БЕЛОРУССКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ
ENGLISH
FOR FUTURE
MATHEMATICIANS
АНГЛИЙСКИЙ
ДЛЯ БУДУЩИХ
МАТЕМАТИКОВ
Рекомендовано
Учебно-методическим объединением
по гуманитарному образованию в качестве
учебно-методического пособия для студентов
учреждении высшего образования, обучающихся по специальностям
1-31 03 01 «Математика (по направлениям)»,
1-31 03 02 «Механика и математическое моделирование»,
1-31 03 08 «Математика и информационные технологии
(по направлениям)», 1-31 03 09 «Компьютерная математика
и системный анализ»
МИНСК
БГУ
2017

УДК 811.111 ’276.6:51 (075.8)
ББК 81.432.1-324я73
А64
Авторы:
Л. К. Бизюк, В. А. Зенченко, Н. Л. Потапова,
Е. Ю. Столярова, С. Н. Тригубкина, И. Н. Шарко
Рецензенты:
кандидат филологических наук В. В. Яскевич-,
кандидат филологических наук А. И. Долгорукова
Английский для будущих математиков = English for Future Mathe-
A64 maticians : учеб.-метод. пособие / Л. К. Бизюк [и др.]. - Минск : БГУ,
2017.-207 с.
ISBN 978-985-566-428-5.
Учебно-методическое пособие «English for Future Mathematicians» предна¬
значено для обучения студентов математических специальностей профессио¬
нально ориентированному английскому языку.
В издании содержатся тексты по алгебре и геометрии, различные виды
упражнений, грамматический справочник.
УД К 811.111 ’276.6:51 (075.8)
ББК 81.432.1-324я73
ISBN 978-985-566-428-5
©БГУ, 2017

CONTENTS
ПРЕДИСЛОВИЕ	5
PART I
UNIT 1 Глагол to be. Оборот there is/are.
Личные и притяжательные местоимения 	6
Text “About My Family and Myself”	12
UNIT 2 Времена группы Indefinite. Глагол to have.
Типы вопросов	18
Text “A Letter to a Friend”	28
UNIT 3 Времена группы Continuous.
Местоимения some, any, no	32
Text “Numbers”	38
UNIT 4 Модальные глаголы и их эквиваленты 	41
Text “Four Basic Operations of Arithmetic”	53
UNIT 5 Времена группы Perfect (Действительный залог) 	58
Text “Rational Numbers and Decimal Numerals” 	62
UNIT 6 Степени сравнения.
Времена группы Perfect Continuous	68
Text “Hie Nature of Algebra” 	77
UNIT 7 Времена группы Indefinite в страдательном залоге 	81
Text “Equations and Identities”	84
UNIT 8 Времена группы Continuous и Perfect
в страдательном залоге	88
Text “Polynomials” 	92
SUPPLEMENTARY READING	96
PRACTICE THE WAY OF PUTTING QUESTIONS
IN ENGLISH	100
3

PART II
UNIT 9 Согласование времен (Sequence of Tenses).
Косвенная речь (Reported Speech) 	106
Text “Mathematical Propositions”	114
UNIT 10	Причастие (Ute Participle) 	124
Text “Points, Lines, Planes and Angles”	129
UNIT 11	Герундий (The Gerund) 	137
Text “Regular Polygons. Special Quadrilaterals” 	146
UNIT 12	Инфинитив (The Infinitive) 	156
Text “Triangles”	160
UNIT 13	Инфинитивные обороты (The Infinitive Constructions)	169
Text “The Circle”	174
UNIT 14	Условные предложения (Conditional Sentences)	182
Text “Matrices”	186
SUPPLEMENTARY READING	194
NUMERALS	202
IRREGULAR VERBS	206

ПРЕДИСЛОВИЕ
Основная задача учебно-методического пособия - вырабо¬
тать у студентов навыки и умения, необходимые для практиче¬
ского использования английского языка в профессиональной
деятельности.
Предлагаемый лексический и грамматический материал по¬
может студентам читать и переводить научно-техническую ли¬
тературу, делать устные сообщения и вести беседу по специаль¬
ности.
Книга состоит из двух частей (Part I, Part И). Первая часть
содержит восемь разделов, вторая - шесть.
Структура разделов унифицирована: грамматический спра¬
вочник с комплексом упражнений, тематический текст для фор¬
мирования навыков изучающего чтения, текст для развития на¬
выков просмотрового и ознакомительного чтения. Все тексты
взяты из оригинальной научно-технической литературы.
Каждый раздел включает послетекстовые упражнения, ори¬
ентированные на закрепление пройденного материала, и упраж¬
нения на перевод с русского языка на английский, цель кото¬
рых - контроль усвоения лексико-грамматических явлений.
В конце первой части представлен блок упражнений для
выработки навыков постановки всех типов вопросов.
Для развития и закрепления навыков письменного и устно¬
го реферирования каждая часть имеет дополнительные науч¬
но-технические тексты (часть 1 - тексты по алгебре, часть 2 -
тексты по геометрии).
Содержащийся в издании учебный материал поможет сту¬
дентам математических специальностей реализовать цели обу¬
чения, сформировать навыки и умения в различных видах ре¬
чевой деятельности.

UNIT 1
Глагол to be. Оборот there is/are.
Личные и притяжательные местоимения
Местоимения (Pronouns)
Personal (личные)
Possessive (притяжательные)
Nominative
(именительный n.)
Objective
(объектный n.)
Dependent
(зависимая форма)
Independent
(независимая форма)
I
you
he
she
it
we
they
me
you
him
her
it
us
them
my
your
his
her
its
our
their y
> + noun
mine
yours
his
hers
its
ours
theirs
Ex. 1. Analyse the following sentences and translate them into Russian.
1.	They live together with their parents. 2. Oxford is famous for its University.
3.1	didn’t have an umbrella, so Ann gave me hers. 4. Are those people friends of
yours7. 5. Michael had an argument with a neighbour of his. 6. We went on holiday
with some friends of ours. 7. I am meeting a friend of mine this evening. 8. He
wants to see us at his place. 9.1 cant find my keys. Where are they7 10. We need the
photographs. Can you give them to us7
Ex. 2. Put in my/our/your/his/her/their/its/mine/yours/ours.
1. We want to go to the cinema after the exam. Do you like ... idea?
2.1	know Mr Watson but I don’t know ... wife.
3.	Mr and Mrs Baker live in London. ... son lives in Australia.
4.	We’re going to have a party and invite some friends of....
5.	This is a nice camera. Is it... ?
6.1	like tennis. It’s ... favourite sport.
7. Is that... car? - No, I haven’t got a car.
6

8.1	want to phone Ann. Do you know ... phone number?
9.	Do you think most people are happy in ... jobs?
10.	Whose books are these? - They are ....
11.	Ulis is a beautiful tree.... leaves are of green colour.
12.	On Sunday John usually has dinner with a friend of....
Местоимение it
Ex. 3. а) Мы используем личное местоимение it как подлежащее или как до¬
полнение вместо существительных, обозначающих неодушевленные предме¬
ты, животных.
1. Where is your pen? - It is in the bag. 2. Your translation is good. I like it. 3. This
is our classroom. It is large. 4. Where is my ticket? I cant find it. 5. Your new car is
nice. Is it expensive? 6.1 want that book. Please give it to me.
b) Мы используем личное местоимение it как формальное подлежащее в
предложениях, в которых говорится о времени, состоянии погоды, явлениях
природы, температуре и расстоянии.
1. It is ten miles to the nearest petrol station. 2. It is Monday again. 3. It is thirty
degrees. 4. It is half past ten. 5. It is a nice day today. 6. It was very windy yesterday.
7. It was my birthday yesterday. 8. How far is it to the nearest restaurant?
с) Мы используем личное местоимение it как формальное подлежащее в
безличных предложениях:
It
is
was
will be
easy/difficult/impossible/dangerous/safe/
expensive/interesting/nice/wonderful/
terrible/a pleasure
to do
1. It is nice to talk to you. 2. It was good of you to phone me. 3. It is impossible to
understand her. 4. It wasn’t easy to find your house. 5. It is difficult to get up early
in the morning. 6. It was a pleasure to listen to her.
Глагол to be
Present
I
am
he
she
it
is
Past
I
he
she
was
it
Future
I
we
shall be
will be
7

Present
we
you
they
are
Past
we
you
they
were
Future
he
she
it
you
they
will be
Analyse the interrogative and negative forms.
1.	Jane is at home at the moment, but her mother isn't. Where is Jane’s mother? Is
her brother at home?
2.	When I HWS a child, I afraid of dogs and my sister wasn't. Why were you
afraid of that little dog yesterday?
3.	Today Sue is in Madrid. Tomorrow she will be in Rome. She won't be in Tokyo
on Friday. Where will Sue be on Saturday?
Ex. 4. Make the following sentences interrogative and negative.
a)	1. Twenty miles is a long way to walk. 2. My native city is very large. 3. The pair
of black trousers is cheap. 4. Phonetics is a branch of linguistics. 5. The family are
fond of their house. 6. The students’ books are on their desks. 7. The capital of
my homeland is Minsk. 8. We are at the English lesson now. 9. The man in this
photograph is my brother. 10. My father is interested in politics.
b)	1. The boy was at home three days ago. 2. My lessons were over at 2 p.m. yesterday.
3. She was ready for the seminar last week. 4. Hie trip was exciting. 5. Hie girl’s
family was very large. 6. The film at the Odeon was long and dull. 7. We were at the
university yesterday. 8. Kate and Olga were pupils last year. 9. A nice play was on at
the theatre last week. 10. My favourite subject at school was mathematics.
c)	1. This book will be interesting. 2. I shall go to the cinema tomorrow. 3. The
house will be ready soon. 4. The text of the next lesson will be short. 5. Mike will be
a student next year. 6. My sister will be a post-graduate in two years. 7. She will be
present at the meeting on Monday. 8. There will be a nice block of flats here. 9. The
students will be free in some minutes. 10.1 shall be ready for the next seminar.
Ex. 5. a) Say the following sentences in the past.
1. Hie facts from the newspaper article are out of date. 2. The boy’s family is at
home. 3. The students from group 2 are in the next room. 4. He is ready to read
this text. 5. The sentences are simple and short. 6. The weather is good at this time
of the year. 7. Bob and James are foreign students. 8. Those people aren’t English.
They are Australian. 9. The article in the newspaper is interesting. 10. Sport is an
essential part of Mike’s life.
8

