UNIVERZITET CRNE GORE INSTITUT ZA STRANE JEZIKE

MILICA VUKOVIĆ

SKRIPTA ZA ENGLESKI JEZIK 2 (JEZIK STRUKE)

ZA STUDENTE PRIMJENJENE I TEORIJSKE MATEMATIKE PRIRODNO-MATEMATIČKOG FAKULTETA

PODGORICA, 2012.

UNIT 1 I PRE-READING TASK

Why do you like studying maths? Where do you see yourself in 10 years from now? What is your favourite field in modern maths? Do you know the word “algebra”? Do you know the adjective of the noun “algebra”? Can you name a new division of algebra?

II READ THE FOLLOWING TEXT

MY FUTURE PROFESSION

When a person leaves high school, he understands that the time to choose his future profession has come. It is not easy to make the right choice of future profession and job at once. Leaving school is the beginning of independent life and the start of a more serious examination of one’s abilities and character. As a result, it is difficult for many school leavers to give a definite and right answer straight away. This year, I have managed to pass the entrance exam and now I am a “freshman” at Moscow Lomonosov University’s Mathematics and Mechanics Department, world-famous for its high reputation and image. I have always been interested in maths. In high school my favourite subject was Algebra. I was very fond of solving algebraic equations, but this was elementary school algebra. This is not the case with university algebra which is far more complex. Now I am a first term student and I am studying the fundamentals of calculus. I haven’t made up my mind yet which field of maths to specialize in. I’m going to make my final decision when I am in my fifth year busy with my research diploma project and after consulting with my scientific supervisor. At present, I would like to be a maths teacher. To my mind, it is a very noble profession. It is very difficult to become a good maths teacher. Undoubtedly, you should know the subject you teach perfectly, you should be well-educated and broad-minded. An ignorant teacher teaches ignorance, a fearful teacher teaches fear, a bored teacher teaches boredom. But a good teacher develops in his students the burning desire to master all branches of modern maths, its essence, influence, wide–range and beauty. All our department graduates are sure to get jobs they would like to have. I hope the same will hold true for me.1

1 Adapted from: H.T. Phuong and L.T. Van, English for Mathematics, Minh City University of Education, 2003.

3

III COMPREHENSION CHECK 1. Are these sentences True (T) or False (F) or is there no evidence (NE)? Correct the false sentences. a. The author has successfully passed an entrance exam to enter the Mathematics and Mechanics Department of Moscow Lomonosov University. ________ b. He liked all subjects when he was at high school. ________ c. This year he’s going to choose a field of maths to specialize in. ________ d. A good teacher of maths will inspire students to study maths. ________ e. His maths degree course lasts four years. ________ 2. Complete the sentences below. a. To enter a college or university and become a student you have to pass __________________. b. Students are going to write their __________________ in the final year at university. c. University students show their essays to their __________________. 3. Answer the following questions. a. Why does the author want to be a maths teacher? b. When did his interest in maths start? c. Why is it difficult to be a good maths teacher? IV DISCUSSION

a. What personal qualities should a maths teacher have? b. Put the following in order of importance.

sense of humour good knowledge of maths sense of adventure children – loving patience intelligence reliability good teaching method kindness interest in maths

c. Do you expect to be a good teacher? d. Describe the best / the worst teacher who taught you. e. What difficulties do freshmen come across in their studies? f. What do you like the most about mathematics? g. When did you start to think of yourself as good at math?

4

UNIT 2 – BASIC TERMS

Writing and saying numbers Numbers over 20 are written with a hyphen: 35 thirty-five 67 sixty-seven

Numbers over 100 329 three hundred and twenty-nine The and is pronounced /n/ In AmE the and is sometimes left out

Numbers over 1000 1100 one thousand one hundred (also informal: eleven hundred)

2500 two thousand five hundred (also informal, esp. in AmE twenty-five hundred) Note: these informal forms are most common for whole hundreds between 1100 and 1900.

A comma or (in BrE) a space is often used to divide large numbers into groups of three figures: 33,423 or 33 423 (thirty three thousand four hundred and twenty-three) 2,768,941 or 2 768 941 (two million seven hundred and sixty-eight thousand nine hundred and forty-one)

Ordinal numbers

1st first 2nd second 3rd third 4th fourth

5th fifth 9th ninth 12th twelfth 21 twenty-first etc.

Fractions

½ a / one half 1/3 a / one third ¼ a / one quarter (in AmE also a / one fourth) 1/12 one twelfth

1/16 one sixteenth 2/3 two thirds ¾ three quarters 9/10 nine tenths

More complex fractions: use over 19/56 nineteen over fifty-six 31/144 thirty-one over one four four Whole numbers and fractions: link with and 2 ½ two and a half 5 2/3 five and two thirds

5

Fractions/percentages and noun phrases: use of

three quarters of the population 75% of the population with half do not use a, and of can sometimes be omitted:

Half (of) the work is already finished. do not use of in expressions of measurement or quantity:

half an hour half a litre use of before pronouns

half of us

Decimals write and say with a point (.) (not a comma!) say each figure after the point separately

79.3 seventy-nine point three 3.142 three point one four two 0.67 zero point six seven

Mathematical expressions + plus - minus x times / multiplied by ÷ divided by = equals / is % per cent (AmE percent) 32 three squared

53 five cubed 610 six to the power of ten √ square root of > is greater than < is less than ≥ is greater than or equal to ≤ is less than or equal to

I Write the following numbers in words:

348 _________________________________________________ 3,356 _________________________________________________ 5,412,312 _________________________________________________ 49/71 _________________________________________________ 0.54 _________________________________________________ 12th _________________________________________________

6

II Write the following mathematical statements in formulae: x equals one over two hundred and thirty-three ____________________________ a over b equals y to the power of five ____________________________ b squared equals five over six ____________________________ a plus or minus b ____________________________ three times five makes fifteen ____________________________ x to the power of seven ____________________________ y to the power of minus a ____________________________ a plus b, in brackets, all squared ____________________________ two times square root of three ____________________________ cube root (of) x ____________________________ n-th root (of) x ____________________________ the square root of four hundred and fifty divided by three plus seven ___________ six point five times ten to the minus three ____________________________ n factorial ____________________________ III Write the following mathematical statements in words:

20 + 17= 37 _________________________________________________ 48 -17= 31 _________________________________________________ 14∙ 2 = 28 _________________________________________________ 100:10=10 _________________________________________________ 6 < 7 _________________________________________________ 8 > 7 _________________________________________________

z ≤ 9 _________________________________________________ 45 ≥ x _________________________________________________ 93 _________________________________________________ √256 _________________________________________________

7

II SOME BASIC MATHEMATICAL TERMS

PART I

As you know, mathematics is the science that deals with space and number. Arithmetic, algebra, geometry and trigonometry are branches of mathematics. In this text you will learn the basic arithmetical and geometrical terms as they are used in English-speaking countries. The numbers 1,3,5,7,9 are called odd numbers, 2,4,6,8 are even numbers. They are also termed cardinal numbers, and they tell us how many things or persons there are in a set. Ordinal numbers, however, define the position of things in a series. These are formed by adding –th to the cardinal number, except for the first three, which run as follows: 1st (first), 2nd (second), 3rd (third). Note also the slight modifications in the ordinals: 5th (fifth), 9th (ninth), 12th (twelfth) and 20th (twentieth). The four elementary rules, or arithmetical operations, are addition, subtraction, multiplication, and division. The following expressions should be read like this: 10+3=13 ten plus three is thirteen and are equals 35-5=30 thirty-five minus five is thirty less equals 7∙9=63 seven times nine is sixty-three multiplied by equals 40:8=5 forty divided by eight is five equals The results of the four operations in the above mentioned equations are called the sum (or sum total), the remainder, the product, and the quotient, respectively. The number 50 368 or any other whole number is called an integer, and it contains five digits. A digit is any numeral from 0 to 9. On the other hand, the quantities ½ (one half), 2/3 (two thirds), ¾ (three quarters) are called vulgar fractions. They are also known as proper fractions because their numerators are smaller than their denominators. But if it is the other way round, they are called improper fractions, e.g. 3/2 (three halves), 5/4 (five fourths) etc. Fractions such as x/y and a/b should be read as x over y and a over b. A decimal fraction, however, is written and read in this manner: 2.7 (two point seven), 5.063 (five point zero six three), or .01 (point zero one).

8

You should also know how a number or mathematical expression is raised to a certain power. We speak of the second, third, fourth, etc. power of a number. For example, 42 is the second power of four, or simply, four squared (also four to the power of two). 53 is the third power of five or five cubed or five to the power of three. Similarly, a5 is a to the fifth power or the power of five, x-5 is x to the power of minus five. The small numbers (2,3,5,-5) are called indices or exponents, and they indicate the power to which a quantity is raised. The root of a number, as you know, represents a completely reversed procedure. Thus, the number 3 is the square root of 9 (√9), the cube root of 27 (3√27), the fourth root of 81 (4√81), etc. I Write the following mathematical expressions in words:

15 + 17= 32 _________________________________________________ 17 -17= 0 _________________________________________________ 13∙ 2 = 26 _________________________________________________ 100:4=25 _________________________________________________ 2 < 7 _________________________________________________ 8 > 5 _________________________________________________

x ≤ 9 _________________________________________________ 7 ≥ x _________________________________________________ 83 _________________________________________________ √64 _________________________________________________ 2 2/3 _________________________________________________ 21st _________________________________________________ x/y _________________________________________________

II Which mathematical operations are represented by the following statements:

3+5=8 _____________________ 8-5=3 _____________________ 8x5=40 _____________________ 40:5=8 _____________________

III Provide examples for the following:

a number containing 6 digits ______________________ a proper fraction ______________________ an improper fraction ______________________ an odd number ______________________ an even number ______________________ an ordinal number ______________________ a cardinal number ______________________

9

IV Complete the sentences with appropriate words: even exponent remainder quotient root digit sum equal branch multiplication

1. The number 57306 contains five __________________. 2. This ________________ of computer science is known as 'artificial

intelligence'. 3. Most of our employees work in New York; the _________________ are in

London. 4. A raised figure or symbol that shows how many times a quantity must be

multiplied by itself is called an __________________. 5. ________________ numbers can be divided by 2. 6. The cube __________________ of 64 is 4. 7. A number which is the result when one number is divided by another is called

a ___________________. 8. The ________________ of 7 and 12 is 19. 9. __________________ of cells leads to rapid growth of the organism. 10. A metre ___________________ 39.38 inches.

