mathematicians and amateurs alike even today. 2. It is difficult, often impossible, to judge the value of a problem correctly in advance, for the final award depends upon the gain which science obtains from the problem. 3. The deep significance of certain problems for the advance of math science, in general, and the important role which they play in the work of the individual investigator are not to be denied. 4. It is by the solution of problems that the researcher tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon. 5. A math theory is not to be considered completed until you made it so clear that you can explain it to the first man whom you meet in the street. 6. It often happens that the same special problem finds application in the most diverse and unrelated branches of math. 13. Discussion 1. In mathematics the conviction that a definite math problem can necessarily be solved must be supported by a proof either in the form of a direct answer to the question posed or by the proof of the impossibility of the solution. What about other sciences? 2. It is one thing to say that a problem is not solved yet and another thing to say that it is impossible to solve it. How is it possible to prove a thing impossible? 3. Among professional mathematicians asking questions rates almost as high as answering them. Why? 4. There are two kinds of math problems: one is so easy that it is not worth doing and the other so difficult that it can’t be done. Give some examples. 5. What is more difficult to prove: the possibility (the existence) of a solution of some problem or the impossibility (the nonexistence) of the solution sought? 110

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UNIT 3 Before you read 1. Match these words with their definitions. 1) break through 2) annual 3) induce 4) feasible 5) incompatible 6) ingenuity

a) coming or happening every year b) to persuade or influence c) unable to exist in harmony; inconsistent d) cleverness and skill e) that can be done f) a major achievement

2. Match the words to make phrases. 1) millennium 2) promising 3) major 4) impressively sounding 5) housing 6) feasible 7) lack of

a) way b) ingenuity c) mathematicians d) breakthrough e) accomodation f) prizes g) problems

Reading 3. Read the text. The Millennium Prize Problems One down, six to go. That is the current situation with the famous seven mathematical problems, known as the millennium problems. They were formulated by the Clay Mathematics Institute (CMI) in 2000, and the Institute reserved a prize of 1,000,000 US$ for the correct solution of each of these problems. The CMI is a private non-profit foundation founded in 1998 by the Boston businessman Landon T. Clay. The purpose of the CMI is to promote the science of mathematics, mainly by awarding research fellowships to promising mathematicians. Besides the seven prizes for the millennium problems, they also give two annual awards: the Clay Research Award for 111

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major breakthroughs in mathematics and the Clay Olympiad Scholar Award for the most creative solution to a problem on the United States of America Mathematical Olympiad. Many people ask what induced the CMI to offer the millennium prizes. A hundred years before, the mathematicians entered the 20th century with 23 Hilbert problems. They were open mathematical problems stated by David Hilbert, one of the greatest mathematicians of (not only) that time, in 1900. Hilbert chose them as those of the open problems in mathematics that were by his judgment the most important ones for the future development of mathematics. Similarly, mathematicians at the turn of the millennium decided to formulate a number of hard enough to solve, but at the same time important open mathematical problems and these became the Millennium Prize Problems. There are many other open problems in mathematics, and most are hard to solve, but the choice fell on the ones we shall name a few lines below because their solution would have the largest impact not only in mathematics, but also in other sciences. Another reason to announce the impressively sounding prizes was also to call the attention of a more general public to the fact that mathematics is a science that is very much alive, with many open, difficult and important problems the mathematicians are trying to solve. • The Riemann Hypothesis; • The Birch and Swinnerton-Dyer Conjecture; • Navier-Stokes Equations; • Yang-Mills theory; • P vs. NP; • The Hodge Conjecture; • The Poincaré Theorem. P vs NP Problem Suppose that you are organizing housing accommodations for a group of four hundred university students. Space is limited and only one hundred of the students will receive places in the dormitory. To complicate matters, the Dean has provided you with a list of pairs of incompatible students, and requested that no pair from this list appear in your final choice. This is an example of what computer scientists call an NP-problem, since it is easy to check if a given choice of one hundred students proposed by a coworker is satisfactory (i.e., no pair taken from your coworker's list also appears on the list from the Dean's office), however the task of generating such a list from scratch seems to be so hard as to be completely impractical. Indeed, the total number of ways of choosing one hundred students from the four hundred 112