b) Say the following sentences in the future.
1. Our street is very green in spring. 2. Children are already in bed at 9 o’clock.
3. Saturday morning is a very busy time for shopping. 4. It is not very cold in winter
here. 5. Hie chief method of teaching is the lecture method. 6. Travelling by sea is
very interesting. 7. The leaves are not green in autumn. 8. Our trip to Moscow is
very fantastic. 9. The workers are at the meeting at this time of the day. 10. At the
end of the lecture she is very tired.
Оборот there is/there are...
(имеется, есть, существует, находится)
Read the sentences and compare the information given.
1.	The teacher’s desk is in the room.
Стол учителя стоит в комнате.
2.	The film on TV was very interesting
last night. Фильм no телевизору был
вчера интересный.
3.	A lot of people will be present at
the party on Saturday. Много людей
придет на вечеринку в субботу.
1.	There is a teacher’s desk in the room.
В комнате стоит стол учителя.
2.	There was an interesting film on TV
last night. Вчера no телевизору шел
интересный фильм.
3.	There will be a lot of people at the
party on Saturday. В субботу на вече¬
ринке будет много людей.
Ex. 6. Analyse the following sentences and translate them into Russian.
a)	1. There are some big trees in the garden. 2. There is a seminar on philosophy
today. 3. There are comfortable apartments in this block of flats. 4. There are a lot
of accidents on this road. 5. There is something in my eye. 6. There are 11 players
in a football team. 7. There is a book and two pens on the desk.
b)	1. There were a lot of children in the yard an hour ago. 2. There was an important
meeting of students with the Dean last week. 3.1 know there were some letters for
me yesterday. 4. When I got home, I was hungry but there wasn’t anything to eat.
5. There was a swimming pool at our hotel last summer.
c)	1. There will be one more department at the University next year. 2. When you
arrive tomorrow, there will be somebody at the station to meet you. 3.1 don’t think
there will be any problems at the exam. 4. There will be dictionaries on every
table. 5. There will be no underground service between Vostok and Kupalovskaya
stations tomorrow.
9

Ex. 7. Make the sentences interrogative and negative. Follow the models.
a)	Are there any new messages for me today?
Yes, there are. / No, there aren't.
There are no (not any) new messages for you today.
How many new messages are there for me today?
1. There are twelve students in my group. 2. There are twenty six letters in the
English alphabet. 3. There is only one chair in this room. 4. There are thirty days in
September. 5. There is one button on my jacket.
b)	Was there little time for this work?
Yes, there was. / No, there wasn't.
There wasn't much time for this work.
How much time was there for this work?
1. There was little water in the bottle. 2. There was only one circus in the city
50 years ago. 3. There were a lot of lights in my Christmas-tree last year. 4. There
was a shop at the end of the street. 5. There were seven flowers in the vase. 6. There
were three pictures on the wall near the door. 7. There was little food in the fridge.
c)	Will there be a new tube station in our street?
Yes, there will. / No, there won't.
There won't (will not) be a new tube station in our street.
What will there be in your street?
1. There will be a meeting in the hall. 2. There will be a bridge across the river.
3. There will be one more flight to London next month. 4. There will be four
seatbelts in my car. 5. There will be a traffic jam in your area.
Pre-Reading Activity
Read and learn the basic vocabulary terms:
sense (n) [sens]
support (v, n) [ss'pc :]
seem (v) [si ~i]
common (adj) pkcmsn]
an only child
fairly large
impressive (adj) [ini'psesiv]
research (adj) [:Css:’]
take after (mother, father)
determined (adj) [dCts mins]
ощущение, чувство
содержать (семью); поддержка
казаться
общий; обыкновенный
единственный ребенок
довольно большой
производящий глубокое впечатление
исследовательский
быть похожим на (мать, отца)
решительный
10

strong-willed (adj) ['szrcrwkd]
trust (v)
be interested in
rely on (v)
diligent (adj) [4 2±z*sr.z]
lively (adj) [zla:v>]
cheerful (adj) [':ksf-1]
restless (adj) [4rss_Iis]
enter a university (college)
sociable (adj) ['ss'jfsbl]
honest (adj) [zcr.:s:J
helpful (adj) [zhelprZ]
stubborn (adj)
deal with (v) [21 ■]
occupation (n) [ckrx'pszjsr.]
play the piano (the guitar, the violin)
be fond of smth
remote (adj) [rz'rriszz]
aunt (n) [2 s.z]
uncle (n) [zA2kl]
cousin (n) ['клгг.]
delicious (adj) [2:'кjss]
get along
united (adj) [;~ 'r.azzz2]
решительный, волевой
доверять
интересоваться
полагаться на
прилежный, старательный
живой, веселый
веселый, жизнерадостный
непоседливый, неугомонный
поступить в университет (колледж)
общительный
честный
услужливый, готовый помочь
упрямый
иметь дело с
род занятий, профессия
играть на пианино (на гитаре, скрипке)
любить
отдаленный, далекий
тетя
дядя
двоюродный брат (сестра)
очень вкусный, приятный
ладить, относиться друг к другу
хорошо
сплоченный
xMemorise the following word combinations:
I believe - полагаю
to go in for trade - заняться торговлей
to look for a better paid job - искать более оплачиваемую работу
to try one’s luck - попытать счастья
to be similar in character - иметь похожие характеры
in appearance - по внешнему виду
to devote to the children - посвящать детям
a grown-up person - взрослый человек
to give up working - бросить работу
to enjoy respect from - пользоваться уважением у
to keep the house - вести хозяйство
11

to be concerned about - беспокоиться о
to be in good shape - быть в хорошей форме
according to the latest vogue - согласно последней моде
to do well at school - хорошо учиться в школе
to make a hell of the house - превратить дом в ад кромешный
to be of a strong built - быть крепкого телосложения
Reading Activity
ABOUT MY FAMILY AND MYSELF
I believe that everything has its beginning in the family. Family is very
important for every person, because it gives you a sense of stability and tradition,
a feeling of having support and understanding. It seems a bit sad that families are
getting so small these days. A family with three or four children is not a common
thing. More often you will find many families where there is just mummy, dad,
one kid and may be a dog. I don’t know what it feels like being an only child in
the family. There are three children in our family. So by modern standards we are
considered to be a fairly large family.
I think I’d better start my story with my dad. His name is Ivan Petrovich. He
is in his late forties, but he looks powerful and impressive. He is tall with dark
hair and brown eyes and is of a strong built. My father is a research worker by
profession. But about 10 years ago he had to look for a better paid job to support
us. He went in for trade. At present he is trying his luck in several trade aspects. My
father is determined, strong-willed, energetic. He looks very businesslike and at
the same time he tries not to lose the sense of humour. And though he is very busy,
he always devotes his free time to the children, mainly to my younger brother and
sister because he takes me for a grown-up person, he trusts me and relies on me. He
is even sure that I can be his partner in business translating some business papers
and documents for him. Generally speaking my father and me are very similar in
character though in appearance I take after my mother.
My mother’s name is Larisa Ivanovna. She has turned 40 this year. But if you
look at her you won’t give her a year older than 30. My mother is a programmer by
profession. Though there were no grandparents around to help my mother when
we were small, she practically never gave up working. She is very interested in her
work, she is a good professional and she enjoys respect from her colleagues. Mum’s
life is not easy, of course, because she has to keep the house in addition to her work.
My mother is a quiet and charming person. She is very kind and she does a good
job of being a mother. She is concerned about her appearance, tries to be in good
shape, elegant and dressed according to the latest vogue, that’s why she looks so
good for her age.
12

My younger brother Sasha is only 7 years old. He studies at school. He does
well at school, which makes all of us happy. He is a diligent, kind and intelligent
boy. He is tall for his age, sporty and we hope he’ll make a good basketball player
one day.
As for my younger sister Kate, she is only 4 years of age. She is a very pretty,
lively, cheerful and energetic little thing. She is very restless and it’s hard for her to
stay in one and the same place for more than a minute, so when the two of them
are playing they make a hell of the house.
Now a few words about me. My name is Denis. I am seventeen. This year I
entered the Belarusian State University. At present I am a first-year student at
the Mechanics and Mathematics Faculty. I have always liked mathematics. My
friends say that I am sociable, honest, helpful and cheerful, but my parents think
that sometimes I am stubborn and hard to deal with. My favourite occupation is
playing the guitar and reading. I am fond of sport as well.
Of course I have many remote relations: two grandmothers and a grandfather,
aunts, uncles and cousins. But only my grandfather and grandmother on the
mothers side live in Minsk. Though my grandma is already an elderly woman,
she often visits us, helps my mother to look after the children and always brings
something delicious to eat. We all enjoy her visits.
There is no “fathers-and-sons” problem in our family. We all are getting along
all right and I think we are a united family. That’s all I can say about my family.
Additional Vocabulary
Ex. 8. Look through the vocabulary below that may be useful when speaking
about your family.
Looks and Appearance
beautiful (adj) ['b1“ ::f“l]	красивый (о женщинах)
blond/fair [fes]/ginger [chinchs]/ dark светлые/русые/рыжие/черные
hair	волосы
blue/grey/hazel eyes	голубые/серые/карие глаза
grey/green/dark-eyed	серо/зелено/черноглазый
curly ['ks l:]/straight ] hair кудрявые/прямые волосы
handsome (adj) [T.sr.ssrri]	красивый (о мужчинах)
height (n) [ha::]	рост
look like (smb.)	быть похожим на (кого-либо)
of medium/short/tall height ['rm d1srri] среднего/низкого/высокого роста
13

plain (adj) [plszrz]
обыкновенный, некрасивый
plump (adj) [р1л22р]
полноватый
pretty (adj)
симпатичный
slender (adj) ['slszds]
стройный
slim (adj)
худой
straight [s:zs:t]/snubbed nose
прямой/курносый нос
Features of Character
be in good/bad mood [~" d]
быть в хорошем/плохом настроении
brave (adj) [bre:v]
смелый
devoted (adj) [d:zvs?t:d]
преданный
faithful (adj) [zfe:9f"l]
дружеский, дружественный
gentle (adj) [Zzherzi]
нежный
gloomy (adj)
мрачный
hard-working (adj) ['hz d'ws kz~ ]
трудолюбивый
kind-hearted (adj) ['kaizd'hz ::d]
добрый, добросердечный
lazy (adj) ['Isiz:]
ленивый
open-minded (adj)	did]
открытый, искренний
polite (adj) [ps'ls.::]
вежливый
re;e:vs£	[”'zsvS
сдержанный
rude (adj) [r" d]
грубый
shy (adj) [/az]
робкий, застенчивый, тихий
Interests and Ambitions
ambition (n) [srr/Zz'sz]
стремление
be keen [ki z] on smth
увлекаться чем-либо
desire (n, v) [d/zais]
желание, желать
dislike (v) [d:s'la:k] smth
не нравится что-либо
(to do smth)
(делать что-либо)
do sports
заниматься спортом
dream (of) (v, n) [dn m]
мечтать о, мечта
hate (v) [heir]
ненавидеть
intend (v) [ir/rend] to do smth
намереваться делать что-либо
intention (n) [:nz:er/=r.]
намерение
make a career [ks'izs]
делать карьеру
play football (chess)
играть в футбол (шахматы)
wish (v)
желать
14