Part II

Geometry is that branch of mathematics which deals with the properties of lines, angles, surfaces and solids. While plane geometry is concerned with plane figures as triangles, squares, rectangles, circles, etc., solid geometry deals with solid figures, that is those having length, breadth and thickness. The triangle is a plane figure bounded by three sides and having three angles, whose total sum is 180°(degrees). A triangle containing a right angle is called a right-angled triangle. An angle that has less than 90° is an acute angle, whereas one having more than 90° is called an obtuse angle. A square is a quadrilateral figure, all the four sides of which are equal. The area of a rectangular figure is computed by multiplying the two adjacent sides. If one side is 5m long, and the other 6m, the area will be 5m by 6m, which makes 30 m2 (square metres). You should also distinguish between straight lines (horizontal, vertical and slanting) and curved lines or curves. The most common curves are the circle, ellipse, parabola and hyperbola. Some of the terms usually associated with the circle are: centre, circumference, radius, diametre, sector and arc. The cube is a solid body with six equal square sides. The sphere is a solid figure where every part of its surface is equidistant from the centre. The cone is a solid with a circular base tapering to a point. The pyramid is a solid figure with a polygonal or square base, whose sloping sides meet at an apex. The cylinder is a solid generated by a straight line moving parallel to itself and describing any fixed curve, especially a circle.

10

I Provide definitions for the following: plane geometry ______________________________________________________ solid geometry ______________________________________________________ a triangle _______________________________________________________ a square _______________________________________________________ a cube _______________________________________________________ an acute angle _______________________________________________________ II Match the synonyms: deal with calculate property usual breadth limited bounded width compute be about sth adjacent neighbouring distinguish produce common characteristic generate differentiate III Find the antonyms of the following words in the text: different _____________ straight _____________ divide _____________ rare _____________ dissociate _____________ IV Draw the following shapes and solids: a cube an acute angle a curved line a cylinder a cone a circle

11

V Name these figures and bodies:

_____________________ __________________ _______________ ______________________

_______________ ________________ VI Match the geometric terms with their definitions:

obtuse triangle ray regular polygon

rhombus equilateral triangle or equiangular triangle isosceles triangle

intersection point complementary angles trapezoid scalene triangle acute triangle

supplementary angles endpoint parallel right triangle parallelogram the Pythagorean theorem

1. A ______________ is one of the basic terms in geometry. We may think of a it as a "dot" on a piece of paper. We identify it with a number or letter. It has no length or width.

12

2. ______________ - This term is used when lines, rays, line segments or figures meet, that is, they share a common point. Example: In the diagram below, line AB and line GH intersect at point D:

3. A ______________ is one of the basic terms in geometry. We may think of it as a "straight" line that begins at a certain point and extends forever in one direction. The point where it begins is known as its endpoint.

4. An ______________ is a point used to define a line segment or ray. A line segment has two of them; a ray has one.

5. Two lines in the same plane which never intersect are called ______________ lines. (line 1 || line 2)

6. Two angles are called ______________________ if the sum of their degree measurements equals 90 degrees. One angle is the complement of the other.

13

7. Two angles are called __________________ if the sum of their degree measurements equals 180 degrees. One angle is the supplement of the other.

8. A ______________ is a polygon whose sides are all the same length, and whose angles are all the same. The sum of the angles of a polygon with n sides, where n is 3 or more, is 180° × (n - 2) degrees.

9. ______________ - A triangle having all three sides of equal length. Its angles all measure 60 degrees.

10. ______________ - A triangle having two sides of equal length.

11. ______________ - A triangle having three sides of different lengths.

12. ______________ - A triangle having three acute angles.

13. ______________ - A triangle having an obtuse angle. One of the angles of the triangle measures more than 90 degrees.

14. ______________ - A triangle having an angle which measures 90 degrees. The side opposite the right angle is called the hypotenuse. The two sides that form the right angle are called the legs. This triangle has the special property that the sum of the squares of the lengths of the legs equals the square of the hypotenuse.

The hypotenuse has length 5, and we see that 32 + 42 = 52 according to 15. ________________________.

16. ______________ - A four-sided polygon with two pairs of parallel sides. The sum its angles is 360 degrees.

17. ______________ - A four-sided polygon having all four sides of equal length. The sum of its angles is 360 degrees.

14

18. ______________ - A four-sided polygon having exactly one pair of parallel sides. The two sides that are parallel are called the bases. The sum of its angles is 360 degrees.

VII Complete the table:

noun adjective verb intersection

segmented supplement complementary

parallel pointy side

VIII Identify the triangle type:

___________ ___________ ___________ ___________ ___________

15

UNIT 3

I PRE-READING TASK

What is autism? How would you describe a genius? Do you know anyone whom you would

call a genius? What makes them special? Who are autistic savants:

a) people with great knowledge and ability b) people who are less intelligent than others but who have

particular unusual abilities that other people do not have Have you seen the film Rain Man? Read the following extracts from the film Rain Man. Raymond is an autistic savant. What is he like?

Charlie: I'm going to see you in 2 weeks. Now, how many days is that before we'll be together? Raymond: 14 days from today, today's Wednesday. Charlie: Hours? Raymond: 336 hours. Of course that's 20,160 minutes. 1,290,600 seconds. Charlie: Who took this picture? Raymond: D-A-D. Charlie: And you lived with us? Raymond: Yeah, 10962 Beachcrest Street, Cincinnati, Ohio. Charlie: When did you leave? Raymond: January 12, 1965. Very snowy that day. 7.2 inches of snow that day. Charlie: Just after Mom died. Raymond: Yeah. Mom died January 5, 1965. Charlie: You remember that day. Was I there? Where was I? Raymond: You were in the window. You waved to me, "Bye bye Rain Man", "Bye bye." [after Ray spills a box of toothpicks on the floor] Raymond: 82, 82, 82. Charlie: 82 what? Raymond: Toothpicks. Charlie: There's a lot more than 82 toothpicks, Ray. Raymond: 246 total. Charlie: How many were there in the box? Sally Dibbs: 250. Charlie: Pretty close. Sally Dibbs: There's four left in the box. Charlie: That's amazing. He should work for NASA or something like that. Doctor: Ray, if you had a dollar and you spent fifty cents, how much would you have left over? Raymond: About seventy. Doctor: Seventy cents? Raymond: Seventy cents. Charlie: So much for the NASA idea.

16

II READING (Adapted from the Guardian)

Daniel Tammet is an autistic savant. He can perform extremely complex mathematical calculations at breakneck speeds. But unlike other savants, Tammet can describe how he does it. He speaks seven languages and is even devising his own language. Now scientists are asking whether his exceptional abilities are the key to unlock the secrets of autism.

Interview by Richard Johnson Daniel Tammet is talking. As he talks, he studies my shirt and counts the stitches. Ever since the age of three, when he suffered an epileptic fit, Tammet has been obsessed with counting. Now he is 26, and a mathematical genius who can figure out cube roots quicker than a calculator and recall pi to 22,514 decimal places. He also happens to be autistic, which is why he can't drive a car or tell right from left. He lives with extraordinary ability and disability.

Tammet is calculating 377 multiplied by 795. Actually, he isn't "calculating": there is nothing conscious about what he is doing. He arrives at the answer instantly. Since his epileptic fit, he has been able to see numbers as shapes, colours and textures. The number two, for instance, is a motion, and five is a thunder. "When I multiply numbers together, I see two shapes. The image starts to change and evolve, and a third shape emerges. That's the answer. It's mental imagery. It's like maths without having to think."

Tammet is a "savant", an individual with an astonishing, extraordinary mental ability. An estimated 10% of the autistic population - and an estimated 1% of the non-autistic population - have savant abilities, but no one knows exactly why. Professor Allan Snyder, from the Centre for the Mind at the Australian National University in Canberra, explains why Tammet is of particular, and international, scientific interest. "Savants can't usually tell us how they do what they do," says Snyder. "It just comes to them. Daniel can. He describes what he sees in his head. That's why he's exciting. He could be the Rosetta Stone."

There are many theories about savants. Snyder, for instance, believes that we all possess the savant's extraordinary abilities - it is just a question of us learning how to access them. "Savants have usually had some kind of brain damage. And it's that brain damage which creates the savant. I think that it's possible for a perfectly normal person to have access to these abilities, so working with Daniel could be very instructive."

Scans of the brains of autistic savants suggest that the right hemisphere might be compensating for damage in the left hemisphere. While many savants

struggle with language, comprehension and logic (skills associated primarily with the left hemisphere), they often have amazing skills in mathematics and memory (primarily right hemisphere skills). Typically, savants have a limited vocabulary, but there is nothing limited about Tammet's vocabulary.

Tammet is softly spoken, and shy about making eye contact, which makes him seem younger than he is. He lives on the Kent coast, but never goes near the beach - there are too many pebbles to count. The thought of a mathematical problem with no solution makes him feel uncomfortable. Trips to the supermarket are always difficult. "There's too much mental stimulus. I have to look at every shape and texture. Every price, and every arrangement of fruit and vegetables. So instead of thinking, 'What cheese do I want this week?', I'm just really uncomfortable."

Autistic savants have displayed a wide range of talents, from reciting all nine volumes of Grove's Dictionary of Music to measuring exact distances with the naked eye. The blind American savant Leslie Lemke played Tchaikovsky's Piano Concerto No1, after he heard it for the first time. And the British savant Stephen Wiltshire was able to draw a highly accurate map of the London skyline from memory after a single helicopter trip over the city. Even so, Tammet could still turn out to be the more significant.

