polunina_ln_bobrovskaia_im_priroda_nauka_tekhnologii

Chaos theory makes precise the ways in which many of dynamical systems.

LANGUAGE IN USE

7. Read the following text and fill in the gaps with the proper words from the brackets.

Numerals

The earliest forms of number notation were simply groups of straight lines, (1) … (neither/either/whether) vertical or horizontal, each line corresponding to the number 1. Such a system is inconvenient when dealing (2) … (on/to/with) large numbers, and as early as 3400 BC in Egypt and 3000 BC in Mesopotamia a special symbol was adopted for the number 10. The addition of this second number symbol made it possible to express the number 11 with 2 (3) … (instead/rather/more) of 11 individual symbols. Later numeral systems introduced extra symbols for a number (4) … (among/with/between) 1 and 10, usually either 4 or 5, and additional symbols for numbers greater (5) … (then/than/that) 10. In Babylonian cuneiform notation the numeral used for 1 was also used for 60 and for powers of 60; the value of the numeral was indicated by its context. This was a logical arrangement (6) … (from/in/on) the mathematical point of view because 600 = 1, 601 = 60, and 602 = 3600. The Egyptian hieroglyphic system used special symbols for 10, 100, 1000, and 10,000.

The ancient Greeks had two parallel systems (7) … (in/to/of) numerals. The earlier of these was based (8) … (on/into/for) the initial letters of the names of numbers: The number 5 was indicated by the letter pi; 10 by the letter delta; 100 by the antique form of the letter H; 1000 by the letter chi; and 10,000 by the letter mu. The later system, which was first introduced (9) … (along/in/next to) the 3rd century BC, employed all the letters of the Greek alphabet plus three letters borrowed (10) … (in/from/on) the Phoenician alphabet as number symbols. The first nine letters of the alphabet were used for the numbers 1 to 9, the second nine letters for the tens from 10 to 90, and the last nine letters for the hundreds from 100 to 900. Thousands were

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indicated by placing a bar (11) … (in/by/to) the left of the appropriate numeral, and tens of thousands by placing the appropriate letter over the letter M. The late Greek system had the advantage that large numbers could be expressed (12) … (with/while/when) a minimum of symbols, but it had the disadvantage of requiring the user to memorize a total of 27 symbols.

(Microsoft Encarta, 2008)

8. Read the text and fill in the gaps with the appropriate words from the table below.

Game Theory

Game Theory is (1) … analysis of any situation involving a conflict of interest, with the intent of indicating the optimal choices that, under given conditions, will lead to a desired outcome. Although game theory has roots in the study of such well-known (2) … as checkers, tick-tack- toe, and poker – hence the name – it also involves much more (3) … conflicts of interest arising in such fields as sociology, economics, and political and military science.

Aspects of game theory were first (4) … by the French mathematician Émile Borel, who wrote several papers on games of chance and theories of play. The acknowledged father of game theory, however, is the Hungarian-American mathematician John von Neumann, who in a series of papers in the 1920s and ’30s established the mathematical framework for all subsequent (5) … developments.

Applications of game theory are wide-ranging and account for (6) … growing interest in the subject. Von Neumann and Morgenstern indicated the (7) … utility of their work on mathematical game theory by linking it with (8) … behavior. Models can be developed, in fact, for markets of various commodities with differing numbers of buyers and sellers, fluctuating values of supply and demand, and seasonal and cyclical variations, as well as significant structural (9) … in the economies concerned. Here game theory is especially relevant to the (10) … of conflicts of interest in maximizing profits and promoting the widest distribution of goods and services.

(11) … have developed an entire branch of game theory devoted to

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the study of issues involving group decision making. Epidemiologists also make use of game theory, especially with (12) … to immunization procedures and methods of testing a vaccine or other medication. Military strategists turn to game theory to study conflicts of interest resolved through “battles” where the outcome or payoff of a given war game is either victory or defeat. Some uses of game theory in analyses of political and military events have been criticized as a dehumanizing and potentially (13) … oversimplification of necessarily complicating factors. Analysis of economic situations is also usually more complicated than zero-sum games because of the (14) … of goods and services within the play of a given “game.”

