Учебник по НТП

tional number is a non-repeating and non-terminating decimal number. For example, the irrational number ∏ called “pi” cannot be written as a quotient of two integers, and its decimal form goes on forever and never repeats.

∏ = 3, 14159265

Symbols, called operators, signify relationships between numbers and/ or variables. Some example operators in the language of algebra are:

+, -, /, *, =, >, <, ³

Delimiters are the punctuation marks in algebra. They let the reader know where one phrase or sentence ends and another begins. Example delimiters used in algebra are:

( ), [ ], { }.

A number of rewriting rules exist within algebra to simplify a with a shorthand notation. Exponential notation is an example of a shorthand notational scheme. If a series of similar algebraic phrases are multiplied times one another, the expression may be rewritten with the phrase raised to a power (the number of times the phrase is multiplied by itself and is written as a superscript of the phrase). For example:

8 * 8 * 8 * 8 * 8 * 8 = 86

(X - 4Y) * (X - 4Y) * (X - 4Y) * (X - 4Y) = (X - 4Y)4

Some special rules apply to exponents. A negative exponent may be transformed to a positive exponent if the base is changed to one divided by the base. A numerical example follows:

5-3 = (1/5)3 = 0.008

When two phrases that have the same base are multiplied, the product is equal to the base raised to the sum of their exponents. The following example illustrate this principle:

182 * 185 = 185+2 = 187 (X+3)8*(X+3)-7 = (X+3)8-7 = (X+3)1 = X+3

A special form of exponential notation, called binomial expansion occurs when a phrase connected with addition or subtraction operators is raised to the second power. Binomial expansion is illustrated below:

(X + Y)2 = (X + Y) * (X + Y) = X2+ XY + XY + Y2 = X2+ 2XY + Y2

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When two expressions are connected with the multiplication operator, it is often possible to “multiply through” and change the expression. In its simplest form, if a number or variable is multiplied by a phrase connected at the highest level with the addition or subtraction operator, the phrase may be rewritten as the variable or number times each term in the phrase.

a * (x + y + z) = ax + ay + az

A corollary to the previously discussed rewriting rule for multiplication of phrases is factoring, or combining like terms. The rule may be stated as follows: If each term in a phrase connected at the highest level with the addition or subtraction operator contains a similar term, the similar term(s) may be factored out and multiplied times the remaining terms. It is the opposite of “multiplying through.” Two examples follow:

ax + az - axy = a * (x + z - xy)

(a+z) * (p-q) - (a+z) * (x+y-2z) = (a+z) * (p-q-x-y+2z)

IV. Knowing Ins And Outs

The language of algebra varies from simple math like 2 + 2 to rather complicated algebraic expressions using variables, operators, integers, quotients, to name but a few. All of us have been taught to write mathematical formulae, but how about reading them aloud? To do this, you will have to use special verbal expressions like «is equal to» or «equals» for the equality sign, «divided by» or «over» for division and fractions, «multiplied by» or «times» for multiplication, «the square root of», «cubed», or «to the power» for roots, etc. For example, the well-known quadratic formula:

can be read like this—x equals the opposite of b plus or minus the square root of b squared minus four a times c all over two times a. For further information on math speaking see Appendix 2.

Work in pairs. Read aloud the following algebraic equations to let your partner write them down. Check the writing for mistakes and change the roles.

ax2 + bx + c = 783

(c - d) * (x + y) = c * (x + y) - d * (x + y) = cx + cy - dx - dy

a5 x5 + a4 x4 + a3 x3 + a2 x2 + a1 x + a0 = 0

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(w3)2 - q (w3) - 1/27 p3 = 0

(x3 - b3) + c (x-b) = (x-b) (x2 + bx + b2 + c) x3 - b3 = (x-b) (x2 + bx + b2)

(x - 1/3 a2)3 = x3 - a2 x2 + 1/3 a22 x - 1/27 a23

9 a2 a1 - 27 a0 - 2 a23

54 - x3 -a1

x = -b ±√ b2 - 4ac 2a

y11 + y1/x + (1 - v2/x2)y = x-v/ x2 * sin (vx)

1/2 (q ±√ q2 + 4/27 p3) = 1/2 q ±√ 1/4 q2 + 1/27 p3 = R ±√ R2 + Q3

A mathematician is a person who says that, when 3 people are supposed to be in a room but 5 came out, 2 have to go in so the room gets empty...