The savant syndrome is more frequently found in males than in females in an approximate ratio of 6 to 1. Savants can recall facts, numbers, license plates, maps, and extensive lists of sports and weather statistics after only seeing them once. Some savants can recall perfectly a very long sequence of music, numbers, or speech. Some, named mental calculators, can do exceptionally fast arithmetic, including prime factorization. Other skills include precisely measuring distances and angles by sight, calculating the day of the week for any given date, and being able to accurately calculate the passing of time without a clock. Most autistic savants have a single special skill while others have multiple skills. Usually these abilities are concrete, non-symbolic, right hemisphere skills as opposed to left hemisphere skills that tend to be more sequential, logical, and symbolic.

17

III COMPREHENSION CHECK 1. Are these sentences True (T) or False (F)? Correct the false sentences. a. All autistic people are savants. _________ b. Savants can usually explain how they do complex calculations. _________ c. There are not many female savants. _________ d. Savants have highly developed left hemisphere skills. _________ e. Left hemisphere is associated with language and comprehension skills. _________ 2. Answer the following questions. a. Why does Tammet avoid going to the beach? b. In what way could Tammet be the “Rosetta stone”? c. How do people become savants? d. What are some of the extraordinary skills that savants have? e. Do you agree with Professor Snyder who says that we all have the savant’s extraordinary skills but we do not know how to use them? 3. Complete the sentences below. a. Comprehension and logic are primarily associated to _____________ hemisphere. b. Most autistic savants have a single special skill whereas some have ______________ ones. c. Scientists are trying to ________________ the secrets of autism. IV VOCABULARY Use the following words to complete the sentences:

recall ability access perform devise

emerge possess display range significant

1. Almost everyone has some musical _________________. 2. The results of the experiment are not statistically __________________. 3. I'm afraid he doesn't _______________ a sense of humour. 4. This material is available in a huge _______________ of colours. 5. No new evidence _________________ during the investigation. 6. A new system has been ______________ to control traffic in the city. 7. She could not ______________ his name. She forgot it. 8. You need a password to get _________________ to the computer system. 9. Her work is ________________ in the gallery. 10. We need to ________________ an experiment to see if this works.

18

V GRAMMAR – REVISION

Zero conditional:

Iron rusts if it gets wet. Ice floats if you drop it in water. Water boils if you heat it to 100 degrees. If you don’t eat, you die. I Complete the rule:

If-clause main clause

If + ………………………… + ……………………………

II Complete the exercise using the zero conditional:

1. If you _______________ (divide) 20 by 5, you _______________ (get) 4. 2. If x ______________ (be) 3, then the final result _______________ (be) 6. 3. If a sequence _______________ (have) a definite number of elements, it

______________ (call) finite. 4. If the size of angle ______________ (be) 90°, we ______________ (call) it a

right angle. 5. If there ________________ (be) at least one element in the set B that is not in

the set A, then B ≠ A. 6. If the sum of the digits of a number ________________ (be) divisible by

three, the number _________________ (be) divisible by three. 7. Whenever I __________________ (not do) my homework, my professor of

maths __________________ (get) angry. 8. If you __________________ (heat) ice, ___________ it ____________

(melt)? 9. People ___________________ (get) hungry, if they __________________

(not eat). 10. When I __________________ (not know) how to solve an equation, I

_________________ (ask) my friend. He knows everything!

First conditional:

You won’t pass the course if you don’t study. If you repair my bike, I will help you with your maths homework. If our professor learns about this, we will be in serious trouble. If he doesn’t stop talking, we won’t be able to solve this equation.

19

III Complete the rule:

If-clause main clause

If + ………………………… + ……………………………..

IV Complete the exercise using the first conditional:

1. If you _______________ (draw) a straight line, you ________________ (divide) the angle.

2. If you _______________ (follow) this rule, you _______________ (find) the solution.

3. If you _______________ (consider) the second example, you ______________ (see) that the greatest common divisor is 2.

4. What _____________ the remainder _______________ (be), if you ________________ (divide) 25 by 4?

5. If you _______________ (perform) these calculations, you ______________ (get) the result.

6. If the exam _________________ (be) tomorrow, what _________________ (you, do)?

7. We ___________________ (invite) Marko to study with us if he __________________ (be) free.

8. You __________________ (fail) if you __________________ (not try) harder.

9. If I ____________________ (be not) busy, I ____________________ (come) to pick you up.

10. If the questions ____________________ (be) easy enough, everyone ___________________ (pass) the test.

Complete the rule:

If I were you, I’d do the test more carefully. He’s so stupid! If he were an animal, he’d be a sheep. We would learn a lot from dolphins if they could talk. If the weather was better, we would study maths in the park.

V Second conditional:

If-clause main clause

If + ………………………… + ……………………………..

20

VI Complete the exercise using the second conditional:

1. What ________________ (happen) if you _______________ (count) to one billion?

2. If I ________________ (be) you, I _______________ (give up) and _________________ (ask) the teacher.

3. If Zeno’s paradoxes ________________ (be not) so subtle and colourful, mathmaticians ________________ (pay not) attention to them.

4. If I ______________ (know) the solution, I _______________ (ask, not) you. 5. What _________________ (you, do) if you _______________ (win) a Nobel

prize for mathematics? 6. If it __________________ (snow) next July, __________________ (you, be)

surprised? 7. If I _________________ (be) an alien, I ____________________ (be able) to

travel round the universe. 8. If I __________________ (win) the lottery, I ___________________ (give)

all money to the charity. 9. If we _________________ (control) our spending a bit better, we

_________________ (save) a lot of money. 10. If I _________________ (be) 18 again, I ____________________ (go) on a

round-the-world tour.

VII Complete the exercise using the correct form of the conditional:

1. If I ______________ (know) maths better, I _______________ (help) you with this problem. Unfortunately, I am not good at it.

2. If you _______________ (multiply) five by six, you ________________ (get) thirty.

3. If I ________________ (be) good at maths, I _____________ (study) it. However, I am very bad at it.

4. If you ________________ (use) this theorem, you ________________ (find) the result soon.

5. If you ________________ (study) hard, you __________________ (pass) your maths exam.

6. If Tammet _______________ (go) to the beach, he _______________ (get) a headache.

7. If a person _______________ (be) an autistic savant, then that person _______________ (possess) some extraordinary ability.

8. When we ________________ (learn) how to use these skills, we _________________ (be able) to perform calculations the same way the savants do.

21

IV UNIT – NUMBER THEORY

I Choose the correct answer to best complete the text:

Number Theory

Number theory is a part of mathematics. It explains 1) ________________ (whole) numbers are, and what properties they have.

The 2) ________________ topics in number theory are prime numbers and factorization. A prime number is a positive, whole number that is special in some ways. For a prime number, there are exactly two whole numbers that 3) ________________ it (with no remainder). These divisors are the number 4) ________________ and 1. No other numbers will divide it 5) ________________. For example, 7 is a prime number, because the only numbers that divide it evenly are 1 and 7. However, 1 is not a prime number, there is only one number that divides it with no leftover. 0 is not a prime number, since divide by zero cannot 6) ________________.

Factorization is taking a composite number apart 7) ________________ numbers that multiply together to get the original number. These smaller numbers are called factors or divisors. 1 is a factor of all numbers. For example, twelve 8) ________________ be factorized as 4 × 3. Since 4 is not a prime number, that is not its prime factorization. 12's prime factorization is in fact 3 × 2 × 2.

If the numbers which 9) ________________ from the factorization are ordered, for example, starting with the smallest number, the factorization of every number is unique. This generalizes to:

1. Every number has a unique prime factorization; 2. Every prime factorization corresponds 10) ________________ a unique

number.2

1. a) that b) which c) what d) how 2. a) many b) main c) leading d) some 3. a) divide b) subtract c) multiply d) share 4. a) himself b) itself c) only d) whole 5. a) precise b) exact c) perfect d) exactly 6. a) do b) done c) was done d) be done 7. a) into b) to c) on d) from 8. a) must b) should c) ought to d) can 9. a) obtained b) are got c) are obtained d) obtain 10. a) in b) to c) into d) towards 2 Adapted from Simple Wikipedia Encyclopedia.

22

II Find the words in the text which have the following meaning:

o separate ________________ o a number that can be divided only by itself and 1 ________________ o characteristic ________________ o complete ________________ o being the only one of its kind ________________ o leftover ________________ o complex ________________ o to be the same as or match sth ________________

III Decide if the following statements are true or false:

1. Prime numbers are composite numbers. _______ 2. Prime numbers can be negative. _______ 3. Zero is not a prime number. _______ 4. The prime factorization of every number is unique. _______ 5. Factorization is a topic of little interest in number theory. _______

IV Translate the following sentences: 1. A prime number is a positive, whole number that is special in some ways. _____________________________________________________________________ 2. The theory of Diophantine equations has even been shown to be undecidable. _____________________________________________________________________ 3. Many questions in number theory require new approaches outside the realm of elementary number theory to solve. _____________________________________________________________________ 4. In elementary number theory, integers are studied using techniques from various mathematical fields. _____________________________________________________________________ 5. The study of perfect numbers is a subfield of elementary number theory. _____________________________________________________________________ 6. Some problems in elementary number theory can only be solved if approached from other mathematical fields. _____________________________________________________________________7. The theory of Diophantine equations has been solved recently. _____________________________________________________________________ 8. Factorization is taking a composite number apart into numbers that multiply together to get the original number. _____________________________________________________________________ 9. If the numbers which are obtained from the factorization are ordered, for example, starting with the smallest number, the factorization of every number is unique. _____________________________________________________________________ 10. However, 1 is not a prime number, there is only one number that divides it with no leftover. _____________________________________________________________________

23

V Complete the exercise with Present Simple, active or passive:

Number theory _____________ (subdivide) into several fields, according to the methods which _____________ (use) and the type of questions which _____________ (investigate).