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applicants is greater than the number of atoms in the known universe! Thus no future civilization could ever hope to build a supercomputer capable of solving the problem by brute force; that is, by checking every possible combination of 100 students. However, this apparent difficulty may only reflect the lack of ingenuity of your programmer. In fact, one of the outstanding problems in computer science is determining whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure. Problems like the one listed above certainly seem to be of this kind, but so far no one has managed to prove that any of them really are so hard as they appear, i.e., that there really is no feasible way to generate an answer with the help of a computer. Stephen Cook and Leonid Levin formulated the P (i.e., easy to find) versus NP (i.e., easy to check) problem independently in 1971. (abridged from http://www.claymath.org/millennium-problems/millennium-prize-problems)

4. Decide if the following statements are true or false, or there is no information in the text. a) true b) false c) no information 1. Six problems were solved. 2. Seven problems were formulated in the 20th century. 3. 23 problems were stated by Hilbert. 4. There are no other open problems in mathematics. 5. Mathematics is a science that is very much alive, with many open, difficult and important problems the mathematicians are trying to solve. 6. The unsolved problems are the very essence of mathematics. 5. Find these numbers in the text. What do they describe? 1. 6. 2. 1900. 3. 20. 4. 400. 5. 100. 6. 1971. 6. Fill in the gaps with the derivative. Poincaré Conjecture (solved: Grigoriy Perelman, 2002–2003) If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by (1) .......... it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate (2) .......... around a doughnut, 113

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then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is “simply connected”, but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this (3) .......... of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult. Nearly a century passed between its formulation in 1904 by Henri Poincaré and its (4) .......... by Grigoriy Perelman, announced in preprints posted on ArXiv.org in 2002 and 2003. Perelman's solution (5) .......... on Richard Hamlton's theory of Ricci flow, and made use of results on spaces of metrics due to Cheeger, Gromov, and Perelman himself. In these papers Perelman also proved William Thurston's Geometrization Conjecure, a special case of which is the Poincaré conjecture. (adapted from http://www.claymath.org/millenium-problems/poincaré-conjecture)

MOVE _________________ 1 DIRECT ________________ 2 PROPER _ ______________ 3 SOLVE _________________ 4 BASE __________________ 5 7. Fill in the missing words in the sentences below. Choose from the following: asserts, equation, property, prime, solutions, frequency, pattern, applications. Riemann Hypothesis Some numbers have the special (1) .......... that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its (2) .......... The distribution of such prime numbers among all natural numbers does not follow any regular (3) .......... , however the German mathematician G.F.B. Riemann (1826–1866) observed that the (4) .......... of prime numbers is very closely related to the behavior of an elaborate function 114

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ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ... called the Riemann Zeta function. The Riemann hypothesis (5) .......... that all interesting solutions of the (6) .......... ζ(s) = 0 lie on a certain vertical straight line. This has been checked for the first 1,500,000,000 (7) .......... A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of (8) .......... numbers. (adapted from http://www.claymath.org/millenium-problems/riemann-hypothesis)

Grammar review 8. Read the text below and choose the correct word for each space. Yang-Mills and Mass Gap The laws of quantum physics stand (1) .......... the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles (2) .......... structures that also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions (3) .......... at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles (4) .......... on a subtle quantum mechanical property called the “mass gap”: the quantum particles have positive masses, even though the classical waves travel (5) .......... the speed of light. This property has been discovered (6) .......... physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the YangMills theory and a mass gap (7) .......... the introduction of fundamental new ideas both in physics and in mathematics. 1. a) to b) on c) by 2. a) to use b) using c) use 3. a) have been tested b) have tested c) were tested 4. a) depends b) depend c) to depend 5. a) at b) to c) in 6. a) with b) by c) to 7. a) will require b) require c) is required 115

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Feedback Did you have any problems with this Module? What are you going to do to help yourself? Activity Problem Solution 1 2 3 Just for Fun A somewhat advanced society has figured how to package basic knowledge in pill form. A student, needing some learning, goes to the pharmacy and asks what kind of knowledge pills are available. The pharmacist says, “Here's a pill for English literature”. The student takes the pill and swallows it and has new knowledge about English literature! “What else do you have?” asks the student. “Well, I have pills for art history, biology, and world history”, replies the pharmacist. The student asks for these, and swallows them and has new knowledge about those subjects. Then the student asks, “Do you have a pill for math?” The pharmacist says, “Wait just a moment”, goes back into the storeroom, brings back a whopper of a pill, and plunks it on the counter. “I have to take that huge pill for math?” inquires the student. The pharmacist replied, “Well, you know math always was a little hard to swallow”.