Family Members and Relations in the Family
average ['sevsrzh] Zsmall/large family
consist of (v) [ksr/szst]
nephew (n)
niece (n) [zi s]
relatives (n) (relations) ['rslstivz]
stepdaughter (n) ['stspdc:s]
stepfather (n) [Zstspfz 5s]
stepmother (n)['s:epzzA5s]
stepson (n) ['stspSAZ]
twins (n) [:w:r.z]
admire (v) [sz'zsais] smb
be attached [s tsft] to
be devoted [zz'vaztzz] to
blame (v) [hlsizs] smb
caring (adj) pkosri-]
close (adj) [klstrs]
cordial (adj) ['ко d;sl]
difficulty (n) [4d:f:kslt:]
distant (adj) ['distsz:]
elderly (adj) ['sidsl:]
frank (adj) [f:ss~k]
friendly (adj) ["frsnzli]
hostile (adj) ['he stall]
impartial (adj) [irr/pz 'si]
intolerant (adj) [:r/tolsrsz:]
loving (adj)plAv:-]
provide (v) for the family [prs'vaid]
reliable (adj) [ts'laishl]
respectful (adj) [:is'psktrnl]
share the domestic chores [ho z]
take care [kss] of smb
warm (adj) [wc m]
wonderful (adj) ['wAZosrhil]
worry (v) about smb [wa::]
средняя/маленькая/болыиая семья
состоять из
племянник
племянница
родственники
падчерица
отчим
мачеха
пасынок
близнецы
любить, обожать кого-либо
быть привязанным к
быть преданным
винить, обвинять кого-либо
заботливый
близкий
сердечный, радушный
трудность
далекий
пожилой
откровенный, искренний
дружеский
враждебный
безразличный
нетерпимый
любящий
обеспечивать семью
надежный
уважительный
делить, разделять домашние
обязанности
заботиться о ком-либо
теплый
прекрасный
беспокоиться о ком-либо
15

Post-Reading Activity
Ex. 9. Answer the following questions.
1.	How old are you? 2. Where are you from? 3. Where and when were you born?
4.	What are your good habits? 5. Who do you most take after? 6. Do you easily make
friends? 7. Are you on friendly terms with all your group mates? 8. What talents
do you think you have? 9. Is it necessary to have a hobby? Why? 10. What is the
right age for young people to get married? 11. Are you for small or large families?
12. What do you do if your parents are not right (in your opinion)? 13. What do your
parents make you do that you don’t like doing? 14. What is your parents’ attitude to
your friends? 15. Why does the fathers-and-sons problem always exist? 16. What is
your idea of a good husband (wife)? 17. What does family happiness depend on?
Ex. 10. Arrange the following words in pairs of antonyms and translate them:
1) hard-working; 2) generous; 3) cheerful; 4) strong; 5) clever; 6) relaxed; 7) nice;
8) optimistic; 9) honest; 10) reserved; 11) sensitive; 12) reliable.
a)	miserable; b) tense; c) horrible; d) pessimistic; e) dishonest; f) emotional; g) lazy;
h) insensitive; i) unreliable; j) mean; k) weak; 1) stupid.
Ex. 11. Study the list of professions in the box and guess the profession of each
person:
vet, plumber, accountant, lawyer, engineer, architect, lecturers, firefighter.
1.	My uncle Jim is an ..., he designs buildings.
2.	An ... is someone who controls the financial situation of people and companies.
3.	There are a lot of... at our faculty.
4.	The person who is afraid of dogs cannot be a ....
5.	It is impossible to imagine our life without a ... who fits and repairs water pipes,
bathrooms.
6.	My father’s friend is a ..., he advises people on legal problems.
7.	Hie man over there plans the construction of roads and bridges, he is an ....
8.	I am very proud of my grandfather, he works in the fire brigade. He is a ....
Ex. 12. Translate into English making use of the words from the box:
slender, beard, shy, short, polite, loving, hazel eyes, pale skin, kind-hearted,
plump cheeks, reserved, caring, ginger-haired, curly hair, broad shoulders,
good-looking, charming, handsome, snubbed nose, take after father, honest.
1. Он занимается спортом, он стройный, с широкими плечами. 2. Моя луч¬
шая подруга симпатичная, но очень застенчивая. 3. У этого известного акте¬
16

ра карие глаза, волнистые волосы и борода. 4. Сегодня у пациента бледная
кожа, он чувствует себя нехорошо. 5. Мой друг не красавец, но он добрый,
честный и надежный парень. 6. У младенца курносый носик и пухлые щеч¬
ки, он очаровательный. 7. Не ленись, делай зарядку - и ты не будешь тол¬
стым. 8. Роберт ирландец, у него рыжие волосы, он сильный и смелый, он по¬
хож на отца. 9. Наш новый секретарь вежливый, сдержанный. Он нам очень
нравится. 10. Любящие и заботливые родственники помогают друг другу в
трудных ситуациях.
Ех. 13. Put in it or there.
1.	... rains a lot in winter. 2. ... was a strong wind yesterday. 3. Is ... a bookshop
near here? 4. ... was a nice day yesterday. 5. We cant go skiing. ... isn’t any snow.
6.... is hot in this room. Open a window. 7.1 was afraid because ... was very dark.
8.	... was a storm last night. Did you hear it? 9. ...is a long way from here to the
nearest shop. 10.... wasn’t anything on television last night. 11. How far is ... from
Milan to Rome?
Ex. 14. Put in him/her/yours etc.
1. Where’s Ann? Have you seen ... ? 2. Where are my keys? Where did I put ... ?
3. That is not my bag, ... is black. 4. This letter is for Bill. Can you give it to ... ?
5.	We wrote to John but he didn’t answer ... letter. 6. I can’t find my pen. Can
I use ... ? - Yes, of course. 7. We’re going to the cinema. Why don’t you come
with ... ? 8. Can we use your washing machine? ... is broken. 9. Did your sister pass
... exams? 10. Tom invited some friends of ... to the restaurant. 11. Some people
talk about ... jobs all the time. 12. Last night I went out for a meal with a friend
of.... 13. We had dinner with a neighbour of....
Ex. 15. Fill in the blanks w ith the necessary form of the verb to be.
1.1 ... at home now. My room ... small. 2. He ... at the University yesterday. 3. We
... in the man’s house last week. 4. Our work ... over tomorrow. 5. The girls ... in
the next room now. 6. Next year she ... a teacher of English. 7. The children ... at
home at this time of the day. 8. My friend ... in bed tomorrow because he is ill.
9.	My brother ... at school at 2 o’clock yesterday. 10. This time last year Jack ... in
Paris. 11. Today the weather ... nice, but yesterday it... very cold. 12. It... a public
holiday yesterday. 13. When I was a child, I ... afraid of dogs.
Ex. 16. Ask special questions.
1. Her desk is in the room (what, where). 2. These houses are old (what). 3. We
are in the classroom (who, where). 4. They will be ready soon (who, when). 5. She
was a post-graduate last year (who, when). 6. The work will be over tomorrow
(what, when). 7. They were ready to begin the work (who, what). 8. I was at the
17

University last week (where, when). 9. The next text is in the note-book (what,
where). 10. They will be students next year (when). 11. This ancient monastery is
a museum now (what).
Ex. 17. Translate into English.
1.	Завтра будет урок английского языка. 2. Эта книга моя или твоя? 3. Сегодня
солнечно, но не тепло. 4. Это не мой пиджак, мой черный. 5. Десять минут
назад дети были в саду. 6. Было приятно послушать ее рассказ. 7. В этом тексте
для меня нет новых слов. 8. Двух студентов не было на уроке английского
языка в прошлую пятницу. 9. Сейчас это их проблема, а не наша. 10. До
ближайшего почтового отделения 500 метров. 11. Будет интересно увидеть
ее в этом новом фильме. 12. Сколько страниц в этой книге?
Ех. 18. Writing. Using the words given in the list for the text do the following
assignments.
1.	Describe the student sitting next to you.
2.	Write a description of someone you know well and like a lot (you may bring a
photo of that person).
3.	Describe someone from your group but do not say who the person is. Other
students listen to your description and must guess the name of the person.
UNIT 2
Времена группы Indefinite.
Глагол to have. Типы вопросов
Глагол to have (have got)
(иметь, обладать)
Present
I
we
you
they
have
have got
he
she
it
has
has got
Past
I
you
he
she
it
we
they
had
Future
I
we
shall have/
will have
you
he
she
it
they
will have
18

В вопросах и отрицательных предложениях используются следующие
формы.
Present
Have you got any money?
Do you have any money?
Have you any money? (less usual)
I haven't got any money.
I don't have any money.
I haven't any money, (less usual)
Has she got a car?
Does she have a car?
Has she a car? (less usual)
She hasn't got a car.
She doesn't have a car.
She hasn't a car. (less usual)
Past
Did they have a car last year?
They didn't have a car last year.
Future
Will the students have a seminar
tomorrow?
The students won't have a seminar
tomorrow.
Глагол to have может входить в состав устойчивых глагольных сочета¬
ний, где он утрачивает свое основное лексическое значение иметь и приоб¬
ретает новый смысл.
have
breakfast/dinner/a cup of coffee/a cigarette/a drink/a meal
a bath/a shower/a swim/a rest/а party/a holiday/a nice time/а good journey/
a good flight/а good trip
an accident/an experience/а dream/a sleep/а lie-down/a look (at something)/
a chat (with somebody)/a talk/a fight/a baby (= give birth to a baby)
difficulty/t rouble/fun
I don't usually have a big breakfast.
What time does Ann have lunch?
Did you have any difficulty at the exam yesterday?
Ex. 1. Make the sentences interrogative and negative.
a)	1.1 usually have a sandwich for my lunch. 2. Students have one English class a
week. 3. We have a shop next to the post office. 4. Most cars have got four wheels.
5. An insect has got six legs. 6. My friend Tim has two little sisters. 7. Tina has got
long blond hair.
b)	1. You had a good teacher of English at school. 2. They had two English lessons
a week last year. 3. He had little time yesterday. 4. She had a few long pencils in her
bag. 5. He had a large family 10 years ago. 6. My friend had a good computer in his
office. 7. We had a very good spring last year. 8. The boy had his fathers blue eyes.
c)	1. They will have all the data next week. 2. He will have a large room in this
house. 3. We shall have a scientific conference next year. 4. We shall have all
modern conveniences next year. 5. My sister will have a baby next month. 6. They
will have a garden in front of their house. 7. Students will have much time for rest
in summer. 8. They will have much information about this event in the papers.
19