The term "arithmetic" _____________ (use, also) to refer to number theory. This is a somewhat older term, which _____________ (be) no longer as popular as it once was. Number theory used to be called the higher arithmetic, but this term _______________ (avoid, now). Nevertheless, it still _____________ (show up) in the names of mathematical fields. Mathematicians working in the field of number theory _____________ (call) number theorists.

VI Complete the exercise with Past Simple, active or passive:

Pythagoras _____________ (live) in the 500's BC, and was one of the first Greek mathematical thinkers. He _____________ (spend) most of his life in the Greek colonies in Sicily and southern Italy. He _____________ (have) a group of followers (like the disciples of Jesus) who _____________ (follow) him around and _____________ (teach) other people what he had taught them. The Pythagoreans _____________ (know) for their pure lives (they _____________ (not eat) beans, for example, because it _____________ (think) beans were not pure enough). They _____________ (wear) their hair long, and only simple clothing _____________ (wear). They _____________ (go) barefoot. Both men and women _____________ (be) Pythagoreans. Pythagoreans were interested in philosophy, but especially in music and mathematics, two ways of making order out of chaos. VII Rewrite the sentences in the passive: 1. Mathematicians investigate perfect numbers within the framework of elementary number theory. _____________________________________________________________________ 2. He proved his last theorem in 1994. _____________________________________________________________________ 3. The students didn’t understand the Chinese remainder theorem. _____________________________________________________________________ 4. They haven’t solved many questions in number theory. _____________________________________________________________________ 5. She didn’t compute the highest common divisor correctly. _____________________________________________________________________ 6. The students will learn the properties of multiplicative functions. _____________________________________________________________________7. Mathematicians have made several important discoveries in this area. _____________________________________________________________________ 8. He is studying maths at the moment. _____________________________________________________________________

24

9. You cannot divide one by zero. _____________________________________________________________________ 10. You may factorize 12 as 4 x 3. _____________________________________________________________________

VIII Put the verbs in brackets into the correct tense (active or passive):

Perfect numbers

In mathematics, a perfect number _____________ (define) as a positive integer which _____________ (be) the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number itself. The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128. Euclid ______________ (discover) that the first four perfect numbers ___________ (generate) by the formula 2n−1(2n − 1):

for n = 2: 21(22 − 1) = 6 for n = 3: 22(23 − 1) = 28 for n = 5: 24(25 − 1) = 496 for n = 7: 26(27 − 1) = 8128

Noticing that 2n − 1 is a prime number in each instance, Euclid _____________ (prove) that the formula 2n−1(2n − 1) ____________ (give) an even perfect number whenever 2n − 1 is prime. It is unknown whether there ____________ (be) any odd perfect numbers. Various results _____________ (obtain), but none that ___________ (help) to locate one or otherwise resolve the question of their existence.

IX Read the text and complete the exercise below:

David Hilbert*

One day when commercial air travel was still in its infancy, the great mathematician David Hilbert was invited to give a talk on any subject he liked. His chosen subject - "The Proof of Fermat's Last Theorem" - came as something of a surprise, particularly given that the famous theorem, as far as anyone knew, remained unproven (see below). Needless to say, the event was eagerly anticipated... Soon enough, the momentous day arrived and Hilbert delivered his lecture. While undeniably brilliant, however, it had nothing to do with Fermat's theorem. After the talk, Hilbert was asked why he had chosen a title which had nothing to do with his lecture. "Oh," he replied, "that was just in case the plane went down."3

3 [Proving Fermat's Last Theorem (that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2) had presented a tempting challenge to mathematicians ever since Fermat's death, whereupon his son Samuel had found a curious marginal note in a copy of Diophantus's Arithmetica: "I have discovered a truly remarkable proof," it read, "which this margin is too small to contain." (The proof was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton.)]

25

Turn direct into indirect speech:

1. ‘I am going to deliver a lecture on Fermat’s last theorem,’ Hilbert said. _____________________________________________________________________ 2. ‘Why did you choose a title that has nothing to do with your lecture?’ other mathematicians asked Hilbert. _____________________________________________________________________ 3. ‘That is just in case the plane goes down,’ Hilbert explained. _____________________________________________________________________ 4. ‘This margin is too small to contain my proof,’ Fermat said. _____________________________________________________________________ 5. ‘I have found a curious marginal note in my father’s book,’ Samuel said. _____________________________________________________________________ 6. ‘I completed the proof in 1993,’ Andrew Wiles boasted. _____________________________________________________________________ 7. ‘David, please come and give a talk on any subject you like,’ the professor said. _____________________________________________________________________ 8. ‘I have chosen "The Proof of Fermat's Last Theorem" for my subject,’ Hilbert said. _____________________________________________________________________ 9. ‘We are eagerly anticipating his lecture,’ the students said. _____________________________________________________________________ 10. ‘I have discovered a truly remarkable proof, which this margin is too small to contain,’ the note read. _____________________________________________________________________

* Hilbert, David (1862-1943) German mathematician and professor

26

V UNIT - APPLIED MATHEMATICS

I Match the subtitles to paragraphs:

a) Applied mathematics vs. applications of mathematics b) History of applied mathematics c) Most successful applications of mathematics d) Definition e) Contemporary applied mathematics

________ Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.

There is no consensus of what the various branches of applied mathematics are. Such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees.

________ Historically, applied mathematics consisted principally of applied analysis, most notably differential equations, approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis), and applied probability. These areas of mathematics were intimately tied to the development of Newtonian Physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a legacy as well; until the early 20th century subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments.

________ Today, the term applied mathematics is used in a much broader sense. It includes the classical areas of analysis such as differential equations, as well as linear algebra, numerical analysis, probability, operations research, and other areas. Recently, fields such as number theory and topology, often thought to be pure mathematics, have become increasingly important as applications, though they are not generally considered to be part of the field of applied mathematics per se.

________ Mathematicians distinguish between applied mathematics, which is concerned with mathematical methods, and applications of mathematics within science and engineering. A biologist using a population model and applying known mathematics would not be doing applied mathematics, but rather using it. However, nonmathematicians do not usually draw this distinction.

________ The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics, computational science, and computational engineering, which use high performance computing for the simulation of phenomena and solution of problems in the sciences and engineering. These are often considered interdisciplinary programs.4

4 Adapted from the Wikipedia Encyclopedia.

27

II Answer the following questions: 1. What is applied mathematics? 2. Explain the relationship between physics and applied mathematics. 3. What is understood under the term pure mathematics? 4. What is the difference between using and doing applied mathematics? 5. What are some of the newest branches within the framework of applied mathematics? III Find the words in the text which have the following meaning: 1. ______________: the practical use of sth, especially a theory, discovery, etc 2. ______________: generally, without considering details 3. ______________: a situation that exists now because of events, actions from the past 4. ______________: clear difference or contrast 5. ______________: to attach or hold two or more things together 6. ______________: especially 7. ______________: in a way that clearly shows the differences between two things 8. ______________: a liquid; a substance that can flow 9. ______________: an area of knowledge studied in a school, college, etc 10. ______________: not mixed with anything else; with nothing added 11. ______________: (from Latin) used meaning 'by itself' to show that you are referring to sth on its own, rather than in connection with other things 12. ______________: a division of an area of knowledge 13. ______________: more and more all the time 14. ______________: to recognize the difference between two people or things 15 . ______________: becoming known, first appearance of sth IV Put in the correct preposition: 1. These areas ______ mathematics were intimately tied ______ the development of Newtonian Physics. 2. They taught classical mechanics ______ applied mathematics departments. 3. There are many applications of mathematics ______ science and engineering. 4. Categorizations are made difficult ______ the way mathematics and science change over time. 5. Mathematicians distinguish ______ applied mathematics and applications of mathematics. 6. There is no consensus ______ what falls into this area.

28

V Translate the sentences. 1. Nonmathematicians do not usually draw this distinction. __________________________________________________________________________________________________________________________________________ 2. The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics. __________________________________________________________________________________________________________________________________________ 3. The distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. __________________________________________________________________________________________________________________________________________ 4. Today, the term applied mathematics is used in a much broader sense. __________________________________________________________________________________________________________________________________________ 5. Recently, fields such as number theory and topology, have become increasingly important. __________________________________________________________________________________________________________________________________________ VI Correct the mistakes. 1. These is often considered interdisciplinary programs. _____________________________________________________________________ 2. It is including the classical areas of analysis such as differential equations. _____________________________________________________________________ 3. Until early 20th century subjects such as classical mechanics were often taught in applied mathematics departments. _____________________________________________________________________ 4. Fluid mechanics may still being taught in applied mathematics departments. _____________________________________________________________________ 5. The distinction between mathematicians and physicists has not been sharply drawn before the mid-19th century. _____________________________________________________________________ VII Expand the following phrases and clauses from the text: 1. This field is concerned with the techniques typically used in the application of mathematical knowledge. _____________________________________________________________________ 2. Recently, these fields, often thought to be pure mathematics, have become important. _____________________________________________________________________ 3. A biologist using a population model and applying known mathematics would not be doing applied mathematics. _____________________________________________________________________

29

4. If the numbers obtained from a factorization are ordered, every factorization is unique. _____________________________________________________________________ 5. The Goldbach conjecture concerning the expression of even numbers as sums of two primes falls into the area of number theory. _____________________________________________________________________6. The Collatz conjecture concerning a simple iteration also belongs into number theory. _____________________________________________________________________ 7. A statement issued by the American Statistical Association says that statistics does not belong to mathematics. _____________________________________________________________________

VIII Reduce the following clauses:

1. Mathematicians distinguish between applied mathematics, which is concerned with mathematical methods, and applications of mathematics. _____________________________________________________________________ 2. The term applied mathematics, which is used in a much broader sense, has changed its meaning. _____________________________________________________________________ 3. Applied mathematics, which includes the classical areas of analysis such as differential equations, is very interesting. _____________________________________________________________________ 4. Any expression like x + 5 or 2x – 3 that contains two or more terms may be called a polynomial expression. _____________________________________________________________________ 5. An axiom is a statement which is generally accepted as true without proof. _____________________________________________________________________ 6. A diametre is a line which passes through the centre of the circle. _____________________________________________________________________ 7. Points A and B that represent the opposite points of a circle are equidistant from the centre. _____________________________________________________________________ IX Put a, an or the where needed: a) _______ Pythagorean Theorem says that in a right triangle, _______ sum of the squares of the two right-angle sides will always be _______ same as the square of the hypotenuse (the long side). A2 + B2 = C2. b) Some mathematicians think that _______ statistics is a part of _______ applied mathematics. Others think it is _______ separate discipline. _______ statisticians in general regard their field as separate from _______ mathematics, and _______ American Statistical Association has issued a statement to that effect. _______ Mathematical statistics provides _______ theorems and proofs that justify statistical

30

procedures and it is based on _______ probability theory, which is in turn based on _______ measure theory. c) _______ Applied mathematics is _______ branch of _______ mathematics that concerns itself with mathematical techniques typically used in _______ application of mathematical knowledge to other domains. d) At some universities there is _______ considerable amount of _______ tension between _______ applied and _______ pure mathematics departments. X Supply the missing articles, where needed:

After receiving a sound education in _______ mathematics, _______ classics, and _______ law at La Flèche and Poitiers, René Descartes embarked on _______ brief career in military service with Prince Maurice in Holland and Bavaria. Unsatisfied with _______ scholastic philosophy and troubled by skepticism of _______ sort explained by Montaigne, Descartes soon conceived _______ comprehensive plan for applying mathematical methods in order to achieve perfect certainty in human knowledge. During _______ twenty-year period of secluded life in _______ Holland, he produced _______ body of work that secured his philosophical reputation. Descartes moved to _______ Sweden in 1649, but did not survive his first winter there.

XI Put the verbs into Past Simple, active or passive: Boring Lecturer Contest In 1971, professor David Coward ______________ (win) the 'Most Boring Lecturer of the Year' contest at Leeds University with a peerless discussion of 'the problems of the urinal'. In March 1986, however, Coward ______________ (leave) without his crown by Exeter University's Frank Oliver, who ______________ (deliver) a brilliantly dull lecture on a subject which he ______________ (call) 'essentially fascinating': Co-efficiency correlations. With his back to the audience, Oliver ______________ (use) a series of unintelligible blackboard diagrams to explain how to 'measure the strength of the relationship between two variables at points between minus one and plus one.' Such ______________ (be) his triumph in this annual competition that the event ______________ (cancel) for several years. When it ______________ (revive) in 1988, Oliver ______________ (win) yet again - by simply repeating his original lecture.

* Oliver, Frank (?- ) British professor

31

VI UNIT - COMBINATORICS

I Read the text and decide where the following sentences/clauses belong:

a) to obtain estimates on the number of elements of certain sets b) and finding algebraic structures these objects may have (algebraic combinatorics) c) These often focus on a partition or ordered partition of a set d) which also has numerous natural connections to other areas e) though it has developed powerful theoretical methods f) It is related to many other areas of mathematics.

Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ________ Aspects of combinatorics include "counting" the objects satisfying certain criteria (enumerative combinatorics), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), ________.

Combinatorics is as much about problem solving as theory building, ________ , especially since the later twentieth century. One of the oldest and most accessible parts of combinatorics is graph theory, ________.

There are many combinatorial patterns and theorems related to the structure of combinatoric sets. ________

An example of a combinatorial question is the following: What is the number of possible orderings of a deck of 52 distinct playing cards? The answer is 52! (52 factorial)5.

Combinatorics is used frequently in computer science ________. A mathematician who studies combinatorics is often referred to as a combinatorialist or combinatorist.

II Match the synonyms:

a) branch connected b) accessible different c) related often d) distinct field e) frequently mention f) refer reachable g) ordering division h) partition assessment i) obtain get j) estimate arrangement

5 Adapted from: www.wikipedia.com

32

III Answer the questions: 1. What theory emerged as one of the earliest within combinatorics? 2. Has combinatorics found any applications? 3. What is enumerative combinatorics? IV Modal verbs. Mark the clause which means the same as the sentence from the text: 1. Aspects of combinatorics include "counting" the objects satisfying certain criteria and deciding when the criteria can be met. a) …when one may accidentally meet the criteria b) … when it is possible to meet the criteria c) … the criteria need to be met 2. It also includes finding algebraic structures these objects may have. a) It is possible that these objects have algebraic structures. b) It is possible to find algebraic structures of these objects. c) …finding algebraic structures which these objects are able to have. V Complete sentences using should, must, have to or can: a. He is required to read his paper. He _________________ read his paper at the seminar. b. Tony was amazing. He ____________ multiply numbers in the thousands with the speed of a calculating machine when he was only seven years old. c. She ______________ summarize the result before she reports it to her boss. (It’s necessary that she does this.) d. The two rays of an angle ______________ not lie on the same straight line. e. I think you ______________ illustrate this problem in the figure. This may be the easiest way. f. Algebraic tools ____________ be used in a number of ways in combinatorics (it is possible to use them). VI Translate the following sentences: 1. Kombinatorika je našla primjenu u informatici. _____________________________________________________________________ 2. Matematičar koji se bavi kombinatorikom se zove kombinatorista. _____________________________________________________________________ 3. Kombinatorika je povezana sa mnogim oblastima matematike. _____________________________________________________________________

33

VII Choose the correct item:

Ramsey theory

Ramsey theory is a 1) ____________ part of extremal combinatorics. It 2) ____________ that any sufficiently large random configuration will contain some sort of order.

Frank P. Ramsey proved that 3) ____________ every integer k there is an integer n, such that every graph on n vertices either contains a clique or an independent set of size k. This is a special case of Ramsey's theorem. For example, 4) ____________ any group of six people, it is always the case that one can find three people out of this group that 5) ____________ all know each other or all do not know each other. The 6) ____________ to the proof in this case is the Pigeonhole Principle: either A knows three of the remaining people, or A does not know three of the remaining people.

Here is a simple proof: Take any of the six people, call him A. Either A knows three of the remaining people, or A does not know three of the remaining people. Assume the former (the proof is identical if we assume the latter). 7) ____________ the three people that A knows be B, C, and D. Now either two people from {B,C,D} know each other (in which case we have a group of three people who know each other - these two plus A) or 8) ____________ of B,C,D know each other (in which case we have a group of three people who do not know each other - B,C,D).

1) a) celebrity b) celebrating c) celebrated d) celebrates 2) a) stated b) is stating c) state d) states 3) a) for b) with c) by d) in 4) a) if give b) appointed c) dedicated d) given 5) a) or b) either c) nor d) whether 6) a) key b) point c) solution d) method 7) a) give b) decide c) let d) should 8) a) anyone b) none c) no d) all * {} – set brackets {a,b,c}- ‘the set of a, b and c’ IX Put the words in brackets into the correct tense:

Chess Chess ______________ (play) on a square chessboard with 64 squares (an eight-by-eight square). At the start, each player ______________ (control) sixteen pieces: one king, one queen, two rooks, two knights, two bishops, and eight pawns. The player should checkmate the opponent's king, whereby the king ______________ (be) under immediate attack and there is no way to remove it from attack on the next move.

34

Chess is interesting from the mathematical point of view. Many combinatorical and topological problems connected to chess ______________ (know) of for hundreds of years. The number of legal positions in chess ______________ (estimate) to be between 1043 and 1050. Typically an average position ______________ (have) thirty to forty possible moves, but there may be as few as zero (in the case of checkmate or stalemate) or as many as 218.

X Put a, an or the where needed:

1+1=1? While dining at Trinity College, ______ Cambridge, one evening, ______ great logician Bertrand Russell claimed that any false argument could be proven from ______ erroneous premise that 1+1=1 ( ______ notion which originated from Aristotle).

Russell was promptly challenged. "If 1+1=1, prove that you're ______ Pope." He thought for ______ moment before proceeding:

"I am one, ______ Pope is one," he declared. "Therefore, ______ Pope and I are one."

* Russell, Bertrand Arthur William (1872-1970) British philosopher, mathematician, social critic and writer

XI Complete the sentences with the correct form of the verbs in brackets:

a. Ever since Galileo _______________ (invent) his telescope men ________________ (study) the motions of the planets with ever increasing interest and accuracy. b. Kepler __________________ (deduce) his famous three laws describing the motion of the planets about the sun. c. The Englishman Thomas Harriot ________________ (be) the first mathematician who _________________ (give) status to negative numbers. d. We knew the solution of this problem because we __________________ (read) it in maths magazine. e. Amalic Emmy Noether ___________________ (publish) a series of papers focusing on the general theory of ideas for four years from 1922 to 1926. f. While W. Hamilton _________________ (walk) along the Royal canal, he ___________________ (discover) the multiplication formula that could __________________ (use) for the quaternions on the stones of a bridge over the canal.

35

VII UNIT - DISCRETE MATHEMATICS I Put a, an, or the where needed: _____ discrete mathematics, also called _____ finite mathematics, is _____ study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring _____ notion of continuity. _____ objects studied in finite mathematics are largely countable sets such as _____ integers, _____ finite graphs, and _____ formal languages. II Read the text and decide if the statements below are true or false:

Algorithms No generally accepted formal definition of "algorithm" exists yet. We can, however, derive an informal meaning of the word from the following quotation from Boolos and Jeffrey: "No human being can write fast enough, or long enough, or small enough to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols". Flowcharts may often used to graphically represent algorithms.