V. Enhancing Skills In English-Russian Interpretation

Render orally the following text:

Zeno’s Paradoxes

The paradoxes of the philosopher Zeno, born approximately 490 BC, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher Parmenides, who believed in monism. Zeno’s paradoxes are perhaps the first examples of a method of proof called reductio ad absurdum, also known as proof by contradiction. They are

also credited as a source of the dialectic method used by Socrates. The three most famous Zeno’s paradoxes deal with counterintuitive as-

pects of continuous space and time:

1. Dichotomy paradox: Before an object can travel a given distance , it must travel a distance . In order to travel , it must travel , etc. Since this

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sequence goes on forever, it therefore appears that the distance cannot be traveled. The resolution of the paradox awaited calculus and the proof that infinite geometric series such as can converge, so that the infinite number of “half-steps” needed is balanced by the increasingly short amount of time needed to traverse the distances.

2.Achilles and the tortoise paradox: A fleet-of-foot Achilles is unable to catch a plodding tortoise which has been given a head start, since during the time it takes Achilles to catch up to a given position, the tortoise has moved forward some distance. But this is obviously fallacious since Achilles will clearly pass the tortoise! The resolution is similar to that of the dichotomy paradox.

3.Arrow paradox: At every moment in time, the arrow is located at a specific position. If the moment is just a single instant, then the arrow does not have time to move and is at rest during that instant. Now, during the following instances, it then must also be at rest for the same reason. The arrow is always at rest and cannot move: motion is impossible.

This paradox may be resolved mathematically as follows: in the limit, as the length of a moment approaches zero, the instantaneous rate of change or velocity (which is the quotient of distance over length of the moment) does not have to approach zero. This nonzero limit is the velocity of the arrow at the instant.

Zeno’s Arrow Paradox is similar to Heisenberg’s Uncertainty Principle because Heisenberg argued that on the subatomic level, the only way to measure a system is to interfere with that system. That is, to measure a quantity, say the position of an electron, the speed of that electron must inevitably be affected. Thus, the very act of observation changes the system. We can be sure of the speed or the position but never both. Either the arrow is where it is or it is where it is not.

The theory of special relativity answers Zeno’s concern over the lack of an instantaneous difference between a moving and a nonmoving arrow by positing a fundamental re-structuring the basic way in which space and time fit together. Zeno was correct that instantaneous velocity in the context of absolute space and absolute time does not correspond to physical reality, and probably doesn’t even make sense. From Zeno’s point of view, the classical concept of absolute time was not logically sound, and special relativity (or something like it) is a logical necessity, not just an empirical fact. It’s even been suggested that if people had taken Zeno’s paradoxes more seriously they might have arrived at something like special relativity centuries ago, just on logical grounds.

One method of dealing with Zeno’s paradoxes has been the claim that matter is not infinitely divisible; that there exist particles of matter so small that further division is not possible, and atomism did develop as a response to these paradoxes. Mathematicians thought they

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had done away with Zeno’s paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century.Nevertheless, Zeno’s paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. In recent time, physicists studying quantum mechanics have noticed that the dynamical evolution (motion) of a quantum system can be hindered (or even inhibited) through observation of the system. This effect is usually called the quantum Zeno effect as it is strongly reminiscent of Zeno’s arrow paradox.

VI. Enhancing Skills In Russian-English Interpretation

Render orally the following text:

Когда Ахиллес Догонит Черепаху ?

Древнегреческий философ Зенон Элейский сформулировал свои знаменитые парадоксы (или апории, как говорили греки) в

5 веке до н. э. Апории Зенона касаются одного из самых трудных и ключевых понятий математики—бесконечности.

Рассмотрим один из самых знаменитых парадоксов Зенона об Ахиллесе и черепахе: быстроногий Ахиллес никогда не догонит черепаху. Почему? Всякий раз, при всей скорости своего бега и при всей малости разделяющего их пространства, как только он ступит на место, которое перед тем занимала черепаха, она несколько продвинется вперед. Как бы ни уменьшалось пространство между ними, оно ведь бесконечно в своей делимости на промежутки и их надобно все пройти, а для этого необходимо бесконечное время. То есть предмет, движущийся к цели, вначале должен пройти половину пути к ней, а чтобы пройти эту половину, он должен пройти ее половину и т.д., до бесконечности. Получается, что достичь цель невозможно?

Пусть первоначальное расстояние между Алиллесом и черепахой 1 метр. Ахиллес бежит со скоростью 1 метр в секунду, а черепаха в два раза медленнее. Тогда после первой секунды Ахиллес пробежит 1 м, а черепаха 0,5 м, и расстояние сократится до 0,5 м; за следующие 0,5 секунд Ахиллес пробежит 0,5 м, а черепаха— 0,25 м, и т. д. Расстояние будет сокращаться в геометрической прогрессии, и полное время T,

T = 1 + 1/2 + 1/4 + 1/8 + ... +1/2n + ...