Ex. 2. Say the following sentences in the past.
1.	They have a few lectures this week. 2. He has much work to do this summer.
3. You have an interesting seminar today. 4. He has good ideas how to spend the
weekend. 5. My parents have a nice little dog. 6. My friend has a reasonable answer.
7. They have lots of visitors in the Art museum. 8. Pete has many bookshelves in
his room. 9. Olga has got a black leather bag. 10. Our students haven’t got much
free time in winter.
Ex. 3. Say the following sentences in the future.
1.	On Sunday my brother has breakfast at 9 o’clock. 2. We have special seminars
on Wednesday and Friday. 3. He has got a lot of publications. 4. Students have
examinations in winter and in summer. 5. She has 3 hours at her disposal. 6. Our city
has many green parks. 7. They haven’t got any money to pay their bills. 8. Our paper
usually has much information about science. 9. First-year students have English
classes two times a week. 10. My friends father has got a fine collection of pictures.
Present Indefinite
The Present Indefinite tense употребляется:
1.	Для обозначения обычных, регулярно повторяющихся или постоян¬
ных действий с использованием следующих словосочетаний и наречий ча¬
стотности: often, always, usually, seldom, rarely, sometimes, never, generally, as a
rule, every day (month), every other day (week, month, etc.), once a week.
He often works till midnight.
My brother plays tennis every other day.
She is never late for classes.
Do you generally speak English in
class?
I sometimes meet your father at the
station.
Он часто работает до полуночи.
Мой брат играет в теннис через день.
Она никогда не опаздывает на заня¬
тия.
Вы обычно по-английски разговари¬
ваете на занятиях?
Я иногда встречаю твоего отца на
станции.
20

Солнце садится на западе.
Она преподает в школе английский.
Тебе нравится дождливая погода?
Его родители живут в Лондоне.
2.	Для формулирования законов природы, утверждения универсальной
истины либо постоянных характеристик.
The Sun sets in the west.
She teaches English at school.
Do you like rainy weather?
His parents live in London.
3.	Для выражения действий или состояний, протекающих в момент речи
с целым рядом статичных глаголов, выражающих:
a)	зрительное, слуховое восприятие: see, hear, notice, taste, smell, etc.
b)	умственные способности: understand, think, believe, remember, recognize,
know, forget, mean, suppose, etc.
c)	чувства и эмоции: like, dislike, hate, love, wish, want, care, prefer, etc.
d)	принадлежность: have, belong, own, possess, etc.
It smells like a hospital in here.
I don’t see anyone in the room.
Do you recognize me?
What does he mean?
Do you know what he is speaking about?
Which of these dresses do you like best?
Do you want anything to drink?
- I want a glass of juice, please.
Jill really hates house work.
Who does this car belong to?
Здесь пахнет как в больнице.
Я не вижу никого в комнате.
Вы меня узнаете?
Что он имеет в виду?
Ты знаешь, о чем он говорит?
Какое из этих платьев тебе нра¬
вится больше всего?
Хочешь что-нибудь выпить?
- Мне, пожалуйста, стакан сока.
Джил очень не любит работу по дому.
Кому принадлежит эта машина?
4.	Если речь идет о расписании мероприятий, движении транспорта, се¬
ансах в кинотеатрах, зачастую с глаголами движения: start, go, come, leave,
arrive, sail, return, etc.
Hie train leaves London next Friday Поезд отправляется из Лондона
at 8 a.m. and arrives at Leeds at 11 a.m. в 8 часов утра в следующую пят¬
ницу и прибывает в Лидс в 11 часов.
Ех. 4. Make the following sentences interrogative and negative.
1.	We do a lot of things in our free time. 2. The shops open at 9 o’clock and close at
5.30. 3. It costs a lot of money to stay in luxury hotels. 4. The Moon goes round the
Earth. 5. They usually sit for hours without saying a word. 6. She keeps her room
tidy as a rule. 7. Mother makes strawberry jam every year. 8. People traditionally
prepare coloured eggs at Easter. 9. A new school opens next week. 10. My father
shaves every other day. 11. Once a week Dave stays in the office till six o’clock.
12. The water in this lake freezes in winter.
21

Ex. 5. Write sentences using these words. Put the verb in the right form.
1.	(always / early / Sue / arrive).
2.	(basketball / I / play / often).
3.	(work I Margaret / hard / usually).
4.	(Jenny / always / nice clothes / wear).
5.	(dinner / we / have / always / at 7.30).
6.	(television / Tim / watch / never).
7.	(like / chocolate / children / usually).
8.	(Julia / parties / enjoy / always).
9.	(never / my brother / breakfast / eat).
10.	(every other day / tennis / my friend / play).
Past Indefinite
Positive
I
we
you
they
he
she
it
played
started
watched
had
saw
did
went
Negative
I
we
you
they
he
she
it
did not
(didn’t)
play
start
watch
have
see
do
	E

Interrogative
Did
I
we
you
they
he
she
it
play?
start?
watch?
have?
see?
do?
go?
The Past Indefinite tense употребляется:
1. Для выражения повторяющихся действий или констатации единично¬
го факта, события, долговременного действия, для описания ряда последо¬
вательных действий, совершенных одним и тем же лицом в прошлом, где ин¬
дикаторами времени могут быть слова: ago, last year (week, month), yesterday,
the other day, in 1997, last (time), for five years, for a week, etc.
Ann spent a lot of money on books
yesterday.
It didn’t rain last night.
When did you go to the cinema last?
She started playing the piano at the
age of five.
They lived in Brest/orfive years before
the war.
I entered the office, looked around and
came up to the secretary.
Анна потратила много денег на
книги вчера.
Прошлой ночью не было дождя.
Когда вы в последний раз ходили
в кинотеатр?
Она начала играть на пианино
в 5 лет.
До войны они 5 лет жили в Бресте.
Я зашел в офис, огляделся и подошел
к секретарю.
22

2.	Для выражения повторяющихся действий или состояний в прошлом,
которые в настоящее время уже не имеют места. При переводе used to +
Infinitive могут употребляться слова: бывало, раньше, обычно, имел обыкно¬
вение.
Не used to smoke forty cigarettes a day
and then he finally gave up smoking.
Do you play golf? - No, but I used to
when I lived in the country.
She used to be such a lively girl (but no
longer now).
The shops didn’t use to open on
Sundays in those days.
Раньше он выкуривал no 40 сигарет
в день, а затем наконец-то бросил
курить.
Ты играешь в гольф? - Нет, но имел
обыкновение играть, когда жил за
городом.
Раньше она была такой
жизнерадостной девочкой.
В то время магазины обычно не ра¬
ботали по воскресеньям.
Ех. 6. Make the following sentences interrogative and negative.
1.	Caroline went to the cinema three times last week.
2.	We did a lot of work yesterday.
3.	The police stopped him on the way home last night.
4.	She passed her examinations successfully in January.
5.	Mozart wrote more than 600 pieces of music.
6.	Peter broke a window last night.
7.	I lived in London for ten years when I was a child.
8.	Janet used to have long hair when she was young.
9.	He gave up his job as a journalist the other day.
10.	Liz used to play tennis a lot when she was a student.
Ex. 7. Write sentences about the past.
1.	Jim always goes to work by car. Yesterday ....
2.	Rachel often loses her keys. She ... last week.
3.	Kate meets her friends every evening. She ... yesterday evening.
4.	I usually buy two newspapers every day. Yesterday I ....
5.	We usually go to the cinema on Sundays. Last Sunday we ....
6.	I eat an orange every day. Yesterday I ....
7.	Tom always has a shower in the morning. This morning he ....
8.	Our friends come to see us every Friday. They ... last Friday.
9.	Pupils pass their A-Level exams every summer. Ann ... last June.
10.	Our faculty holds the Olympiads in mathematics in spring. Last spring our
faculty....
23

Future Indefinite
Positive/Negative
I/we
you/they
he/she/it
shall(‘ll)/shall
not/shan’t
will(‘ll)/will
not/won’t
be
win
eat
come
Interrogative
I/we/you/
be?
win?
eat?
come?
Will
they/he/
she/it
The Future Indefinite Tense употребляется:
1. Для выражения однократных и повторяющихся действий, которые
предположительно будут иметь место в будущем. Индикаторами времени
могут быть слова: tomorrow, the day after tomorrow, next year, in a week (month,
year), in 2018, etc.
Next year I’ll be 18.
Spring will come soon.
Spring has come, so the snow will start
melting, the birds will come back home.
В следующем году мне будет 18.
Скоро наступит весна.
Наступила весна, поэтому
начнется таяние снега и птицы
прилетят домой.
2.	Для выражения предположения, сомнения, вероятности, обещания
после глаголов: to believe, to doubt, to expect, to think, to be sure, to be afraid, a
также наречий probably, perhaps.
Гт sure he’ll get better.
I don't think I’ll go out tonight, I’m too
tired.
No doubt you’ll enjoy the performance.
Do you think they’ll win the match?
I’ll probably be a bit late this evening.
I haven’t seen Carol today. I expect she
will phone this evening.
Я уверен - ему станет лучше.
Я не думаю, что я пойду куда-
нибудь сегодня вечером. Я очень
устала.
Несомненно, тебе понравится
представление.
Как ты считаешь, они выиграют
этот матч?
Я, вероятно, немного опоздаю
сегодня вечером.
Сегодня я не видел Кэрол. Полагаю,
она позвонит вечером.
3.	Для передачи спонтанного решения, к которому пришли в момент
речи.
Don’t lift the suitcase. I’ll help you. He поднимайте чемодан. Я вам
помогу.
It looks like rain. I’ll take my umbrella Похоже, пойдет дождь, я возьму
then.	свой зонтик.
24

