UNIVERZITET CRNE GORE

INSTITUT ZA STRANE JEZIKE

MILICA VUKOVIĆ

ENGLISH FOR STUDENTS OF MATHEMATICS

SKRIPTA ZA ENGLESKI JEZIK 2 (JEZIK STRUKE)

ZA STUDENTE TEORIJSKE I PRIMJENJENE MATEMATIKE

PRIRODNO-MATEMATIČKOG FAKULTETA

PODGORICA, 2015

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 _________________________________________________

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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), 2

nd (second), 3

rd (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 + 4

2 = 5

2 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.

http://www.imdb.com/name/nm0000129/http://www.imdb.com/name/nm0000163/http://www.imdb.com/name/nm0000129/http://www.imdb.com/name/nm0000163/http://www.imdb.com/name/nm0000129/http://www.imdb.com/name/nm0000163/http://www.imdb.com/name/nm0000129/http://www.imdb.com/name/nm0000163/http://www.imdb.com/name/nm0000129/http://www.imdb.com/name/nm0000163/http://www.imdb.com/name/nm0000129/http://www.imdb.com/name/nm0000163/http://www.imdb.com/name/nm0000129/http://www.imdb.com/name/nm0000163/http://www.imdb.com/name/nm0000129/http://www.imdb.com/name/nm0732152/http://www.imdb.com/name/nm0000163/http://www.imdb.com/name/nm0732152/http://www.imdb.com/name/nm0000163/http://www.imdb.com/name/nm0000129/

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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) the

Fields Medal 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.

http://simple.wikipedia.org/wiki/Mathematicshttp://simple.wikipedia.org/wiki/Numberhttp://simple.wikipedia.org/wiki/Propertieshttp://simple.wikipedia.org/wiki/Numberhttp://simple.wikipedia.org/wiki/Divisionhttp://simple.wikipedia.org/wiki/Division_by_zerohttp://simple.wikipedia.org/wiki/Composite_numberhttp://simple.wikipedia.org/wiki/Divisorhttp://simple.wiktionary.org/wiki/unique

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.

_____________________________________________________________________

http://simple.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Diophantine_equationhttp://en.wikipedia.org/wiki/Decision_problemhttp://en.wikipedia.org/wiki/Diophantine_equationhttp://simple.wikipedia.org/wiki/Composite_numberhttp://simple.wiktionary.org/wiki/unique

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.

_____________________________________________________________________

http://en.wikipedia.org/wiki/Arithmetichttp://en.wikipedia.org/wiki/Mathematicianshttp://www.historyforkids.org/learn/bc.htmhttp://www.historyforkids.org/learn/religion/christians/jesus.htmhttp://www.historyforkids.org/learn/food/index.htmhttp://www.historyforkids.org/learn/greeks/clothing/index.htmhttp://www.historyforkids.org/learn/greeks/philosophy/index.htmhttp://www.historyforkids.org/learn/greeks/art/music/index.htmhttp://www.historyforkids.org/learn/greeks/science/math/index.htmhttp://www.historyforkids.org/learn/greeks/philosophy/rationality.htmhttp://en.wikipedia.org/wiki/Chinese_remainder_theoremhttp://en.wikipedia.org/wiki/Multiplicative_function

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(2

2 − 1) = 6

for n = 3: 22(2

3 − 1) = 28

for n = 5: 24(2

5 − 1) = 496

for n = 7: 26(2

7 − 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.)]

http://en.wikipedia.org/wiki/Negative_and_non-negative_numbershttp://en.wikipedia.org/wiki/Divisorhttp://en.wikipedia.org/wiki/6_%28number%29http://en.wikipedia.org/wiki/28_%28number%29http://en.wikipedia.org/wiki/496_%28number%29http://en.wikipedia.org/wiki/8128_%28number%29http://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Prime_number

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.

http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Differential_equationshttp://en.wikipedia.org/wiki/Approximation_theoryhttp://en.wikipedia.org/wiki/Representation_%28mathematics%29http://en.wikipedia.org/wiki/Asymptotichttp://en.wikipedia.org/wiki/Calculus_of_variationshttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Newtonian_Physicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Fluid_mechanicshttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Differential_equationshttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Operations_researchhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Engineering_mathematicshttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Matrix_population_modelshttp://en.wikipedia.org/wiki/Computational_mathematicshttp://en.wikipedia.org/wiki/Computational_sciencehttp://en.wikipedia.org/wiki/Computational_engineeringhttp://en.wikipedia.org/wiki/High_performance_computinghttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Engineering

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.

http://en.wikipedia.org/wiki/Newtonian_Physics

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.