36

Algorithms are essential to the way computers process information, because a computer program is essentially an algorithm that tells the computer what specific steps to perform (in what specific order) in order to carry out a specified task, such as calculating employees’ paychecks or printing students’ report cards. Thus, an algorithm can be considered to be any sequence of operations that can be performed by a system. 1. Algorithms are quite easy to define. ________ 2. Algorithms represent a set of operations or instructions that a system can perform. ________ 3. Algorithms have found useful applications in many areas of everyday life and business. ________ 4. Computers couldn’t function without the application of algorithms. ________ 5. According to Boolos and Jeffrey, people could list members of infinite sets if they used some notation. ________ 6. The flowchart suggests that one should never buy a new lamp if the old one does not work. ________ II Find the words in the text that have the following meanings: - a diagram that shows the connections between the different stages of a process or parts of a system: ______________ - something that has no end: ______________ - completely necessary; extremely important in a particular situation or for a particular activity: ______________ - not seeming to be based on a reason, system or plan: ______________ - to do and complete a task: ______________ - to get sth from sth: ______________ - to calculate sth exactly: ______________ - a system of signs or symbols used to represent information, especially in mathematics, science and music: ______________ III Find the antonyms of the following words: refuse ______________ informal ______________ infinite ______________ explicit ______________

specific ______________ capable ______________ specified ______________ elementary ______________

37

IV Choose the option which expresses the same meaning as the proposed clauses and sentences: 1. We can, however, derive an informal meaning of the word from the following quotation. a) we have the possibility to derive… b) we are supposed to derive… c) we are allowed to derive… 2. No human being can write fast enough… a) No human has the possibility to write fast enough… b) No human is able to write fast enough… c) No human is allowed to write fast enough… 3. Flowcharts may often used to graphically represent algorithms. a) Flowcharts can often be used to graphically represent algorithms. b) We can maybe often use flowcharts to graphically represent algorithms. c) We are capable of often using flowcharts to graphically represent algorithms. 4. An algorithm can be considered to be any sequence of operations that can be performed by a system. a) An algorithm is any sequence of operations that can be performed by a system. b) An algorithm is maybe any sequence of operations that can be performed by a system. c) An algorithm is capable of being any sequence of operations that can be performed by a system. V The Language of Proof.

A theorem and its proof are typically laid out as follows:

Theorem (name of person who proved it and year of discovery, proof or publication). Statement of theorem. Proof. Description of proof.

The end of the proof may be signalled by the letters Q.E.D. or by one of the tombstone marks "□" or "∎", introduced by Paul Halmos following their usage in magazine articles.

Example:

If and then .

38

To show that we need to show that So we suppose By hypothesis, so Also by hypothesis, , so Since this was true for any arbitrary we have shown that

* A is a subset of C

For any/each x which is an element of A, x is an element of C

VI Complete the following proofs with appropriate items:

a) which concludes the proof if also then let __________ A and B are finite sets such that A = B __________ |A|=|B|.

Here we take advantage of the fact that A is a finite set. __________ n be the integer such that |A| = n. You should then index the elements of A so that

Now , so we see that B

has at least n elements, that is __________, every element of B is in A, so it follows that there are no more elements in B than there are in A, so , thus |B| = n = |A|, __________.

* {a, b, c} - the set of a, b and c

| | the cardinality of the set A

b) and if then either so

let assume consider this shows __________ A and B are finite sets __________ __________

.

We __________ that we have two finite sets A and B and that they do not have the same number of elements. __________ n = | A | and m = | B | . Then, number the

elements in A and B, so and . Since , __________ n < m or m < n. Without loss of generality, we assume that n < m. __________ the set B − A. Since A has only n elements, we can take out at

39

most n elements from B, leaving at least m-n elements in B-A. __________ that there is at least one element in B that is not in A, __________ .

* ≠ is not equal to; does not equal VII Fill the texts using only one word per gap:

Failed Mathematician

________ great mathematician David Hilbert ________ once asked ________ a certain former student. "For ________ mathematician he ________ not have enough imagination," Hilbert remarked. "But he ________ become a poet and now he ________ doing fine."

* Hilbert, David (1862-1943) German mathematician and professor

Proof of Death "My friend, G. H. Hardy, who ________ professor ________ pure mathematics," Bertrand Russell recalled, "told me once that if he could find a proof ________ I was going ________ die in five minutes he would ________ course be sorry to lose me, but this sorrow ________ be quite outweighted ________ pleasure in the proof. I entirely sympathised with him and was not at ________ offended."

* Russell, Bertrand Arthur William (1872-1970) British philosopher, mathematician, social critic and writer

40

UNIT 9 – GREAT MATHEMATICIANS I Pre-reading task. Complete the following quiz on famous mathematicians6: 1. This great Greek mathematician from 287-212 BC, is very famous for his attempts at the measurement of the circle. Aristotle Pythagoras Plato Archimedes 2. Greek philosopher whose theorem about the length of the hypotenuse is famous to many students. Pythagoras Euclid Archimedes Einstein 3. This family had eight great mathematicians. One of the sons, Daniel, had a very famous theorem. Erdos Fibonacci Bernoulli Riemann 4. Name one of the two men credited with discovering Calculus. Answer: ______________________________ 5. This young genius made contributions to group theory, but was killed in a duel battling for the heart of a lover at the age of 20. Carl Gauss Evariste Galois Leonhard Euler Augustin Cauchy 6. He is the founder of set theory and introduced the concept of infinite numbers and cardinal numbers. Leonard Euler Carl Gauss Georg Cantor Leonardo Fibonacci 7. The French philosopher born in 1596 whose work, 'La Geometrie,' led to Cartesian geometry. Pierre de Fermat Blaise Pascal Henri Poincare Rene Descartes 8. A French lawyer and government official, he had us stumped with his last theorem for centuries. Rene Descartes Pierre de Fermat Galileo Galilei Augustin Caucy 9. Born in Italy in 1564, a pioneer of modern applied mathematics, physics and astronomy, he is called the Father of Modern Science. Most students will know him for his work with pendulums and the dropping of different sized weights. Evariste Galois Galileo Galilei Bernhard Riemann Leonardo Fibonacci

6 Adapted from www.funtrivia.com and http://www.liz.richards.btinternet.co.uk/webpage4a.htm

41

10. The schizophrenic American mathematician who lived on to win a Nobel Prize and was portrayed by Russel Crowe in the movie, 'A Beautiful Mind.' Paul Erdos John Nash Albert Einstein Isaac Newton 11. He was a Greek Mathematician known for shouting "Eureka!" in his bathtub Galileo Pythagoras Archimedes Euclid 12. He dropped different weight balls from the tower of Pisa to show that they would hit the ground at the same time. Isaac Newton Galileo Pythagoras Euclid 13. He is known as the "Father of Geometry". Isaac Newton Galileo Pythagoras Euclid 14. This "Prince of Mathematicians" once quickly solved a problem where he was asked to add the first 100 numbers together. Gauss John Nash Isaac Newton Galileo 15. This Italian mathematician made telescopes to look at the moon. Gauss Isaac Newton Pythagoras Galileo II READING:

AN INTERVIEW WITH LEONARDO

FIBONACCI7

Fibonacci was the greatest mathematician of his age. He did not simply master the arts of geometry, arithmetic, trigonometry, and algebra, but also made his knowledge useful to all the businesses involving math. He eliminated use of complex Roman numerals and made mathematics more accessible to the public because he brought the Hindu-Arabic system (including zero) to Western Europe.

7 Adapted from: http://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/fibonacci2.html

42

Q. What is your name and its origin? A. “In actuality my original name was Leonardo, and back then people named each other according to location so I was Leonardo of Pisa. Yes, it is the same city as the famous leaning tower. Anyway, I decided to adopt the more professional name of Fibonacci, or “son of Bonacci”, as Guilielmo Bonacci was my father. Q. When were you born? A. “Sometime around 1175. My memory clouds now that I am around 800 years old; please forgive my vague personal knowledge. I don't even remember my wife's name. Anyway, I was born during the Dark Ages. I was a merchant and I traveled to the East and North Africa and was the one who popularized these new systems and modified them slightly. I remember that my interest in the Arabs and their strange numbers was part of what gave me so much advantage back home.” Q. Where were you raised and how did this affect you? A. “I was raised in Pisa, Italy. It was already an independent republic, a small city-state with a pretty large commerce and seaport. My father found work there and instructed me a bit in accounting.” Q. What do you think people see you as? A. “Most know me as a very serious scholar. I suppose that everyone currently knows about what kind of scholar; namely a mathematician, but I am also very interested in the laws and patterns of nature.” Q. Can you give any examples of how your mathematics are seen in nature? A.“The Fibonacci Sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. The formula basically is a guide to adding the previous two numbers in the Hindu-Arabic system to get a new number ad infinitam. Interestingly enough, this is found everywhere in living things because of the way things grow exponentially in nature. Also, did you notice how much art and music have to do with the sequence? If you look at piano keys or famous works of art you will always see recurring patterns obviously of the "Golden numbers.” We take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13) and we divide each by the number before it, we will find the following series of numbers: = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538... The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. It has a value of approximately 1.61804.” Q. What are your basic achievements? A.” My most important was my role in bringing Eastern mathematics into Western mathematics. You may even be familiar with the fact that I introduced the fractional bar because the numbers were otherwise rather confusing in accountant notation.” Q. What do you think your greatest contribution has been to the world? A. “I believe that my book, Liber Abbaci, was the most important thing I put into creation. It seriously aided the introduction of Hindu-Arabic numerals to Western Europe and set a solid example for arithmetic, geometry, and algebra, but more importantly a sturdy foundation for purely theoretical applications of math. The cool thing is that it was noticed by the more common people and actually used. That is the greatest thing a mathematician can hope for- the integration of his work into the systems of the world!”

43

I Decide whether the following statements are true or false:

1. Fibonacci’s works found useful applications in everyday life. 2. Fibonacci invented the Arabic numeral notation. 3. He was named after his father. 4. He lived in the 11th century in Italy. 5. Fibonacci was instructed in accounting and travelled widely, which helped him to bring his mathematical work into creation. 6. He was interested only in mathematics. 7. He introduced the fractional bar in the notation system. 8. This is a true interview with one of the greatest mathematicians of all times.