4.	Для выражения будущего времени в главной части сложноподчинен¬
ного предложения, а в придаточном предложении употребляются Present
Indefinite, Present Perfect после союзов: if, when, while, after, before, as soon as,
until/till.
I’ll phone you as soon as I arrive.
When you return home, you’ll notice
a lot of changes.
It’s pouring down. We’ll get wet
through if we go out.
When you see Jane again, you won’t
recognize her.
Come on! Mum will be worried if we
are late again.
As soon as Bob and Alice have got
married, they’ll move to California.
I shan’t phone you until I have done
my homework.
Я позвоню тебе, как только приеду.
Когда ты вернешься домой, ты за¬
метишь много изменений.
Идет проливной дождь. Мы промок¬
нем, если выйдем на улицу.
Когда ты снова увидишь Джейн, ты
ее не узнаешь.
Поторопись! Мама будет волно¬
ваться, если мы снова опоздаем.
Как только Боб и Алиса поженятся,
они переедут в Калифорнию.
Я не буду звонить тебе до тех пор,
пока не выполню домашнее задание.
Следует запомнить
1.	Мы используем shall I... ?/shall we...? для выражения просьб,
предложений.
Shall I open the window?	Открыть окно?
I’ve got no money. What shall I do? У меня нет денег. Что делать?
Where shall we go this evening? Куда мы пойдем сегодня вечером?
2.	Мы используем won't, чтобы сообщить, что кто-то отказы¬
вается (не желает) выполнить какое-либо действие или какое-либо
устройство, механизм, машина не работает.
The car won't start. I wonder what’s Машина никак не заводится.
wrong with it.	Интересно, что с ней случилось.
Ех. 8. Make the following sentences interrogative and negative.
1. They will knowr the results in a week. 2. Clothes will be different in many years.
3.	Sally will phone you when she gets home from work. 4. The weather will be
much warmer tomorrow. 5. Everybody will have a computer in the nearest future.
6.1 w'ill remember this day all my life. 7. You will become a well-qualified specialist
25

in 5 years. 8. Jack will be back in a minute. 9. He will like our new house when he
sees it. 10. I will pay him a lot if he works well. 11. Helen will stay in bed till the
clock strikes seven.
Типы вопросов
(Types of Questions)
Общий (General)
Auxiliary verb
Subject
Predicate
or part of it
Object
Adverbial Modifier
Do
you
watch
TV
in the evening?
Have
they
got
free time
in summer?
Does
she
wash
her hair
in the morning?
Специальный (Special)
(кроме подлежащего)
Interrogative
Pronoun
Auxiliary
Verb
Subject
Predicate
or part of it
Object
Adverbial
Modifier
Why
does
she
get up
so early?
Who
did
you
borrow
the money from?
Where
shall
we
g°
now?
Специальный (Special)
(к подлежащему)
Interrogative Pronoun
(+ a noun)
Predicate
Object
Adverbial Modifier
Who
invited
you
to the theatre?
Which bus
goes
to the city centre?
Whose friends
will visit
him
in hospital?
Pre-Reading Activity
Guess the meaning of the following words.
Mathematician (n) [,ms9:ms4zzj'.V2], founder (n) [4fa'2r.dsj, progress (n)
information (n) [mfs'ms:;'sV]>privilege (n) [4p”v:kq<], union (n)
[\1“ szsr.J, especially (adv) [:'spsj Vl:]> comfortable (adj) [4k.\mf s :sbl].
26

Read and learn the basic vocabulary terms:
delay (n) [d:4Is:]
recent (adj) [4r: s.Y':]
manage (v) ['22^2:25]
belong (v) [bflc-]
outstanding (adj) [22:'s:ss2d:r]
room-mate (n)	22,22s::]
describe (v) [d:'skra:d]
routine (n) [r'2	2]
flash by (v) press']
drag (v) [' dreg]
instructive (adj) [:2/i:r.xk::v]
compete (v) [ks224 p: :]
facilities (n) [fs's:l:::z]
area (n) [' s Vr:s ]
various (adj) [4ve's^r:ss]
society (n) [ss4s5.:s::]
depend on (v) [d:'ps2.d]
imagine (v) [:422ssd<:2]
nowadays (adj) [42a”sds:z]
отсрочка, промедление, задержка
последний, недавний
руководить, управлять, суметь (сделать)
принадлежать, быть частью группы
выдающийся, знаменитый
товарищ по комнате
описывать
определенный режим, заведенный
порядок
пронестись, промчаться
тянуться, затягиваться
содержательный, поучительный
соревноваться
благоприятные условия, возможности
район, область, сфера
различный, всевозможный
общество, объединение
зависеть от
воображать, представлять себе
в наше время, теперь
Memorise the following word combinations:
to be busy - быть занятым
the students’ hall of residence - студенческое общежитие
as a matter of fact - в самом деле, в действительности
to face South - выходить окнами на юг
to have a shower - принять душ
to be available - быть в наличии
to be for/against - быть за/против
to press a button - нажимать кнопку
to have a nap - вздремнуть
to have a late night - поздно лечь спать
it usually takes me - мне обычно требуется
to attend lectures - ходить на лекции
on weekdays - в рабочие дни
27

to be up to one’s neck in work - быть по горло загруженным работой
to have a lie-in - оставаться в постели (позже обычного)
recreational facilities and entertainments - места отдыха и развлечения
my best regards to your parents - наилучшие пожелания твоим родителям
on the one hand - с одной стороны
on the other hand - с другой стороны
Reading Activity
A LETTER TO A FRIEND
Dear Linda,
I’m very sorry for the delay in answering your recent letter. I was so awfully
busy. In the spring I passed my А-Level examinations and tests at school and
managed to get good results. Now I am a first-year student at the Mechanics
and Mathematics Department at the Belarusian State University. It certainly is a
great privilege to belong to the faculty among whose founders were outstanding
mathematicians.
I have so much to tell you about my life in Minsk. I live in the students’ hall of
residence and share the room with two other girls. One of them is from Gomel, the
other girl is from Brest. As a matter of fact they are not only my room-mates but
also my good friends. We have a nice and comfortable room, it faces South.
And now I would like to describe the routine I more or less follow every day.
During the week I usually wake up at 6.30 a.m. I sometimes lie in bed for five
minutes but then I have to get up. I have a shower, clean my teeth and at 7 a.m.
I’m ready for breakfast. In the morning I don’t bother to cook very much, so I
have a light breakfast. We usually have lectures and seminars in the morning
and sometimes in the afternoon. On some days lessons flash by very quickly, but
sometimes they drag more slowly, especially when we write tests or have some
colloquium. In general I enjoy my university hours because they are instructive
and interesting. After classes we have dinner at the students’ dining room and
sometimes go to the reading-room, where the computer-based information is
available six days a week.
Oh, I haven’t told you yet that I learned how to use a computer and bought one
for myself. But one thing worries me greatly. Once people managed to write and
think using their brains, but now people can’t do anything without these machines.
On the one hand, I am for progress. It is impossible to imagine our life without
computers nowadays. But on the other hand, I’m against everything depending on
pressing a button.
28

If I don’t go to the library, I get back to the hall of residence and try to have a
nap, especially if I had a late night. Then I am busy doing my homework. It usually
takes me two or three hours.
I don’t only attend lectures and read books here. I have the chance of developing
myself as a person. Students organize clubs and societies covering various areas
such as sport, drama, music, dances, etc. Every university has a students’ union
which organizes recreational facilities and entertainments. As you remember,
I used to play tennis, but I don’t anymore. Here I joined the Athletics Club. We
compete in area, regional and national competitions. So on weekdays I am up to
my neck in work and often very tired by the end of the day. Most evenings I go to
bed at about 11.30 p. m. and fall asleep very quickly.
Hie weekends are different. On Sunday I have a lie-in. In the evening I often go
out. There is so much to see and so many places to go to in Minsk. Why don’t you
come for two or three days? I’d love to see you.
My best regards to your parents.
Yours, Kati.
Post-Reading Activity
Ex. 9. Answer the follow ing questions.
1.	Do you take a cold or a hot shower in the morning? 2. What time do you leave
home to get to the university? 3. Have you ever been late for classes? 4. How many
classes do you have every day? 5. Eating is such a waste of time and effort, isn’t it?
It would be better if we could simply take pills. 6. What is your idea of a good rest
after classes? 7. How long does it take you to prepare your homework? 8. What
sports are you good at? 9. Do you take part in any organized sporting activities?
10.	Do you prefer to stay in or go out in the evening? 11. What is your favourite
pastime? 12. Do you make any plans for the weekend? 13. Do you like to spend
your free time with your friends or on your own? 14. How often do you go to
the cinema (to the theatre)? 15. Why do some people prefer to watch films at the
cinema instead of relaxing in front of their TV sets? 16. Are theatres as popular
now as they used to be? 17. What is the best time for you to go to bed?
Ex. 10. Find the Russian equivalents for the following English word
combinations:
1) to belong to a group; 2) to describe the routine; 3) to depend on the weather;
4) to have a late night; 5) on the other hand; 6) to have a lie-in; 7) to attend lectures;
8) as a matter of fact; 9) to be available; 10) a students’ hall of residence; 11) a recent
letter; 12) recreational facilities and entertainments; 13) to be up to one’s neck in
work; 14) an outstanding mathematician.
29