(Abridged from Dauben, J. W. Game Theory, Microsoft Encarta, 2008)

9. Read the text and write the numbers given in brackets in figures.

An investment firm is hiring mathematicians. After the first round of interviews, three hopeful recent graduates – a pure mathematician, an applied mathematician, and a graduate in mathematical finance – are asked what starting salary they are expecting.

The pure mathematician: “Would (1) … (thirty thousand dollars) be

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too much?”

The applied mathematician: “I think (2) … (sixty thousand dollars) would be OK.”

The math finance person: “What about (3) … (three hundred thousand dollars)?”

The personnel officer is flabberghasted: “Do you know that we have a graduate in pure mathematics who is willing to do the same work for (4) … (a tenth) of what you are demanding!?”

“Well, I thought of (5) … (one hundred and thirty five thousand dollars) for me, (6) … (one hundred and thirty five thousand dollars) for you – and (7) … (thirty thousand dollars) for the pure mathematician who will do the work.”

(From Runde, V. Math Jokes, www.math.ualberta.ca)

10. Try to solve the following math puzzle.

Bridge crossing

A group of four people has to cross a bridge. It is dark, and they have to light the path with a flashlight. No more than two people can cross the bridge simultaneously, and the group has only one flashlight. It takes different time for the people in the group to cross the bridge:

Annie crosses the bridge in 1 minute, Bob crosses the bridge in 2 minutes, Clifford crosses the bridge in 5 minutes, Dorothy crosses the bridge in 10 minutes.

How can the group cross the bridge in 17 minutes?

(From www.math.utah.edu)

10.Just for fun.

A pure and an applied mathematician are asked to calculate 2 x 2.

The applied mathematician’s solution:

We have

2 x 2 = 2 x 1/(1-1/2).

The second factor on the right hand side has a geometric series

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expansion

1/(1-1/2) = 1 + 1/2 + 1/4 + 1/8 + …

Cutting off the series after the second term yields the approximate solution

2 x 2 = 2 x (1 +1/2) = 3.

The pure mathematician’s solution:

We have

2 x 2 = (-2) x 1/(1-3/2).

The second factor on the right hand side has a geometric series expansion

1/(1-3/2) = 1 + 3/2 + 9/4 + 27/8 + …, which diverges.

Hence, the solution to 2 x 2 does not exist.

(From Runde, V. Math Jokes, www.math.ualberta.ca)

SPEAKING

11. Choose one of the following topics for discussion. Work in pairs to represent opposite standpoints. Use the phrases from the table to agree or disagree with your classmate.

Are people who are really good at maths very interesting? Are all intelligent people good at maths?

Are girls or boys better at maths?

12. Think of your maths teacher at school. What can you say about her/him?

-describe her/his lessons,

-teaching style,

-personal qualities,

-her/his treatment of pupils,

-her/his attitude towards pupils’ problems etc.

WRITING

13. Solve the following problems. Write your answers using numerals and mathematical symbols.

Mary has 2 more books than Joe. Mary has 6 books. How many books does Joe have?

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Our class has 27 children and Mrs. Johnson’s class has 26. How many cupcakes will we need for our joint party?

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A pizza is divided into six pieces. Mary eats two pieces. What fraction of the pizza did Mary eat? What fraction is left?

_________________________________________________________

Is $20.00 enough to buy items priced at $12.97, $4.95, and 3.95?

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About how much would 4 cans of beans cost if each costs $0.79?

_________________________________________________________

How many cars will we need to transport 19 people if each car holds 5?

_________________________________________________________

How many more packages of 5 ping-pong balls can be made if there are 19 balls left in the bin?

_________________________________________________________

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How much does each of 5 children have to contribute to the cost of a $19 gift?

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The Florida’s Best Orange Grove has 15 rows of 21 orange trees. Last year's yield was an average of 208.3 oranges per tree. How many oranges might they expect to grow this year? What factors might affect that number?

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Is the new Nintendo game, Action Galore, cheaper at Sears where it is 20% off their regular price of $49.95 or at Macy’s where it’s specially priced at $41.97?