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за которое отставание Ахиллеса от черепахи станет равным нулю— то есть Ахиллес догонит-таки шуструю черепаху,—займет по формуле суммы бесконечно убывающей геометрической прогрессии:

T = 1/1 -q

где q—знаменатель геометрической прогрессии q=1/2, откуда выводим T=2. То есть Ахиллес догонит черепаху за 2 секунды: черепаха пробежала 1 метр, Ахиллес—2, и никакой вечности ему не потребовалось.

Неужели все так просто? Нет, конечно. В апории «Ахиллес и черепаха» Зенон рассматривает пространство как сумму конечных отрезков и противопоставляет ему бесконечную непрерывность времени. Неадекватность создается Зеноном посредством рассмотрения способности Ахиллеса покрыыть путь, ограниченный не точкой, где он нагоняет черепаху (как того требует условие реального процесса успешной погони), а точкой, где черепахи по условию реального процесса уже быть не может в принципе. Здесь Зенон очень остро поставил проблему актуальной бесконечности.

Когда мы говорим, что отрезок есть бесконечное множество точек, мы принимаем на веру то, что это бесконечное множество мы можем охватить сразу, но Зенон утверждает: если точка не имеет размера, то сколько бы мы нулей ни складывали, в результате будет ноль. Если точка имеет линейный размер, то, взяв сразу бесконечное множество точек, мы получим бесконечность. Таким образом, точка имеет линейный размер, и точек в отрезке конечное число.

Именно к этому выводу и пришел Аристотель, который возражал против использования самого понятия актуальной бесконечности, заменив его бесконечностью потенциальной. Можно сколь угодно долго делить отрезок пополам, но в любой момент времени образуется только конечное число частей. Аристотель фактически признал правоту Зенона: актуальная бесконечность внутренне противоречива; и, значит, пользоваться этим понятием в математике нельзя.

А когда мы с вами взяли бесконечный ряд геометрической прогрессии, мы работали с актуальной бесконечностью, решив, что за время, стремящееся к нулю, объект может покрыть ненулевое расстояние. Методы работы с бесконечно малыми величинами, то есть такими, которые меньше любого числа, но больше нуля, первыми применили Ньютон и Лейбниц (с анализа бесконечно малых началась «высшая математика»). Но сами методы внутренне противоречивы. Они покоятся на понятии актуальной бесконечности, а оно было и осталось источником новых и новых парадоксов, с которыми столкнулись ученые уже в двадцатом веке, попытавшись

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выстроить безупречно корректные обоснования математики. Рассмотрим множество натуральных чисел не больших N:

1, 2, 3, … m, … N.

Оно состоит из N членов. Пусть N—четное. Тогда множество четных чисел, не больших N: 2, 4, …, 2m, … N, состоит из M = N/2 членов. Ясно, что сколь большим ни было бы N, четных чисел, не больших его, будет всегда в два раза меньше, чем N.

Рассмотрим теперь целиком весь натуральный ряд и целиком все четные числа. Вот тут и проявляется противоречивость актуальной бесконечности. На первый взгляд, четных чисел явно меньше. Но так ли это? Мы уже не можем пересчитать все члены множества, и поэтому нам придется прибегнуть к другому методу сравнения множеств. Если каждому члену одного множества мы поставим в соответствие один и только один член другого, мы будем считать эти множествами равными, или (математически более строго) равномощными. Но сопоставить весь натуральный ряд и множество четных чисел очень просто: каждому m соответствует 2m. Значит, множества равномощны. Этот парадокс рождает именно актуальная бесконечность, о которой и говорил Зенон Элейский в своих апориях.

VII. Solving Translation Problems

Let‘s start with a joke: “Two trucks loaded with thousands of copies of Roget’s Thesaurus collided as they left a New York publishing house last Thursday, according to the Associated Press. Witnesses were aghast, amazed, astonished, astounded, bemused, benumbed, bewildered, confounded, confused, dazed, dazzled, disconcerted, disoriented, dumbstruck, electrified, flabbergasted, horrified, immobilized, incredulous, nonplussed, overwhelmed, paralyzed, perplexed, scared, shocked, startled, stunned, stupified, surprised, taken aback, traumatized, upset...“

This impressive list of synonyms shows that words used to describe the same situation can not only represent nuances of meaning, but also differ in their language status belonging to bookish, neutral or colloquial style. While neutral words are welcome in all styles, bookish words, especially those of terminological character, are frequent in formal speech, and colloquialisms are the words highly improper in scientific, business, academic and technical texts.

Read the text below. Copy out the underlined words and define whether they belong to the bookish, neutral or colloquial layer of the language. Find synonyms for each word from the other layers and explain differ- ences in their usage. Translate your list into Russian.