а) зависеть от погоды; Ь) ходить на лекции; с) места отдыха и развлечения;
d) быть студентом группы; е) быть в наличии; f) выдающийся математик;
g) быть по горло загруженным работой; h) студенческое общежитие; i) опи¬
сывать определенный режим дня; j) оставаться в постели позже обычного;
к) поздно лечь спать; 1) с другой стороны; т) в действительности; п) послед¬
нее письмо.
Ех. 11. Read what Sharon says about a typical working day.
I usually get up at 7 o’clock and have a big breakfast. I walk to work, which takes me
about half an hour. I start work at 8.45.1 never have lunch. I finish work at 5 o’clock
I’m always tired when I get home. I usually cook a meal in the evening. I don’t
usually go out. I go to bed at about 11 o’clock. I always sleep well.
Yesterday was a typical working day for Sharon. Write what she did or didn't do
yesterday.
1.	She got up at 7 o’clock.
2.	She ... a big breakfast.
3.	She ....
4.	It... to get to work.
5.	... at 8.45.
6.... lunch.
7.	... at 5 o’clock.
8.	... tired when ... home.
9.	... a meal yesterday evening.
10..	.. out yesterday evening.
11..	.. at 11 o’clock
12..	.. well last night.
Ex. 12. Complete the sentences. Use I'll (I will) + one of these verbs:
carry, do, eat, send, show, sit, stay.
1.	My bag is very heavy.
2.	Enjoy your holiday.
3.1	don’t want this banana.
4.	Do you want a chair?
5.	Did you phone Jenny?
6.	Are you coming with me?
7.	How do you use this camera?
-	... it for you.
-	Thank you. ... you a postcard.
-	Well, I’m hungry. ... it.
-	No, it’s OK.... on the floor.
-	Oh, no, I forgot.... it now.
-	No, I don’t think so.... here.
-	Give it to me and ... you.
Ex. 13. Write sentences beginning I think... or I don't think
1.	(Diana will pass the exam)
2.	(Diana won’t pass the exam)
3.	(we’ll win the game)
4.	(I won’t be here tomorrow)
5.	(Sue will like her present)
6.	(they won’t get married)
7.	(you won’t enjoy the film)
8.	(it won’t rain this afternoon)
9.	(the exam will be difficult)
10.	(she won’t be up to her neck in work)
30

Ex. 14. Ask special questions.
I.	They had an important paper in the desk, (what, where) 2. Five girls from our
group live in the hall of residence, (how many) 3. Paul and Jim played tennis
yesterday, (when) 4. This student has got three lectures today, (how many) 5. His
friends work hard all day. (whose) 6. Professor Smirnov will hold a seminar
tomorrow, (what, when) 7. These men have a logical plan, (who) 8. It took me two
hours to do my homework yesterday, (how long) 9. We will probably go to Scotland
for our holiday, (where) 10. We usually have our meals in the kitchen, (where)
II.	1 like a big breakfast in the morning, (who, when) 12. Sally goes to the theatre
once a month, (how often) 13. My cousin won one million rubles in the lottery,
(how much)
Ex. 15. Translate into Russian.
1.1	overslept this morning because I had a late night yesterday. 2. Helen will stay in
this evening. 3. Did you go out last Sunday? 4. At the end of each term students are
always up to their neck in work. 5. Jennifer and her room-mate get on well because
they respect each other. 6. Jack left his house a bit late in the morning, missed the
bus and was late for classes. 7. We have a long lunch break but I never go to the
university canteen. 8. At times there are other things to do like going shopping,
doing sports and so on. 9. Later in the evening Kate does a bit of painting which is
a sort of a hobby for her. 10.1 used to go fishing when at school, but now I haven’t
got any time to do that.
Ex. 16. Translate into English.
1.	Обычно я принимаю душ по утрам. 2. В Минске есть много мест отдыха и
развлечений. 3. Твоя квартира выходит окнами на юг или на север? 4. Я за¬
нимаюсь английским в выходные дни, так как я по горло загружен работой в
течение недели. 5. К сожалению, у меня очень мало времени на отдых. 6. Моя
подруга никогда не опаздывает на первую лекцию. 7. В наше время не так уж
легко поступить в университет. 8. Если вечером я ложусь поздно спать, то на
следующий день я стараюсь поспать немного днем. 9. Здесь у меня есть воз¬
можность играть в теннис два раза в неделю. 10. Как правило, рабочая неделя
пролетает очень быстро. 11. Катя живет в студенческом общежитии № 7, и с
ней в комнате живут еще две девушки.
Ех. 17. Writing.
1.	Write a letter to your mother describing your weekdays at the University.
2.	Write a letter to your friend in which you tell him/her how you enjoy your life
at weekends.
31

UNIT 3
Времена группы Continuous.
Местоимения some, any, no
Present
Past
Future
I
am working
I/he/she/it
was working
I
he/she/it
we/you/they
will be
working
he/she/it
is working
we/you/they
were working
we/you/they
are working
Hie Present Continuous tense употребляется для:
1. Обозначения действия, происходящего в момент речи со следующими
словами: at this moment, at the time, now, at present, just now, still или время
действия задается контекстуально.
We are all waiting for you outside.
What are the children doing now? -
They are playing in the park.
Listen attentively! The teacher is
explaining a new grammar rule.
Мы все ждем тебя на улице.
Чем сейчас занимаются дети? -
Они играют в парке.
Слушайте внимательно! Учитель
объясняет новое грамматическое
правило.
2.	Выражения действия, совершающегося в более широкий период вре¬
мени или изменяющейся ситуации с глаголами: get, develop, increase, change,
improve, etc.
I am travelling a lot these days.
В настоящее время я много путе¬
шествую.
Мой брат изучает физику в
Кембридже.
Из года в год наша жизнь меняется.
My brother is studying physics in
Cambridge.
Our life is changing from year to year.
3.	Передачи запланированных событий (особенно социальных или свя¬
занных с поездками).
What are you doing tonight?	Что ты делаешь сегодня вечером?
- We are having a party.	- У нас будет вечеринка.
I’m going to the dentist on Monday.	В понедельник я иду к зубному врачу.
32

Ex. 1. Compare and analyse the usage of the Present Indefinite Tense and the
Present Continuous Tense.
1.1	usually watch TV in the evenings. I am watching TV now. 2. Are you looking for
a key now? You always look very smart. 3. Why are you not wearing your new dress
now? She usually wears fashionable clothes. 4. Today it is raining heavily outside. It
rains heavily in autumn in this part of the country. 5. He walks very slowly as a rule
but now he is walking fast. 6.1 usually don't rest after university, but today I’m very
tired and I am having a rest. 7.1 think I will be on holiday next month. I am going
on holiday next month. 8. We live in Washington, though we are staying in London
at the moment. 9. I play tennis every week. Where are the children? - They are
playing tennis on the court now. 10. Hie foreign scientists are flying back to Europe
tomorrow. Some passenger planes fly faster than sound.
The Past Continuous tense употребляется для:
1. Выражения незаконченного временного действия, протекавшего
в определенный момент в прошлом с индикаторами времени: at 5 o'clock
yesterday, from ... till, all day yesterday, the whole evening, when he came, while, all
day long, at noon.
What were you doing at 2 o'clock
yesterday7.
- I was having a shower when the
phone rang.
I don’t know what he said. I wasn’t
listening to him.
When I got home, water was running
down the kitchen walls.
Чем ты занимался вчера в 2 часа7
- Я как раз принимал душ, когда
зазвонил телефон.
Я не знаю, что он сказал. Я не слушал
его.
Когда я пришел домой, на кухне по
стенам струилась вода.
2.	Выражения длительного действия, протекавшего в более широком
отрезке времени, но которое не является непрерывным в течение всего от¬
резка.
We were bathing in the sea during the Летом мы купались в море.
summer.
Before I came here, I was taking a post- До моего приезда сюда я учился
graduate course at Berlin university. в аспирантуре Берлинского
университета.
3.	Описания двух или более одновременно протекавших действий в про¬
шлом (со словом while).
I was doing my homework while Mom Я делал домашнее задание, в то
was cooking lunch.	время как мама готовила обед.
33

Ex. 2. Compare and analyse the usage of the Past Simple Tense and the Past
Continuous Tense.
1. I dropped my bag when I was running for a bus. 2. I played computer games
yesterday. I was playing computer games at 3 o’clock yesterday. I was playing
computer games the whole evening yesterday. 3. When I came into the kitchen,
mother was cooking. 4. Did you do your homework yesterday? - I ws doing my
homework from 6 till 9 o’clock yesterday. 5. She always looked very smart. When
I met her in Rome, she was wearing a long beautiful dress. 6. Father came home at
5 o’clock yesterday. Then he was reading a newspaper while mother was watching TV.
Hie Future Continuous Tense употребляется для:
1.	Выражения незаконченного длительного действия, которое будет
протекать в определенный момент или более широкий период времени в бу¬
дущем с индикаторами времени: at 5 o'clock tomorrow, this time next week, soon,
tonight, all day long.
On Friday night we will be celebrating В пятницу вечером мы будем
my brother’s birthday.	праздновать день рождения моего
брата.
This time next week I’ll be lying on the В это же время на следующей
beach in Philadelphia.	неделе я буду загорать на пляже в
Филадельфии.
2.	Выражения запланированного действия, которое неизбежно или с
большей степенью вероятности состоится в ближайшем будущем.
I’ll be meeting him at the office	Я встречусь с ним в офисе завтра,
tomorrow.
Will you be going into town today? Ты сегодня поедешь в город?
Ex. 3. Compare and analyse the usage of the Future Simple Tense and the Future
Continuous Tense.
1. Tomorrow I will begin decorating my flat as soon as I come home from the
market. I'll be decorating it from 2 till 6 o’clock. 2. I think 17/ use a bike to get
to school tomorrow. Will you be using your bike this evening? 3. Richard will be
cleaning the house while Sue is cooking dinner. 4. Hi is time tomorrow evening my
friends will be flying over France. Hiey’// probably go to the UK too. 5.1 think 17/
take an umbrella because it is still raining. Hiey say it will be raining the whole
week-end.
Ex. 4. Make the following sentences interrogative and negative.
1. Hiis time next month I will be sitting on the beach. 2. Mr. Molden was driving a
car at the time of the accident. 3. Peter will be working the whole evening tomorrow.
34

4.	The little boy was swimming in the sea the whole morning. 5. Yesterday the
teams were playing football from 2 till 5 o’clock. 6. It is getting cooler and cooler
day by day. 7. Mr. Pitt is talking on the phone at the moment. 8. The Browns are
moving house next week. 9. In two years’ time he will be living in the country.
Местоимения some, any, no
Утвердительные предложения (Affirmative sentences)
Some употребляется перед
существительным в значении
несколько, немного,
какие-то, какие-нибудь,
какой-то, некоторые
There is some milk in the fridge.
В холодильнике есть молоко.
Some people enjoy jogging in the morning.
Некоторые люди любят бегать по утрам.
I’d like to put you some questions.
Мне бы хотелось задать тебе несколько
вопросов.
Any употребляется в значении любой
You can come any day you like.
Вы можете прийти в любой день.
Some употребляется перед
числительными в значении около,
приблизительно
There were some ninety people at the concert.
На концерте было около девяноста
человек.
Any и его производные
употребляются в общих вопросах в
косвенной речи.
The police asked John if he had seen
anybody in a dark coat in the bank.
Полиция спросила Джона, видел ли он
кого-нибудь в темном пальто в банке.
Any и его производные
употребляется с такими
отрицательными словами, как:
never, hardly, without, little, seldom,
rarely, few, to refuse, to deny, to fail, to
prevent, etc.
I hardly know anybody in the neighborhood.
Я едва знаю кого-либо из соседей.
Вопросительные предложения (Interrogative sentences)
Any (какие-то, какие-нибудь,
немного, всякий, любой) и его
производные употребляются
в общих вопросах
Have you got any English books?
Есть ли у вас какие-нибудь английские
книги?
Some и его производные
употребляются для выражения
просьбы или предложения
May I have some more coffee?
Могу ли я взять еще кофе?
Would you like some tea?
He хотели бы вы чаю?
35