II Match the synonyms:

ordinary ad infinitam

infinitely master

strong slightly

a little sturdy

learn vague

at the moment currently

unclear common

III Match the words with their meanings:

recurring value pattern approximately contribution

achievement ratio successive accessible foundation

1. the relationship between two groups of people or things that is represented by two numbers showing how much larger one group is than the other _________________

2. a regular arrangement of lines, shapes, colours, etc. as a design on material, carpets, etc _________________

3. how much sth is worth _________________ 4. following immediately one after the other _________________ 5. to be similar or close to sth in nature, quality, amount, etc., but not exactly the same

_________________ 6. an action or a service that helps to cause or increase sth _________________ 7. that can be reached, entered, used, seen, etc _________________ 8. happening again _________________ 9. a principle, an idea or a fact that sth is based on and that it grows from _________________ 10. a thing that sb has done successfully, especially using their own effort and skill _________________

44

IV Use the following words from the text to best complete the sentences:

achievement accounting fractional bar contribution

ratio value pattern approximately

1. Fractions can be expressed as a decimal or as two numbers separated by a ________________. 2. The symbol π is used to show the _______________ of the circumference of (= distance around) a

circle to its diametre (= distance across). Its _______________ is _______________ 3.14159. 3. One of Fibonacci’s greatest _______________ was to bring Eastern mathematics into Western

mathematics.

GRAMMAR AND VOCABULARY REVISION8

I Complete the texts using the correct form of the verb in brackets:

A) Charles Babbage The English mathematician Charles Babbage, famed for his invention of an early mechanical computer (the so-called "Analytical Engine"), once ____________ (take) issue with one of Tennyson's poems. The poet soon ______________ (receive) a letter from the logician: "In your otherwise beautiful poem," Babbage wrote, "one verse ______________ (read), Every moment dies a man, Every moment one ______________ (be) born. "If this ______________ (be) true, the population of the world ______________ (be) at a standstill. In truth, the rate of birth is slightly in excess of that of death. I would suggest: Every moment ______________ (die) a man, Every moment 1 1/16 is born. "Strictly ______________ (speak)," Babbage ______________ (add), "the actual figure is so long I cannot get it into a line, but I ______________ (believe) the figure 1 1/16 ______________ (be) sufficiently accurate for poetry." B) Isaac Newton "The popular idea of mathematics is that it ______________ (concern, largely) with calculations. What many people ______________ (not, realize) - and mathematicians at parties ______________ (give up) correcting them - is that mathematicians ______________ (be) often no better calculators, and sometimes worse, than the average non-mathematician... "Even the giants of mathematics ______________ (suffer) from this minor disability: 'Sir Isaac Newton,' ______________ (say) one observer, 'though so deep in algebra and fluxions, ______________ (not, can) make up a common account; and, he used to get somebody else ______________ (make up) his accounts for him.'"

II Rewrite the sentences in the italics using indirect speech:

Gauss The brilliant mathematician Karl Friedrich Gauss once visited his professor and claimed to have constructed a septendecagon (a seventeen-sided figure). "Nonsense," the professor replied. "That is impossible." "Well, then,"

8 All texts in this section have been adapted from: www.anecdotage.com

45

Gauss persisted. "I have just figured out how to resolve a seventeenth degree polynomial." "Bah, trivial," the professor replied. "I've done it myself." Gauss later repaid this professor, an amateur poet, with a dubious compliment: "He is the finest poet among mathematicians, and the finest mathematician among poets."

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

III Put articles where needed:

Simple Arithmetic Incredibly, _________ great number theorist Ernst Kummer was so inept at _________ simple arithmetic that he often asked _________ students to help him in class. On one occasion, Kummer sought _________ result of a simple multiplication. "Seven times nine," he began. "Seven times nine is er - ah - ah - seven times nine is..." "Sixty-one," _________ mischievous student suggested and Kummer wrote the "answer" on _________ blackboard. "Sir," another one interrupted, "it should be sixty-nine." "Come, come, _________ gentlemen, it can't be both," Kummer exclaimed. "It must be _________ one or _________ other!" IV Choose the correct option: A) Paul Erdos 1. _____________ was Paul Erdos's mathematical aptitude 2. _____________ he was hailed as a genius even by his most gifted colleagues: "Erdos, who 3. _____________ the year at the Institute of Advanced Study, was in the audience but he half-dozed through most of my lecture," Mark Kac once 4. _____________. "The subject matter was too far removed 5. _____________ his interests. 6. _____________ the end I described briefly my difficulties with the number of prime 7. _____________. At the mention of number theory Erdos perked up and asked me 8. _____________ once again what the difficulty was. 9. _____________ the next few minutes, even before the lecture was over, he interrupted to announce that he had the 10. _____________!"

1. a) so b) such c) that d) as 2. a) what b) which c) that d) so 3. a) has spent b) was spending c) spent d) spends 4. a) had recalled b) recalling c) has recalled d) recalled 5. a) to b) on c) in d) from 6. a) towards b) wards c) onward d) forward 7. a) divisors b) numerators c) divisibility d) division 8. a) explained b) to explain c) explaining d) explain 9. a) after b) at c) within d) in 10. a) solvution b) solution c) solving d) solve

46

B) John von Neumann The 1. __________ mathematician (and father of computing) John von Neumann 2. __________ in the habit of simply 3. __________ the answers to homework assignments on the blackboard (the solution 4. __________, of course, 'obvious'). One day, 5. __________ wily (=cunning) student tried 6. __________ some useful information from the professor by 7. __________ whether there was another way of solving a certain problem. Von Neumann 8. __________ for a moment before 9. __________ his reply: "Yes." 1. a) fame b) notorious c) famed d) popular 2. a) is b) has been c) was d) had been 3. a) write b) being written c) to write d) writing 4. a) being b) has been c) will be d) to be 5. a) the b) a c) an d) – 6. a) to get b) get c) to have got d) got 7. a) ask b) to ask c) asked d) asking 8. a) think b) has thought c) was thought d) thought 9. a) give b) gave c) giving d) to give

V Complete the text using the given words:

David Hilbert

foundations expert created mathematician area contribution invented work theory results

The notoriously absent-minded _________________ David Hilbert once found himself talking with Helmut Hasse. When Hasse began to talk about his recent _________________ to class-field _________________, Hilbert interrupted him, insisting that he explain the theory's basic concepts and _________________. Hasse agreed and Hilbert grew progressively enthusiastic. "This is extremely beautiful," he finally exclaimed. "Who _________________ it?" Hasse's reply was as great a shock to Hilbert as the question was to Hasse; the theorem's creator, of course, was none other than David Hilbert!

47

MOCK TEST (COLL.)

I Choose one of the four options to best complete the text. (7)

Numbers

A number is a concept from mathematics, used to count or measure. Depending on the field of mathematics where numbers 1) ___________________, there are different definitions:

People use symbols to represent numbers, they call them numerals. Common places where numerals are used are for labelling, for example as telephone numbers, for ordering, as in serial numbers or to put a unique identifier (ISBN is a unique number that 2) ___________________ identify a book)

3) ___________________ numbers are used to measure how big a set is (0, 1, 2,..) 4) ___________________ numbers are used to specify a certain element in a set or sequence. (First, Second, Third)

Integers are the 5) ___________________ numbers. Integers are the numbers used for counting and for identification, for example, house numbers, together with the negative numbers.

These are the integers:

Integers are whole numbers that have a positive sign (+) or a negative sign (−) in front of it. 0 is also an integer, even though it has no positive or negative signs. While integers are a 6) ___________________of the rational numbers and real numbers, they are smaller set because they don't have decimal 1) ___________________. The integers have no end in either direction, which means that no number is the largest and no number is the smallest.

1. a) use b) can use c) are used d) are being used 2. a) can b) might c) should d) must 3. a) ordinal b) integral c) odd d) cardinal 4. a) ordinal b) even c) odd d) whole 5. a) complete b) entire c) integrated d) whole 6. a) set b) sequence c) subset d) division 7. a) commas b) dots c) places d) points

II Read the text and then do the exercises given below (8).

Autistic savants have displayed a wide range of talents, from reciting all nine volumes of Grove's Dictionary of Music to measuring exact distances with the naked eye. The blind American savant Leslie Lemke played Tchaikovsky's Piano Concerto No1, after he heard it for the first time. And the British savant Stephen Wiltshire was able to draw a highly accurate map of the London skyline from memory after a single helicopter trip over the city. Even so, Tammet could still turn out to be the more significant.

The savant syndrome is more frequently found in males than in females in an approximate ratio of 6 to 1. Savants can recall facts, numbers, license plates, maps, and extensive lists of sports and weather statistics after only seeing them once. Some savants can recall perfectly a very long sequence of music, numbers, or speech. Some, named mental calculators, can do exceptionally fast arithmetic, including prime factorization. Other skills include precisely measuring distances and angles by sight, calculating the day of the week for any given date, and being able to accurately calculate the passing of time without a clock. Most autistic savants have a single special skill while others have multiple skills. Usually these abilities are concrete, non-symbolic, right hemisphere skills as opposed to left hemisphere skills that tend to be more sequential, logical, and symbolic.

a) Find synonyms in the text for the following words:

show _________________ precise ____________________ estimated, rough, not precise ___________________ remember __________________ ability __________________ b) Decide whether the following statements are true or false:

1. Autistic savants are very talented people who are often very successful in their careers. _____ 2. All savants can measure angles and distances by sight. ________ 3. Autistic savants have a highly developed left hemisphere. __________

48

III Translate the following sentences. (10) 1. An obtuse triangle has one obtuse angle. ________________________________________________________________________________ 2. Today, the term applied mathematics is used in a much broader sense. ________________________________________________________________________________ 3. Two lines in the same plane that never intersect are called parallel lines. ________________________________________________________________________________ 4. Combinatorics is connected to many areas of mathematics. ________________________________________________________________________________ 5. This year I managed to pass the entrance exam and now I am a student at Moscow Lomonosov University. _________________________________________________________________________________ V Complete the sentences using the correct form of the conditional. (5) 1. If I _______________ (be) as good as you at maths, I ___________________ (study) it. However, I almost failed it last year. 2. What __________________ (happen) if you ___________________ (write) all numbers on a piece of paper? 3. If you __________________ (follow) this rule, you __________________ (soon, get) the correct result. 4. If a person ___________________ (hold) a degree in mathematics, that person __________________ (call) a teacher of mathematics. 5. If ________________ (read) this book, I _________________ (get) a good grade. VI Write the following formulae in words. (5)

25 + 14 = 39 _________________________________________________ 345 – 45 = 300 _________________________________________________ 11 ∙ 2 = 22 _________________________________________________ 14 : 2 = 7 _________________________________________________ 15 < 27 _________________________________________________ 87 > 5 _________________________________________________ a ≤ 45 _________________________________________________ 16 ≥ y _________________________________________________ 63 _________________________________________________ √16 _________________________________________________

VII Fill in with appropriate words. (5)

The numbers 1, 3, 5, 7, and 9 are called ____________ numbers. 0, 2, 4, 6, 8 are _________ numbers. They are also termed as _____________ numbers and they tell us how many things or persons there are in a set. _____________ numbers define the position of things in a series. The four elementary rules or arithmetical operations are _____________, ______________, __________________ and _________________.