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References 1. Activities in algebra. URL: http://www.math.wichita.edu/history/ activities/algebra-act.html. 2. BBC.com. URL: http://www.bbc.co.uk/. 3. Clay Mathematics Institute. URL: http://www.claymath.org/poincare/. 4. Clay Mathematics Institute. URL: http://www.claymath.org/sites/ default/files/millenniumprizefull. 5. English Today: Cambridge Journal on line. URL: /www.journals. cambridge.org/action/display. 6. Estevas, R. English for computer users / R. Estevas. – Cambridge: Cambridge University Press, 2002. – 210 p. 7. Glendinning, E. and McEwan, J. English in Computing / E. Glendinning and J. McEwan. – Edinburgh : Nelson Edinburgh, 1989. – 130 p. 8. Hilberts 23 problems. URL: http://www.cmi.ac.in/~smahanta/hilbert. html. 9. Hornby, A. S. Oxford Advanced Learner’s Dictionary of Current English / A. S. Hornby. – Oxford : Oxford University Press, 1987. – 1042 p. 10. Hornby, A. S. Oxford Advanced Learner’s Dictionary of Current English – 6th edition / A. S. Hornby. – Oxford : Oxford University Press, 2004. – 1540 p. 11. Internet TESL Journal. URL: http://iteslj.org/. 12. Journal of English for Academic purposes. URL: http://www. elsevier.com/wps/find/journaldescription.agents/622440/authorinstructions. 13. Maths-logic-puzzles. URL: http://www.learn-with-math-games. com/support-files/math-logic-puzzles-1.pdf. 14. Oxford Journals. URL: http://www.oxfordjournals.org/our_journals/. 15. Science Clarified. URL: http://www.scienceclarified.com/Ga-He/ Geometry.html. 16. Science daily. URL: http://www.sciencedaily.com/releases/2013/ 01/ 130125103931.html. 17. Science Museum. URL: http://www.sciencemuseum.org.uk/ onlinestuff/stories/ada_lovelace.aspx. 18. Scientific Journals. URL: http://www.dir.yahoo.com. 19. Scientific Journals International. URL: http://www.scientific journals.org/.

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20. The  Encyclopedia Britannica, Ninth Edition and Tenth Edition. URL: http://www.1902encyclopedia.com/A/ALG/algebra-01.html. 21. The greatest mathematicians of the past. URL: http://fabpedigree. com/james/mathmen.htm#Leibniz. 22. The Guardian. URL: http://www.guardian.co.uk/science/blog/2010/ oct/27/millennium-prize-problems-mathematics. 23. The Millennium Problems. URL: http://www.fgcsic.es/lychnos/ en_EN/articles/the_millennium_problems. 24. The story of mathematics. URL: http://www.storyofmathematics. com/glossary.html. 25. Wired. URL: http://www.wired.com/wiredenterprise/2012/06/ berkeley-programming-languages. 26. Wolfram Maths World. URL: http://mathworld.wolfram.com/ topics/Geometry.html. 27. Workjoke. Professional jokes. URL: http://www.workjoke.com/ projoke42.html. 28. Википедия. Свободная энциклопедия. URL: http://en.wikipedia. org/w/index.php. 29. Электронный словарь Abby Lingvo. URL: / www.lingvo.ru/.

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Appendix 1 Mini-Dictionary A abstract algebra – the area of modern mathematics that considers algebraic structures to be sets with operations defined on them, and extends algebraic concepts usually associated with the real number system to other more general systems, such as groups, rings, fields, modules and vector spaces. algebra, n – a branch of mathematics that uses symbols or letters to represent variables, values or numbers, which can then be used to express operations and relationships and to solve equations. algebraic expression – a combination of numbers and letters equivalent to a phrase in language, e.g. x2 + 3x – 4. algebraic equation – a combination of numbers and letters equivalent to a sentence in language, e.g. y = x2 + 3x – 4. algorithm, n – a step by step procedure by which an operation can be carried out. analytic (Cartesian) geometry – the study of geometry using a coordinate system and the principles of algebra and analysis, thus defining geometrical shapes in a numerical way and extracting numerical information from that representation. analysis (mathematical analysis) – grounded in the rigorous formulation of calculus, analysis is the branch of pure mathematics concerned with the notion of a limit (whether of a sequence or of a function). arithmetic, n – the part of mathematics that studies quantity, especially as the result of combining numbers (as opposed to variables) using the traditional operations of addition, subtraction, multiplication and division (the more advanced manipulation of numbers is usually known as number theory). associative property – property (which applies both to multiplication and addition) by which numbers can be added or multiplied in any order and still yield the same value, e.g. (a + b) + c = a + (b + c) or (ab)c = a(bc). axiom, n – a proposition that is not actually proved or demonstrated, but is considered to be self-evident and universally accepted as a starting point for deducing and inferring other truths and theorems, without any need of proof. 119