Any употребляется перед
прилагательными в сравнительной
степени в значении немного
Can you go any faster?
He мог бы ты идти немного быстрее?
Some употребляется в специальных
вопросах
Why haven’t you given me something to
cover with?
Почему ты не дал мне чем укрыться?
Отрицательные предложения (Negative sentences)
No эквивалентно not any/not a
I have по free time left. = I don’t have any free
time left.
У меня не осталось свободного времени.
No (not а) или none of, а не not any
употребляется в значении ни один,
никакой перед существительным в
функции подлежащего.
None обычно согласуется со
сказуемым в единственном числе.
No information has been received from him.
От него не получено никаких сведений.
Not a newspaper wrote about it.
Ни одна газета не писала об этом.
None of my friends lives near me.
Никто из моих друзей не живет рядом со
мной.
No употребляется перед
прилагательными в сравнительной
степени в значении нисколько не
I'm afraid the weather today is no better than
it was yesterday.
Я боюсь, что сегодня погода нисколько не
лучше, чем вчера.
Ex. 5. Analyse the following sentences and translate them into Russian.
1.	Are there any English books in the library? 2. I cant find any mistakes in
your dictation. 3. I’d like to have some more jam. 4. Can you give me some more
information? 5. There is some sugar in the cake but there is no salt. 6. What book
shall I take? - Any you like. 7. Take some juice, please. Its very tasty. 8. Would you
like some time to finish your work? 9.1 know someуйппу jokes. 10. No students are
happy to have extra seminars. 11.1 can do it without anybody's help. 12. Once I ate
some ten ice-creams a day. 13. I want to know if you have done anything good in
your life. 14. Can I have some of these books?
Ex. 6. Choose the right word.
He left without saying something/anything to somebody/anybody. 2. Suddenly
anyone/someone entered the room. 3. Is there something/anything good on TV
tonight? 4. Do you want anything/something to eat? 5. Is there any/some coffee in
the coffee-pot? 6. Someone/anyone can take part in the competition. 7. There is
anything/nothing in the bag. 8. There are any/no matches left. 9. Will you have any/
some more jam? 10.1 can’t find the pen nowhere/anywhere.
36

Ex. 7. Answer the following questions using the pattern below.
-	Have you got any sisters?
-	Yes, I have some.
-	No, I have no sisters. I -No, I haven't any sisters.
1. Do you want something to eat? 2. Have you got any news? 3. Do you know
anybody in the village? 4. Have you invited anybody to the party? 5. Do you
understand anything? 6. Was there anything interesting at the exhibition? 7. Do
you have any energy left? 8. Have you seen John anywhere? 9. Is there any coffee
in the coffee-pot?
Pre-Reading Activity
Guess the meaning of the following words.
System (n) ['szstsm], symbol (n) [simbl], positive (adj) ['pczztzv], diagram (n)
['], complex (adj) ['kcmplek-s], rational (adj) [ks 'sr/J, fundamental
(adj) [глг.2='тегЛ1], fact (n) [fsskt], express (v) [zks'pz’ss], negative (adj)
[k.scs:zv], start (n) [sta t], position (n) [ps'zz'r.J, direction (n) [dz'zsk t_],
occupy (v) pckrzoaz], zero (n)	different (adj) ['dzfsrerz], basic (adj)
['bs-kj.
Read and learn the basic vocabulary terms:
number (n) [ч г.лггЛ=]
число, количество, номер
date back to (v) [4 dszz]
датироваться, вести начало от
antiquity (n) [szz'zzkwzzz]
древность, античность
integer (n) [4zs:zz<s]
целое число
aid (n) [4 ezd]
помощь
complete (v) [k=~/p:z t]
завершать, делать полным
fraction (n) [4 rrskk]
дробь
imaginary (adj) [z'zzsszhzr.szz]
мнимый
count (v) [4 ka~~:J
считать
real (adj) [zzsl]
действительный
unity (n) [\1~ zzztz]
единица, единство
establish (v) [zs'zsbk ]
устанавливать
ratio (n) ['zezjsz]
отношение, пропорция
negative (adj) pzzecsrzv]
отрицательный
division (n) [dz'vz^z]
деление
either (conj.) [' az*s ]
любой, каждый
allow (v) [s'k~]
позволять, допускать
divisor (n) [dz'vazzs]
делитель
37

quotient (n) ['kwc^/sr.:]	частное, отношение
include (v) [:r/k?x 2]	заключать, содержать в себе
special (adj) ['sps 'si]	особый, специальный
compose(v) [ksrr/ps”z]	составлять
Memorise the following word combinations:
zero is neither positive nor negative - ноль не является ни положительным, ни
отрицательным
to label a point on the line - отметить точку на прямой
to each point on the line we assign a number - каждой точке на кривой мы
ставим соответствующее число
Reading Activity
NUMBERS
The beginning of our number system dates back to antiquity where symbols,
which we call positive integers, were used as an aid in counting, and only in the
nineteenth century the system, which we know today, was completed. As an aid in
studying this number system, let’s use the diagram.
{Integers
Fractions
(Rational
Irrational
Real s'
Imaginary
Complex
The first numbers we use are the positive integers, and the fundamental fact
that there is a first integer, unity, but not a last is soon established. Later positive
fractions, or numbers, which can be expressed as the ratio of two of these integers,
are used and understood. Then it is seen that these integers and fractions can be
negative as well as positive. The division point between the positive and negative
numbers which is the position from which we start to count in either direction, is
occupied by the number zero. This number is different from all others in that we
are not allowed to use it as a divisor.
The positive integers are often written without the phis sign, thus we may write
789 instead of + 789. Since zero is neither positive nor negative, it has no sign.
If we take a straight line and label a point on the line 0 and another point +1,
we impose a scale on the line in terms of which we can mark off the line with the
positive numbers to the right of 0 and the negative numbers to the left.
38

To each point on the line we assign a number whose length is the distance of
the point from zero and whose sign + or - is determined whether the point is to
the left or right of zero. Hie numbers in this uncountable set are known as the real
numbers. The integers correspond to a small subset of the reals. Hie positive and
negative integers and fractions, together with zero, are called rational numbers.
Besides rational numbers we find irrationals, which are defined as numbers
that cannot be expressed as the quotient of two integers. The ^2, -V3 and л are
examples of such numbers.
Hie two classes of numbers, rational and irrational, form the real number
system, which we shall use in the first part of our course. Later we shall study such
numbers as ^-2, -V-l, etc., which are called imaginaries; and finally it will be
seen that the basic system of all numbers is the complex, in which the reals and
imaginaries are included as special cases. 2 + V-3 is such a number and we see, that
it is composed of a real and an imaginary parts.
To denote the part of a complex number we use the notation
R(a + bi) = a
for the imaginary part.
Arithmetic is performed on complex numbers in the same way as on real
numbers, except that i2 is replaced by -1 whenever it occurs.
Post-Reading Activity
Ex. 8. Answer the following questions.
1. What were positive integers used for? 2. When was the number system completed?
3.	Are the first numbers which we use the positive integers or the negative ones?
4.	Is unity a first or a last integer? 5. By what is the division point between positive
and negative numbers occupied? 6. What are rational numbers? 7. Can irrational
numbers be expressed as quotients of two or three integers? 8. What numbers are
called imaginaries? 9. What do we assign to each point on the line? 10. What do we
use to denote the real part of a complex number?
Ex. 9. Find the Russian equivalents for the following English word combinations:
1) in either direction; 2) imaginaries; 3) is composed of; 4) real and imaginary
parts; 5) a quotient of two integers; 6) the number system; 7) to denote the real
part; 8) is replaced by; 9) the basic system; 10) in terms of; 11) to correspond to
a subset.
а) основная система; b) система чисел; с) действительные и мнимые части;
d) в любом направлении; е) заменяется; f) обозначать действительную часть;
g) состоит из; h) мнимые числа; i) частное двух чисел; j) через, посредством;
к) соответствовать подмножеству.
39

Ex. 10. Give the proper English equivalents for the Russian expressions:
древность, соотношение, разделительная точка, делитель,
рациональные и иррациональные числа, частное двух чисел,
мнимые числа, запись R (а + bi) = ау
действительная и мнимая части.
1. The ... is occupied by the number zero. 2. Hie irrationals cannot be expressed
as a .... 3. Positive fractions or numbers can be expressed as the ... of two of these
integers. 4. We are not allowed to use the number zero as a ... .5. Hie beginning
of our number system dates back to ... . 6. Hie real number system is formed
of.... 7. Numbers ^-2, -V-l are called .... 8. Hie expression 2 + "7-3 is composed
of a .... 9. To denote the real part of a complex number we use the ....
Ex. 11. Open the brackets and put the verbs in the proper tenses.
1. Why you (to look at) me like that? 2. We (to read) a book while he (to cook)
lunch at that time yesterday. 3. Hiis time next month I (to cross) the Pacific Ocean.
4. I (to wait) for you when you come out. 5. He (to sit) in a cafe when I saw him.
6. I (to go) to the cinema tonight. 7. Look! She (to wear) the same dress as me.
8.	Yesterday at 8 o’clock we (to watch) the football match. 9. He (to drive) his car
himself today. 10. Great news! Jack (to come) in a week.
Ex. 12. Fill in the blanks with the pronouns from the box:
any, anything, no, some, nobody, anyone, anything, any, some,
anybody, anywhere, something, anyone.
1. Is there ... juice left in the fridge? 2. Could I have ... coffee? 3. Has ... called
me? 4. Is there ... I can do for you? 5. She said ... very interesting. 6. Don’t go ...
tonight. 7. When I came home there was ... there. 8. Does ... know ... funny jokes?
9.	Would you like ... more tea? 10. Did you notice ... strange about him? 11. Hiere
is ... time left at all. 12. We can rarely meet... as brave as he is.
Ex. 13. Ask special questions.
1.	Hie students are applying irrationals in the given mathematical problems, (who)
2.	Hie teacher is explaining how to mark off the line with numbers, (what) 3. Hiere
are some positive numbers in the expression, (what) 4. She was trying to find the
common ratio of the three fractions when he came, (what) 5. We will be using this
notation to introduce imaginaries next term, (when) 6. The students are doing the
research now. (who) 7. At 4 o’clock yesterday Professor Clarkson was delivering a
lecture, (when) 8. This time next month we will be having a winter session, (what)
40