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KEY WORDS

abstraction, add, addition, algebra, analysis, angle, area, arithmetic, axiom, calculation, calculus, change, conjecture, counting, curve, derivative, divide, division, equation, figure, fraction, function, geometry, infinitesimal, integer, integral, limit, logic, mathematics (applied, pure), measurement, multiply, multiplication, number theory, number, numeral, plane, quantity, set theory, space, structure, subtract, subtraction, theorem, topology, trigonometry, variable, volume

SUPPLEMENTARY READING

Development of Calculus

(1) Isaac Newton and Gottfried Wilhelm Leibniz invented calculus in the 17th century, but isolated results about its fundamental problems had been known for thousands of years. For example, the Egyptians discovered the rule for the volume of a pyramid as well as an approximation of the area of a circle. In ancient Greece, Archimedes proved that if c is the circumference and d the diameter of a circle, then 31/7d<c<310/71d. Archimedes used the same technique for his other results on areas and volumes. Archimedes discovered his results by means of heuristic arguments involving parallel slices of the figures and the law of the lever. Unfortunately, his treatise The Method was only

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rediscovered in the 19th century, so later mathematicians believed that the Greeks deliberately concealed their secret methods.

(2)By the early 17th century, mathematicians had developed methods for finding areas and volumes of a great variety of figures. In his Geometry by Indivisibles, the Italian mathematician F. B. Cavalieri

used what he called ‘indivisible magnitudes’ to investigate areas under the curves y = xn, n = 1 ...9. Also, his theorem on the volumes of figures contained between parallel planes (now called Cavalieri’s theorem) was known all over Europe. At about the same time, the French mathematician René Descartes’ La Géométrie appeared. In this important work, Descartes showed how to use algebra to describe curves and obtain an algebraic analysis of geometric problems. A codiscoverer of this analytic geometry was the French mathematician Pierre de Fermat, who also discovered a method of finding the greatest or least value of some algebraic expressions – a method close to those now used in differential calculus.

(3)Although many other mathematicians of the time came close to discovering calculus, the real founders were Newton and Leibniz. Newton’s discovery (1665-66) combined infinite sums (infinite series), the binomial theorem for fractional exponents, and the algebraic expression of the inverse relation between tangents and areas into methods we know today as calculus. Newton, however, was reluctant to publish, so Leibniz became recognized as a codiscoverer because he published his discovery of differential calculus in 1684 and of integral calculus in 1686. It was Leibniz, also, who replaced Newton’s symbols with those familiar today.

(4)Despite the advances in technique, calculus remained without logical foundations. Only in 1821 did the French mathematician A. L. Cauchy succeed in giving a secure foundation to the subject by his theory of limits, a purely arithmetic theory that did not depend on geometric intuition or infinitesimals. Cauchy then showed how this could be used to give a logical account of the ideas of continuity, derivatives, integrals, and infinite series. In the next decade, the Russian mathematician N. I. Lobachevsky and German mathematician P. G. L. Dirichlet both gave the definition of a function as a correspondence between two sets of real numbers, and the logical foundations of calculus were completed by the German mathematician

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J. W. R. Dedekind in his theory of real numbers, in 1872.

(Abridged from Berggren, J. L., Singer, J. Calculus, Microsoft Encarta, 2008)

1. Read the text and decide whether the following statements are TRUE, FALSE or there is NO such INFORMATION in the text.

1. Methods for finding areas and volumes of a great variety of figures had been developed by the beginning of the 17th

2. Decide which part of the text (1, 2, 3, 4) contains the following information.

Archimedes showed that the value of π lay between approximately 3.1429 and approximately 3.1408.

One of Descartes most enduring legacies was his development of Cartesian geometry which uses algebra to describe geometry.

3. Give the answer to the following question.

Why was René Descartes’ La Géométrie so important for the development of calculus?

1.Descartes invented the method to determine the number of positive and negative roots of a polynomial.

2.This work provided the foundation for Infinitesimal calculus in Europe.

3.In this book he outlined his views on the universe.

4.He showed that the angular radius of a rainbow is 42 degrees.

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4. What is the main idea of the text?

1.The development of calculus was built on earlier concepts of instantaneous motion and area underneath curves.

2.Leibniz and Newton are usually both credited with the invention of calculus.

3.Many outstanding mathematicians contributed to the continuing development of calculus.

4.A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis.

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