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access, v – connect to, or get (information) from a system or a database. access control – a feature of a computer security system which prevents unauthorized users from accessing a system. access request – a user request for data from a database. address, n – a location within the memory of a computer. address register – a register which stores an address in memory. AL – artificial intelligence. ALGOL – algorithmic language: a language developed for mathematical and scientific purposes. algorithm, n – a prescribed set of well-defined rules or instructions for the solution to a problem. analyst, n – someone responsible for understanding a problem in a business environment and designing a computer system to solve it. application(s) program – a program written in a high-level language, designed to perform a specific function such as calculate a сompany’s payroll. artificial intelligence – the discipline concerned with the building of computer programs that perform tasks requiring intelligence when done by humans. Assembler, n – a program that takes as input a program written in assembly language and translates it into machine code. assembly language – a human-readable representation of machinecode programs. B bandwidth, n – the difference between the lowest and highest frequency in a group of frequencies. BASIC – beginners’ all-purpose symbolic instruction code: a programming language developed in the mid-1960s to exploit the capability (new at that time) of the interactive use of a computer from a terminal. binary number – a number (0 or 1) used in binary arithmetic. bit, n – binary digit holding the value 0 or 1: the smallest unit of information in a computer system. boot, v – reload the operating system of a computer. broadcast – a message-routing algorithm in which a message is transmitted to all nodes in a network. bug, n – an error in a program. byte – a character consisting of 8 binary digits or bits.

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C C – a highly portable programming language originally developed for the UNIX operating system, derived from BCPL via a short-lived predecessor B. C++ – a programming language combining the power of objectoriented programming with the efficiency and notational convenience of C. calculus, (infinitesimal calculus) – a branch of mathematics involving derivatives and integrals, used to study motion and changing values. calculus of variations – an extension of calculus used to search for a function which minimizes a certain functional (a functional is a function of a function). capacity, n – the amount of free unused space left on a disk. cell, n – a location in a spreadsheet capable of holding text, number data, or a formula. central processing unit – the principal operating part of a computer, consisting of the arithmetic unit and the control unit. channel, n – a specialized processor that consists of an information route and associated circuitry to control input/ output operations. More than one I/O device may be attached to a channel for fast accessing and updating of information. check point – a point in a series of programs at which a backup is taken, and the point at which the series of programs will be restarted. circuit, n – a combination of electrical devices and conductors that form a conducting path. click, v – press the button on a mouse to initiate some action or mark a point on the screen. COBOL – common business-oriented language designed for commercial business use. code, n – the representation of information data in symbolic language or a secret fashion. code, v – write a computer program. compile, v – interpret a source program or a list of instructions in symbolic language. compiler, n – a program which converts source programs into machine code. Each high-level language has its own compiler. conference, n – a computer-based system enabling users to participate in a joint activity despite being separated in space or time. control function – a function performed by the control unit of a computer co-ordinating the internal functions and passing commands to the processor. 121