_____________________________________________________________________

http://en.wikipedia.org/wiki/Computational_mathematicshttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Topologyhttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Differential_equationshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Fluid_mechanicshttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Matrix_population_models

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 + B

2 = C

2.

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

http://en.wikipedia.org/wiki/Goldbach%27s_conjecturehttp://en.wikipedia.org/wiki/Even_and_odd_numbershttp://en.wikipedia.org/wiki/Collatz_conjecturehttp://en.wikipedia.org/wiki/American_Statistical_Associationhttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Differential_equationshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/American_Statistical_Associationhttp://en.wikipedia.org/wiki/Mathematical_statisticshttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Proof_%28mathematics%29

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

http://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Measure_theoryhttp://en.wikipedia.org/wiki/Mathematicshttp://www.philosophypages.com/dy/s2.htm#scholhttp://www.philosophypages.com/dy/s5.htm#skephttp://www.philosophypages.com/dy/m9.htm#mont

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

http://en.wikipedia.org/wiki/Algebrahttp://en.wikipedia.org/wiki/Algebraic_combinatoricshttp://en.wikipedia.org/wiki/Algebraic_combinatoricshttp://en.wikipedia.org/wiki/Algebraic_combinatoricshttp://en.wikipedia.org/wiki/Partition_of_a_sethttp://en.wikipedia.org/wiki/Ordered_partitionhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Pure_mathematicshttp://en.wikipedia.org/wiki/Countable_sethttp://en.wikipedia.org/wiki/Finite_sethttp://en.wikipedia.org/wiki/Enumerative_combinatoricshttp://en.wikipedia.org/wiki/Enumerative_combinatoricshttp://en.wikipedia.org/wiki/Enumerative_combinatoricshttp://en.wikipedia.org/wiki/Combinatorial_designhttp://en.wikipedia.org/wiki/Matroidhttp://en.wikipedia.org/wiki/Extremal_combinatoricshttp://en.wikipedia.org/wiki/Extremal_combinatoricshttp://en.wikipedia.org/wiki/Extremal_combinatoricshttp://en.wikipedia.org/wiki/Combinatorial_optimizationhttp://en.wikipedia.org/wiki/Graph_theoryhttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Computer_science

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.

_____________________________________________________________________

http://en.wikipedia.org/wiki/Algebra

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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.

http://en.wikipedia.org/wiki/Ramsey_theoryhttp://en.wikipedia.org/wiki/Sufficiently_largehttp://en.wikipedia.org/wiki/Frank_P._Ramseyhttp://en.wikipedia.org/wiki/Ramsey%27s_theoremhttp://en.wikipedia.org/wiki/Pigeonhole_Principlehttp://en.wikipedia.org/wiki/Chessboardhttp://en.wikipedia.org/wiki/Chess_piecehttp://en.wikipedia.org/wiki/King_%28chess%29http://en.wikipedia.org/wiki/Queen_%28chess%29http://en.wikipedia.org/wiki/Rook_%28chess%29http://en.wikipedia.org/wiki/Knight_%28chess%29http://en.wikipedia.org/wiki/Bishop_%28chess%29http://en.wikipedia.org/wiki/Pawn_%28chess%29http://en.wikipedia.org/wiki/Checkmate

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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.

http://en.wikipedia.org/wiki/Combinatoricshttp://en.wikipedia.org/wiki/Topology

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.

http://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Countable_setshttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Graph_%28mathematics%29http://en.wikipedia.org/wiki/Formal_languagehttp://en.wikipedia.org/wiki/Flowchart

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 all 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 ______________

http://en.wikipedia.org/wiki/Computerhttp://en.wikipedia.org/wiki/Computer_program

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 .

http://en.wikipedia.org/wiki/Flowcharthttp://en.wikipedia.org/wiki/Flowcharthttp://en.wikipedia.org/wiki/Q.E.D.http://en.wikipedia.org/wiki/Tombstone_%28typography%29http://en.wikipedia.org/wiki/Paul_Halmos

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. ______