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The Golden Ratio

The Euclidean Elements is the most known mathematical work of the antique science. This one is written in the 3d century B.C. and contains fundamentals of the antique mathematics: elementary geometry, number theory, algebra, theory of proportions and relations, methods of area and column calculation, etc. Just from the Euclidean Elements the following geometrical problem called the problem of “division of the

line segment in extreme and mean ratio” had came.

The essence of the problem consists of the following: To divide a line segment into two parts so that the length of the smaller part is to the length of the larger part as the length of the larger part is to the length of the entire segment. Or, translating this verbal definition into algebraic language and labeling the shorter part of the segment as x and the longer part as the unit length, 1, we conclude that the whole segment has length 1 + x and then the ratio may be defined in algebraic terms by means of the proportion:

x2 = x + 1

Leonardo da Vinci was the first to call this number of x, approximately equal to 1.6180339887498948482, as the “golden section”. In modern science, the golden ratio is also known as the divine proportion or golden mean.

This number is often encountered when taking the ratios of distances in simple geometric figures such as the pentagram, decagon and dodecagon. It is denoted φ (phi) after the Greek sculptor Phidias who closely studied the Divine Proportion, or sometimes T (tau), which is an abbreviation of the Greek “tome,” meaning “to cut”. Phi has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers. Phi can be found throughout the universe; it crops up in more places in art, music and so on than any number except phi. Claude Debussy used it explicitly in his music and Le Corbusier in his architecture. There are claims the number was used by Leonardo da Vinci in the painting of the Mona Lisa, by the Greeks in building the Parthenon and by ancient Egyptians in the construction of the Great Pyramid of Khufu.

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Given a rectangle having sides in the ratio 1:phi, phi is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio 1:phi. Such a rectangle is called a golden rectangle (an isosceles triangle with a vertex angle of 36 degrees).

The golden ratio is seen in some surprising areas of mathematics. In the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the foundation for an incredible mathematical relationship behind phi. The ratio of consecutive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13..., each number being the sum of the previous two numbers) approaches the golden ratio, as the sequence gets infinitely long. The sequence is sometimes defined as starting at 0, 1, 1, 2, 3... Zero is the zeroth element of the sequence.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

The ratio of each successive pair of numbers in the series approximates phi (1.618033988749895...) , as 5 divided by 3 is 1.666..., and 8 divided by 5 is 1.60. After the 40th number in the series, the ratio is accurate to 15 decimal places.

Because of this ratio, the approximation to φ gets better as the Fibonacci numbers get higher. Interestingly enough, the sum of all of the approximation errors is φ. Stated mathematically:

The golden ratio has been famed throughout history not only for its

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aesthetic properties. The golden ratio is considered a universal natural phenomenon: the beautiful arrangement of leaves in some plants, called phyllotaxis, obeys a number of subtle mathematical relationships. For instance, the florets in the head of a sunflower form two oppositely directed spirals: 55 of them clockwise and 34 counterclockwise. Surprisingly, these numbers are consecutive Fibonacci numbers.

The ratios of alternate Fibonacci numbers are given by the convergents to phi-2, where phi is the golden ratio, and are said to measure the fraction of a turn between successive leaves on the stalk of a plant: 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. A similar phenomenon occurs for daisies, pineapples, pinecones, cauliflowers, and so on.

Lilies, irises, and the trillium have three petals; columbines, buttercups, larkspur, and wild rose have five petals; delphiniums, bloodroot, and cosmos have eight petals; corn marigolds have 13 petals; asters have 21 petals; and daisies have 34, 55, or 89 petals--all Fibonacci numbers.

VIII. Mastering English Grammar

Translate the sentences paying special attention to the equivalent-lack- ing grammatical structures:

1. It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

2.Indeed, some mathematicians devoted much of their life’s work to the pursuit of that goal, and the search for a proof led to the development of whole new branches of mathematics, but it was not until this decade that the English mathematician Andrew Wiles, from Princeton University, finally completed the task.

3.Fermat’s last theorem is a generalization of the Pythagorean theorem stating that in a right triangle (where one angle equals 90°), the sum of the squares of two sides equals the square of the hypotenuse.

4.These solutions are known as Pythagorean triples, and there exist an infinite number of them (even excluding trivial solutions for which a, b and c have a common divisor).

5.The difficulty in proving is that the case revolves around the fact that there is an infinite number of equations, and an infinite number of possible values for a, b, and c. The proof has to prove that no solutions exist within this infinity of infinities.

6.The actual proof is very indirect, and involves sophisticated tools from algebraic geometry (in particular elliptic curves and modular forms), Galois theory and Hecke algebras, the branches of mathematics, which at face value appear to have

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