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control unit – one of the two main components of the CPU. It transmits co-ordinating control signals and commands to the computer. CPU – central processing unit. cracker, n – a person who tries to break security on computer systems – getting through firewalls and password systems. crash, n – a severe failure of a computer system that causes the hardware or software to be restarted. cursor, n – a symbol on a computer screen that indicates the active position. D data, n – information that has been prepared, often in a particular format, for a specific purpose. The term is used in computing to distinguish information from program instructions. database, n – a file or group of files structured in such a way as to satisfy the needs of various users and accessed using the facilities of a database management system. data frame – one of a number of predefined slices into which data may be broken for transmission. data processing – the handling or manipulating of information called data which is specially prepared to be understood by the computer. debug, v – remove bugs from a program. decimal number – a real number which expresses fractions on the base 10 standard numbering system using place value, e.g. 37⁄100 = 0,37. deductive reasoning or logic – a type of reasoning where the truth of a conclusion necessarily follows from, or is a logical consequence of, the truth of the premises (as opposed to inductive reasoning). derivative, n – a measure of how a function or curve changes as its input changes, i.e. the best linear approximation of the function at a particular input value, as represented by the slope of the tangent line to the graph of the function at that point, found by the operation of differentiation. descriptive geometry – a method of representing three-dimensional objects by projections on the two-dimensional plane using a specific set of procedures. differential equation – an equation that expresses a relationship between a function and its derivative, the solution of which is not a single value but a function (has many applications in engineering, physics economics, etc). 122

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differential geometry – a field of mathematics that uses the methods of differential and integral calculus (as well as linear and multilinear algebra) to study the geometry of curves and surfaces. device, n – a piece of hardware that is attached to a computer and is not part of the main central processor. digit, n – a number which has only one character: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. disk, n – a storage device in the form of a circular magnetic plate in which the information is stored via magnetic encoding. diskette, n – a flexible magnetic disk which can be removed from a computer. download, v – send programs or data from a central computer to a remote terminal or PC. E electronic circuit – a combination of electrical devices and semiconductors that form a conducting path. electronic mail – messages sent between users of computer systems, where the system is used to hold and transport messages. execute, v – run a program in a computer. expert system – a system built for problem solving which tries to emulate the skills of a human expert. F formula, n – a rule or equation describing the relationship of two or more variables or quantities, e.g. A = πr2. fraction, n – a way of writing rational numbers (numbers that are not whole numbers), also used to represent ratios or division, in the form of a numerator over a denominator, e.g. 3⁄5 (a unit fraction is a fraction whose numerator is 1). function, n – a relation or correspondence between two sets in which one element of the second (codomain or range) set ƒ(x) is assigned to each element of the first (domain) set x, e.g. ƒ(x) = x2 or y = x2 assigns a value to ƒ(x) or y based on the square of each value of x. file, n – information held on disk or tape in order for it to exist beyond the time of execution of a program. floppy disk – a diskette. flowchart, n – a diagram or a sequence of steps which represent the solution to a problem. 123

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format, v – prepare a disk for use by a computer whereby the structure of the pattern of information to be held on the disk is written to the disk surface. FORTRAN – a programming language widely used for scientific computation. G gateway, n – a device that links two networks in a way that is usually visible to the network users. graphics, n – a non-character based method of displaying information on a screen, usually used for displaying pictures. group, n – a mathematical structure consisting of a set together with an operation that combines any two of its elements to form a third element, e.g. the set of integers and the addition operation form a group. group theory – the mathematical field that studies the algebraic structures and properties of groups and the mappings between them. H hacker, n – a person who attempts to breach the security of a computer system by access from a remote point. hard disk – a fixed disk inside a computer which may not be removed. hardware – the computer equipment and its peripherals. high-level language – a language in which each instruction represents several machine code instructions, making the notation more easily readable by the programmer. I index, n – a set of links that can be used to locate records in a data file. infected – being inhabited by a computer virus. infector, n – something that transmits a computer virus. infinitesimal – quantities or objects so small that there is no way to see them or to measure them, so that for all practical purposes they approach zero as a limit (an idea used in the development of infinitesimal calculus). infinity, n – a quantity or set of numbers without bound, limit or end, whether countably infinite like the set of integers, or uncountably infinite like the set of real numbers (represented by the symbol ∞). integers – whole numbers, both positive (natural numbers) and negative, including zero. 124