49

VIII Name these figures and bodies. (5)

TOTAL: _________________ / 45

50

MOCK EXAM

I Circle one of the given options to best complete the text: (12)

Pigeonhole Principle The Pigeonhole Principle 1. _________________ that if n+1 pigeons fly to n holes, there 2. _________________ be a pigeonhole containing at least two pigeons. This apparently trivial principle is very 3. _________________.

The pigeonhole principle is an example of a counting argument which can be 4. _________________ to many formal problems, including ones involving infinite sets that cannot be put into one-to-one correspondence.

The first statement of the principle is believed to have been made 5. _________________ Dirichlet in 1834 under the name Schubfachprinzip ("drawer principle" or "shelf principle"). In Italian too, the original name "principio dei cassetti" is still in use; in some other languages (for example, Russian) this principle 6. _________________ the Dirichlet principle (not to be confused with the minimum principle for harmonic functions of the same name).

Let us 7. _________________ some examples. Example (Putnam 1978) 8. _________________ A be any set of twenty integers 9. _________________ from the arithmetic progression 1,4, . . . ,100. Prove that there must be two distinct integers in A whose sum is 104. 10. _________________: We partition the thirty four elements of this progression 11. _________________ nineteen groups {1},{52}, {4,100} , {7,97}, {10,94},. . . {49,55}. Since we are choosing twenty integers and we have nineteen sets, by the Pigeonhole Principle there must be two integers that belong 12. _________________ one of the pairs, which add to 104. 1. a) states b) is stated c) is stating d) statement 2. a) may b) might c) ought to d) must 3. a) power b) powerful c) powerless d) powered 4. a) application b) apply c) applied d) applicated 5. a) by b) from c) to d) on 6. a) calls b) call c) is called d) is calling 7. a) to see b) seeing c) see d) have seen 8. a) let b) must c) make d) take 9. a) choose b) chose c) is chosen d) chosen 10. a) solve b) salvation c) solution d) solving 11. a) on b) into c) from d) and 12. a) on b) to c) about d) at

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

51

II Complete the text with the appropriate words. (5)

theorem celebrated order states integer

set random proved special either

Ramsey theory is a ________________ part of extremal combinatorics. It ______________ that any sufficiently large ________________ configuration will contain some sort of ________________.

Frank Ramsey ________________ that for every integer k there is an ________________ n, such that every graph on n vertices ________________ contains a clique or an independent ________________ of size k. This is a ________________ case of Ramsey’s ________________.

III Supply the missing articles where needed. (5) _______ mathematics (colloquially, maths or math) is _______ body of knowledge centered on such concepts as quantity, structure, _______ space, and change, and also _______ academic discipline that studies them. Benjamin Peirce called it "_______ science that draws necessary conclusions".[2] Other practitioners of mathematics maintain that mathematics is _______ science of pattern, and that _______ mathematicians seek out _______ patterns whether found in _______ numbers, space, science, computers, ________ imaginary abstractions, or elsewhere. IV Turn the direct into indirect speech: (5)

1. ‘I think that this theory cannot be proven,’ Gauss said.

_____________________________________________________________________ 2. ‘What is applied mathematics?’ the student asked.

_____________________________________________________________________ 3. ‘I’ll show you how to solve this problem,’ he said.

_____________________________________________________________________ 4. ‘I expect you to hand in you homework by tomorrow,’ the teacher said.

_____________________________________________________________________ 5. ‘I gave an interesting lecture,’ the professor boasted.

_____________________________________________________________________

52

V Complete the text using the correct form of the verbs in brackets: (8) Srinivasa Ramanujan was a mathematical prodigy. "I remember once ______________ (go) to see him when he was lying ill at Putney," the mathematician G. H. Hardy once ________________ (remember). "I ______________ (take) the taxicab number 1729, and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "'No,' he replied, 'it _______________ (be) a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'" 50-50 Proposition While lecturing on probability at Warwick University one day in October 1972, Jeffrey Hamilton, demonstrating the effect of chance, _______________ (take) a coin from his pocket and casually tossed it in the air. The probability that the coin would land face up (heads) was exactly the same as the probability that it _________________ (land) face down (tails); it was, Hamilton explained, a 50-50 proposition. Hamilton and the assembled students then ___________________ (watch) as it hit the floor, bounced, rolled, spun around - and came to rest on its edge. After a stunned silence, a wild applause ______________ (hear) in the room. VI Translate the following text: (10)

Q. Can you give any examples of how your mathematics are seen in nature?

A.“The Fibonacci Sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. The formula basically is a guide to adding the previous two numbers in the Hindu-Arabic system to get a new number ad infinitam. Interestingly enough, this is found everywhere in living things because of the way things grow exponentially in nature. Also, did you notice how much art and music have to do with the sequence? If you look at piano keys or famous works of art you will always see recurring patterns obviously of the "Golden numbers.” We take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13) and we divide each by the number before it, we will find the following series of numbers: = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538... The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. It has a value of approximately 1.61804.”

VII Find synonyms in the text: (5)

How many golf balls can fit in a school bus? Job: Product Manager Answer: This is one of those questions Google asks just to see if the applicant can explain the key challenge to solving the problem. Reader Matt Beuchamp came up with a dandy answer, writing: I figure a standard school bus is about 8ft wide by 6ft high by 20 feet long - this is just a guess based on the thousands of hours I have been trapped behind school buses while traffic in all directions is stopped.

53

That means 960 cubic feet and since there are 1728 cubic inches in a cubic foot, that means about 1.6 million cubic inches. I calculate the volume of a golf ball to be about 2.5 cubic inches (4/3 * pi * .85) as .85 inches is the radius of a golf ball. Divide that 2.5 cubic inches into 1.6 million and you come up with 660,000 golf balls. However, since there are seats and crap in there taking up space and also since the spherical shape of a golf ball means there will be considerable empty space between them when stacked, I'll round down to 500,000 golf balls. Which sounds ludicrous. I would have guessed no more than 100k. But I stand by my math. Of course, if we are talking about the kind of bus that George Bush went to school on or Barney Frank rides to work every day, it would be half that....or 250,000 golf balls.

very good _________________

form an opinion ________________

to prevent someone from escaping from somewhere ___________________

things that are useless or unimportant ___________________

fairly large, especially large enough to have an effect or be important _____________________

54

INFORMACIONI LIST

A + B smjer

Naziv predmeta: Engleski jezik 2 (stručni)

Šifra predmeta Status predmeta Semestar Broj ECTS

kredita Fond časova

Obavezan II 2 2

Studijski programi za koje se organizuje:

Akademski i primijenjeni osnovni studijski programi PRIRODNO-MATEMATIČKOG FAKULTETA - smjer A – Teorijska matematika i B (Primjenjena matematika i računarske nauke) (studije traju 6 semestara, 180 ECTS kredita).

Uslovljenost drugim predmetima: : Nema uslova za prijavljivanje i slušanje predmeta Ciljevi izučavanja predmeta: Predmet ima za cilj osposobljavanje studenta da razumiju i da se razumiju i da se služe engleskim jezikom struke. Ime i prezime nastavnika i saradnika: mr Milica Vuković Metod nastave i savladanja gradiva: : Predavanja i vježbanja. Priprema prezentacije na zadatu temu iz jedne od oblasti sadržaja predmeta. Učenje za kolokvijum i završni ispit. Konsultacije. Sadrzaj predmeta Pripremna nedjelja Priprema i upis semestra

I Uvod u gradivo. Reading: My Future Profession; Basic mathematical terms II Mathematical terms – algebra and geometry III Reading: A Genius Explains; Conditionals IV Reading: Number Theory; Active and Passive V Revision VI Reading: Applied Mathematics; Articles; Transformations VII Slobodna nedjelja VIII Priprema za kolokvijum IX Kolokvijum X Reading: Combinatorics; Modal verbs XI Reading: Discrete Mathematics; The Language of Proof XII Reading: An Interview with Leonardo Fibonacci; Vocabulary Revision XIII Grammar Revision XIV Translation exercises XV Priprema za završni ispit

XVI Zavrsni ispit XVII Ovjera semestra i upis ocjena XVIII Popravni ispitni rok

Obaveze studenta u toku nastave: Student je dužan da redovno pohađa nastavu, uradi prezentaciju na zadatu temu i polaže kolokvijum i završni ispit.

Literatura: Skripta za Engleski jezik 2 (Jezik struke) za studente primjenjene i teorijske matematike

Oblici provjere znanja i ocjenjivanje:

Ocjenjuju se:

Prezentacija – 5 Kolokvijum - 45 Završni ispit – 50

Ime i prezime nastavnika koji je pripremio podatke: mr Milica Vuković