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integral, n – the area bounded by a graph or curve of a function and the x axis, between two given values of x (definite integral), found by the operation of integration. input, n – the information which is presented to the computer. input, v – put information to a computer for storage or processing. input-output – the part of a computer system or the activity that is primarily dedicated to the passing of data into or out of the central processing unit. instruction, n – part of a computer program which tells the computer what to do at that stage. interface, n – a common boundary between two systems, devices, or programs. Internet, n – an informal shared public network linking UNIX and other computers world-wide using the Internet protocol. I/O – input/output. irrational numbers – numbers that cannot be represented as decimals (because they would contain an infinite number of non-repeating digits) or as fractions of one integer over another, e.g. π, √2, e. K keyboard, n – an input device like a typewriter for entering characters. key number – a unique number generated to identify a record. L LISP – a programming language designed for the manipulation of non-numeric data. LOGO – a programming language developed for use in teaching young children. logarithm, n – the inverse operation to exponentiation, the exponent of a power to which a base (usually 10 for natural logarithms) must be raised to produce a given number, e.g. because 1,000 = 103, the log10 100 = 3. logic, n – the study of the formal laws of reasoning (mathematical logic the application of the techniques of formal logic to mathematics and mathematical reasoning, and vice versa). low-level language – a language such as assembly language in which each instruction has one corresponding instruction in machine code. M machine code – the code actually executed by the computer, not easily readable by the programmer. 125

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megabyte – one million bytes: unit of measure for the amount of memory or disk storage on a computer. memory, n – a device or medium that can retain information for later retrieval. Microsoft – a computer software company. modem, n – modulator and demodulator: a device that converts the digital bit stream used by the computer into an analog signal suitable for transmission over a telephone line (modulation) and then converts it back to digital (demodulation). multimedia – an application of computer technology that allows the capture, manipulation, and presentation of different types of data, e.g. text, graphics, video, animation, sound, etc. N network – a system which connects up a number of computers and communication devices to enable messages and data to be passed between those devices. non-Euclidean geometry – geometry based on a curved plane, whether elliptic (spherical) or hyperbolic (saddle-shaped), in which there are no parallel lines and the angles of a triangle do not sum to 180°. notation, n – a system of symbols. O object-oriented – describing a computer architecture in which all processes, files, I/O operations, etc., are represented as objects (i.e. data structures in memory that may be manipulated by hardware and software). operating system – the set of programs that jointly control the system resources and the processes using those resources on a computer. operator, n – someone responsible for running a computer (usually a mainframe). output, n – the result of performing arithmetic and logical operations on data. output, v – transmit processed data to a physical medium such as a printer or disk drive. P PASCAL – a programming language designed as a tool to assist the teaching of programming as a systematic discipline. password, n – a method of security in which the user has to enter a unique character string before gaining access to a computer system. 126

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peripheral – an input or output device attached to a computer. phreaker, n – a person who is basically interested in cracking telephone systems. pirate, v – use software that has been copied in breach of copyright. pixel, n – an individual dot on a computer screen. PL/I – programming language I. a programming language developed by the US IBM users’ group, implementing the best features of COBOL, FORTRAN, and ALGOL. polynomial, n – an algebraic expression or equation with more than one term, constructed from variables and constants using only the operations of addition, subtraction, multiplication and non-negative whole-number exponents, e.g. 5x2 – 4x + 4y + 7. prime numbers – integers greater than 1 which are only divisible by themselves and 1. processing – the performing of arithmetic or logical operations on information which has been input to a computer. program, n – a list of instructions which are used by the computer to perform the user’s requirements. programmer, n – someone who writers computer programs. programming – the act of writing a computer program. programming language – a notation for the precise description of computer programs. Q query, n – a request for information from a database. quadratic equation – a polynomial equation with a degree of 2 (i.e. the highest power is 2), of the form ax2 + bx + c = 0, which can be solved by various methods including factoring, completing the square, graphing, Newton's method and the quadratic formula. R RAM – random-access memory: this is memory which can be read and written to. The basic element is a single cell capable of storing one bit of information. Each cell has a unique address in memory and so can be accessed in a random order. rational numbers – numbers that can be expressed as a fraction (or ratio) a⁄b of two integers (the integers are therefore a subset of the rationals), or alternatively a decimal which terminates after a finite number of digits or begins to repeat a sequence. 127

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real numbers – all numbers (including natural numbers, integers, decimals, rational numbers and irrational numbers) which do not involve imaginary numbers (multiples of the imaginary unit 1, or the square root of – 1), may be thought of as all points on an infinitely long number line. record, n – a collection of data handled together in movements to and from storage. register, n – a group of devices that are used to store information within a computer for high-speed access. S security system – a system which controls access to a computer and maintains the security of that computer. set, n – a collection of distinct objects or numbers, without regard to their order, considered as an object in its own right. software – a general term for any computer program. software developer – someone who writes software. software package – a series of programs written for a generic application, e.g. a payroll package, which can be adapted by the user to meet individual needs. source program – the original high-level language program which has to be converted to machine code before it may be executed. spreadsheet – a program that manipulates tables consisting of rows and columns of cells and displays them on a screen. The value in a numerical cell is either typed in or is calculated from values in other cells. structured programming – a method of programming development that makes extensive use of abstraction in order to factorize the problem and give increased confidence that the resulting program is correct. subset, n – a subsidiary collection of objects that all belong to, or is contained in, an original given set, e.g. subsets of {a, b} could include: {a}, {b}, {a, b} and {}. symmetry, n – the correspondence in size, form or arrangement of parts on a plane or line (line symmetry is where each point on one side of a line has a corresponding point on the opposite side, e.g. a picture a butterfly with wings that are identical on either side; plane symmetry refers to similar figures being repeated at different but regular locations on the plane). systems manager – a person responsible for the management and administration of a computer system. systems program – a program written for a particular type of hard ware. 128

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T template – a pre-shaped pattern used as a guide. theorem, n – a mathematical statement or hypothesis which has been proved on the basis of previously established theorems and previously accepted axioms, effectively the proof of the truth of a statement or expression. transmission – the sending of a message. transcendental number – an irrational number that is “not algebraic”, i.e. no finite sequence of algebraic operations on integers (such as powers, roots, sums, etc.) can be equal to its value, examples being π and e. For example, √2 is irrational but not transcendental because it is the solution to the polynomial x2 = 2. transfinite numbers – cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. trigger, v – set a process in motion. U UNIX – an operating system originally developed by Bell laboratories in 1971 for DEC PDP 11 minicomputers. update, v – modify data held by a computer system. upgrade, v – replace or modernize software with a later version of the same software. user, n – an individual or group making use of the output of a computer system. user-friendly – describing interactive systems that are designed to make the user’s task as easy as possible by providing feedback. user interface – the means of communicating between a human being and a computer. V variable, n – a string of characters used to denote a value stored within a computer which may be changed during execution. vector, n – a physical quantity having magnitude and direction, represented by a directed arrow indicating its orientation in space. vector space – a three-dimensional area where vectors can be plotted, or a mathematical structure formed by a collection of vectors. virus, n – a self-replicating program, usually designed to damage the system on which it lands. virus checking program – a program that is used to detect the presence of a virus in memory or on disk. 129

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virus scanner – a program that detects viruses which have already infected a computer. virus shield – a program that detects viruses as they attempt to infect the computer. virus signature – the particular features of each computer virus that enable it to be recognized. W word processing – the use of a computer to compose documents with facilities to edit, re-format, store, and print documents with maximum flexibility. worm, n – an entirely self-replicating virus which is not hardware dependent.

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Appendix 2

Word and word combinations used for connection of separate parts of a statement 1. Means of connection pointing out the succession of ideas and actions: first / at first / first of all to begin with во-первых, сначала second / secondly во-вторых next / further / then далее, затем finally / lastly / at last (и) наконец 2. Means of connection which express linking with a statement: in addition в дополнение к сказанному, кроме того moreover более того, кроме того, к тому же furthermore далее, более того, к тому же likewise / similarly точно так же, аналогичным образом besides кроме этого, помимо этого, притом 3. Means of connection expressing the contrast: otherwise иначе, в противном случае rather скорее, вернее, пожалуй on the one hand с одной стороны on the other hand с другой стороны however / still / однако, тем не менее, все же nevertheless on the contrary / conversely (и) наоборот, напротив, с другой стороны 4. Means of connection, showing the statement as the consequence of summing up the previous material: hence следовательно, отсюда, из этого следует thus итак, так, таким образом, поэтому there fore / accordingly поэтому, следовательно, таким образом consequently / следовательно, в результате этого as a consequence briefly / in short / короче говоря in a few words to sum up / to summarize итак, суммируя; можно сказать, что 131

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Appendix 2

Words, word combinations and constructions which are used for expressing author’s attitude to the statement 1. Words, word combinations and constructions expressing assurance: of course конечно, разумеется undoubtedly / with out несомненно, бесспорно doubt / no doubt needless to say нечего и говорить, само собой разумеется in reality в действительности in deed на самом деле in fact / as a matter of fact фактически, на самом деле to say (tell) the truth по правде говоря by no means никоим образом 2. Words, word combinations and constructions expressing probability: probably вероятно in all probability по всей вероятности perhaps возможно apparently очевидно, по-видимому presumably по-видимому, предположительно say скажем 3. Words, word combinations and constructions giving appreciation of the statement if it is desirable or undesirable: fortunately к счастью, по счастливой случайности unfortunately к сожалению strange enough удивительно что; как это ни удивительно surprisingly неожиданным образом; как ни странно curiously enough как ни странно 4. Words, word combinations which are used for emphasizing and making more precise of separate parts of a statement: mainly в основном; главным образом largely главным образом; преимущественно essentially по существу; в основном particularly / in particular в особенности, в частности generally speaking вообще говоря strictly speaking строго говоря as a rule как правило 132

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at any rate во всяком случае at best в лучшем случае at least по крайней мере in a sense в известном смысле in a general (broad) sense в широком смысле to a certain degree (extent) до некоторой степени more or less более или менее to say nothing of не говоря уже о (чем-л.) to say the least по меньшей мере 5. Words and word combinations introducing additional explanations to express idea: in other words другими словами, иначе говоря in plain words просто говоря more simply проще говоря to be more exact / говоря точнее; точнее more properly so to speak / so to say так сказать 6. Constructions expressing reference on general acceptance of the idea or the action: It is well (commonly) known that…/ It is common knowledge that… Общеизвестно, что… It has been recognized that… Признано, что… It is generally realized that… Всем известно, что… There is no denying that… Нельзя отрицать того, что…

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Appendix 3 Latin Abbreviations A.C. (ante Christum) A.D. (anno domini) ad inf., ad infin. (ad infinitum) ad int. (ad interim) ad fin. (ad finem) ad loc. (ad locum) ad val. (ad valorem) ae., aet., aetat. (aetatis) A.M. (ante meridiem) B.S., B.Sc. (Baccalaureus Scientiae) c. (cum) ca., cir., circ. (circa) cf. (confer) e.g. (exempli gratia) et al. (et alii, et alia) etc. (et cetera) et seq. (et sequens, et sequentes) ff. (foliis) ib., ibid. (ibidem) id. (idem) i.e. (id est) lb. (libra) loq. (loquitur) m. (meridies) N.B. (Nota bene) no. (numero) non obst. (non obstante) non seq. (non sequitur) n.r. (non repetatur) p.d. (per diem) per an. (per annum) per cent. (per centum) P.M. (post meridiem) p.r.n. (pro re nata) 134

before Christ in the year of the Lord to infinity in the meantime near the end [of the page] to the place according to the value of age, aged before midday Bachelor of Science with about compare for [the sake of an] example and others and the rest, and so forth and the following on the [following] pages in the same place the same that is pound he (she, it) speaks midday Note well by number notwithstanding it does not follow do not repeat by the day by the year per hundred after midday as the need arises

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pro tem. (pro tempore) prox. (proximo mense) P.S. (post scriptum) q.d. (quaque die) seq., seqq. (sequentia) stat. (statim) vs. (versus) v.s. (vide super)

for the time, temporarily next month written after every day following immediately against see above

Nouns of Latin and Greek Origin Singular Plural Modern forms Plural -on (-um) -a continuum continua criterion criteria criterions curriculum curricula datum data equilibrium equilibria infinitum infinita spectrum spectra spectrums maximum maxima maximums minimum minima momentum momenta phenomenon phenomena polyhedron polyhedra quantum quanta vacuum vacua vacuums -is (-ix) -es analysis analyses axis axes basis bases crisis crises hypothesis hypotheses index indices indexes matrix matrices parenthesis parentheses vertex vertices thesis theses synthesis syntheses 135

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-us calculus genius locus modulus nucleus radius rhombus -a abscissa hyperbola formula

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-i calculi genii loci moduli nuclei radii rhombi -ae abscissa hyperbolae formulae

geniuses nucleuses radiuses rhombuses abscissas hyperbolas formulas

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Учебное издание

Свиридова Татьяна Николаевна

MATHEMATICS THROUGH ENGLISH Учебное пособие

Редактор М.В. Саблина Компьютерная верстка О.А. Кравченко

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