9. They are solving some mathematical problems at the moment, (who) 10. We are
going on a summer holiday in a couple of days, (when)
Ex. 14. Translate from English into Russian.
1. We will be considering complex numbers next term. 2. They are trying to
find the quotient of these two numbers. 3. This time last week we were having a
holiday. 4. We are substituting unknowns for irrationals to get a right result. 5. Hie
researchers were using this equipment all month long. 6. Are there any irrational
numbers in this system? 7. The students were finishing the test on mathematics
when the bell rang. 8. Hie delegation of scientists is coming to Minsk in a week.
9.	Hiis time next term the students of MME will be celebrating the birthday of the
faculty. 10.1 don’t see any other ways to arrange this matter. 11.1 can do it without
anybody’s help. 12. You can take any of these books.
Ex. 15. Translate from Russian into English.
1. Студенты выполняют контрольную работу сейчас. 2. Группа аспирантов
отправляется в Германию в декабре. 3. Мы будем использовать комплексные
числа в следующем семестре. 4. В это время через неделю мы будем сдавать
экзамены. 5. Мы решали эту задачу целый урок вчера. 6. Мы стараемся найти
частное этих целых чисел. 7. Что ты делаешь завтра после занятий? 8. В это
время вчера мы обедали в столовой. 9. Что ты делал, когда я позвонил тебе?
10.	Некоторые из этих числовых систем сложны. 11. Студенты выбирают
мнимые и иррациональные числа из данных выше.
UNIT 4
Модальные глаголы и их эквиваленты
Сап
Present
сап
cannot (cant)
am/are/is (not) able to
Future
сап
shall/will (not)
be able to
Past
could
could not (couldn't)
was/were (not) able to
Can (уметь, мочь, знать как сделать, иметь право сделать что-либо) выражает
умственную, физическую способность (или их отсутствие), способность сделать
что-то в силу обстоятельств, реальную возможность;
41

to be able to (быть способным к чему-либо, быть в состоянии, иметь силу,
власть, ум, возможность что-либо сделать) используется вместо отсутствующих
видовременных форм глагола сап, а также как самостоятельный глагол.
Не can swim.
Он умеет плавать.
I am able to help you
now. (particular situation)
Я в состоянии помочь
тебе сейчас.
I can’t hear anything.
Я ничего не слышу.
I can do it tomorrow;
Я могу сделать
это завтра.
I shall be able to
help you.
Я смогу помочь
тебе.
Повторяющиеся действия,
способность делать что-то в
прошлом:
Не could read at the age of 5.
Он умел читать в 5 лет.
Однократное успешное действие в
прошлом (смог, удалось):
Не w’as able to win the game.
Он смог победить в игре.
Просьба
Can you tell те the time? (разг.)
Можешь сказать, который сейчас час?
Could you give те a lift? (вежливо)
Не могли бы вы меня подвезти?
Разрешение/просьба дать разрешение
You can take the book now. (разг.)
Можешь взять книгу сейчас.
Can I go to the cinema, Mum? (разг.)
Можно мне пойти в кино, мама?
-	Could I make a suggestion? (вежливо)
Можно мне внести предложение?
-	Of course, you сап/тау.
Конечно, можно.
Запрет (нельзя) - самая частая и нейтральная форма запрета, не разрешается с
точки зрения закона или правил (в разговорной и письменной речи).
You can’t cross the street here.
Здесь нельзя переходить улицу.
They couldn’t wear jeans at wrork.
Им нельзя было ходить в джинсах
на работу.
Предложение помощи
Can I help you? (разг.)
Я могу вам помочь?
Could I carry that bag for you? (вежливо)
Могу я помочь вам поднести сумку?
42

Сильное сомнение/удивление
(не может быть, чтобы (can’t); вряд ли (can’t); неужели (Сап ... ?)
Употребляется в отрицательных и вопросительных предложениях с любыми фор¬
мами инфинитива. Время совершения действия передается формой инфинитива,
а не формой глагола сап. Форма could делает предложение менее категоричным.
С неперфектным инфинитивом -
предположение о действии, относящемся к
настоящему и будущему:
Can (could) it be so late?
Неужели уже так поздно?
Can (could) they be waiting for us?
Неужели они сейчас нас ждут?
You can’t be thinking of leaving now.
Сейчас ты не можешь думать об отъезде.
Can she dislike me?
Неужели я ей не нравлюсь?
You can’t be cold. Its very hot here.
He может быть, чтобы тебе было холодно.
С перфектным инфинитивом -
предположение о действии,
относящемся к прошлому:
Can (could) it have been so late?
Неужели было так поздно (тогда)?
Can (could) they have been waiting
for us?
Неужели они нас ждали (тогда)?
Не can’t have said it.
He может быть, чтобы он это
сказал (тогда).
She can’t have been lying.
He может быть, чтобы она лгала.
She can’t have failed to see him.
He может быть, чтобы она с ним
не встретилась.
Возможность (отсутствие препятствия к совершению действия)
General possibility:
This road can get very busy.
Эта дорога может быть очень оживленной.
Specific situation:
The roads could get very busy tomorrow because
there is a demonstration.
На этих дорогах может быть очень большое
движение, поскольку завтра состоится
демонстрация.
In those days, everybody could find
a job.
В то время все могли найти
работу.
Ex. 1. Analyse the following sentences and translate them into Russian.
1. She could do sums in her head when she was 6 years old. 2. Peter was able to
carry out the experiment successfully, but Nick couldn’t finish it without any help.
3.1 can speak three languages but I cant spell in any of them. 4. Will you be able
to notice the difference between these two proofs? 5. Can I discuss the subject
with my group-mates? 6. Could you tell me the way to the University? 7. You cant
use these dangerous materials in your research. 8. When she sings you can hear
her all over the house. 9. Can I do anything for you? 10. You can get this book at
43

the library. 11. Anybody can make mistakes. 12. Can (could) you show me these
data? - I’m afraid, I cant. 13. You can speak to Jane now. 14. Could you spell your
name for me?
May
Present
may
may not
Future
may
shall/will (not) be allowed to
Past
might (not)
was/were (not) allowed to
Разрешение/просьба дать разрешение (можно; можете)
You may answer the
questions later, (офиц.)
Можете ответить на
вопросы позже.
-	May I come in? (вежливо)
Можно войти?
-	Yes, you may. Можно.
Отрицательные ответы:
a)	не очень строгий запрет:
-	No, you may not.
Вам не разрешается.
b)	категорический отказ:
-	No, you mustn’t.
Нет, нельзя, запрещено.
c)	вежливый отказ:
-	Гт afraid, you can’t.
Боюсь, что нет.
You may call me tomorrow,
(офиц.)
Можете позвонить мне
завтра.
They will be allowed to
carry out the experiment at
our laboratory.
Им разрешат провести
эксперимент в нашей
лаборатории.
Форма might в
значении разрешения
употребляется только в
косвенной речи:
Не said that I might
borrow his pen.
Он сказал, что я могу
взять (одолжить) его
ручку.
Не was allowed to enter
the country.
Ему разрешили въехать
в страну.
Запрет (нельзя)
You may not talk during the
test, (офиц.)
Нельзя разговаривать во
время теста.
You will not be allowed to
take the exam.
Тебе не разрешат сдавать
экзамен.
We were not allowed to
tell her everything.
Нам запретили
рассказать ей все.
Сомнение/неуверенность (может быть, возможно)
Might указывает на меньшую степень уверенности в совершении действия, чем
may; употребляются все формы инфинитива.
Предположение о действии, которое относится к
настоящему и будущему.
The rain may (might) stop later in the day.
Возможно, попозже днем дождь прекратится.
Предположение о
действии,которое
относится к прошлому.
Не may have been ill.
Возможно, он болел.
44

He may (might) come tomorrow.
Возможно, он придет завтра.
They may (might) be living in the country.
Возможно, они живут за городом (сейчас).
The door may have been
locked.
Возможно, дверь была
заперта.
Возможность (отсутствие препятствия к совершению действия)
Students may (can) express this familiar theorem in terms
of an equation.
Студенты могут выразить эту хорошо знакомую им
теорему через уравнение.
She said that we might
(could) get to the centre
of the city by bus.
Она сказала, что мы
можем добраться до
центра на автобусе.
Упрек (might)
Упрек по поводу действия, которое еще может быть
реализовано:
You might help us.
Ты мог бы нам помочь.
You might visit your friend as he is ill.
Ты мог бы навестить своего друга - он сейчас болеет.
Упрек по поводу нереа¬
лизованного действия:
You might have phoned.
Мог бы и позвонить (но
не позвонил).
You might have told me
about it.
Ты мог бы и сказать мне
об этом (но не сказал).
Ex. 2. Analyse the following sentences and translate them into Russian.
1. The students may not use the calculator at the exam. 2. Professor, may I take the
exam next week? 3. Were students allowed to visit the laboratory? 4. Hie answer
may give the key to the whole problem. 5. Will he be allowed to take part in the
conference? 6. Hie teacher may ask you to stay after the lessons and copy the text.
7. You are not allowed to use the machine without permission. 8. You may do the
rest of the work tomorrow. 9. May I smoke here? - No, you mustn’t. 10. May I take
this map? - No, you may not. 11. May he wait for us in the hall? - Yes, he may.
Must
Present
must
must not (mustn't)
Future
shall/will (not) have to
Past
had to
did not have to
Did you have to ... ?
Необходимость/долг/обязанность (должен, нужно, надо, обязан)
I must... передает необходимость выполнить действие в силу л ичной убежденности.
45

You must obey these rules.
Ты обязан подчиняться
правилам.
I’m afraid I must go now.
Боюсь, я должен идти.
-	Must I do it?
Я должен сделать это?
-	Yes, you must.
Да, должен.
-	No, you needn’t.
Нет, не надо.
She will have to come
later.
Она должна будет
прийти позже. / Ей
придется прийти позже.
Му parents had to work
very hard to build up their
business.
Моим родителям пришлось
усердно работать, чтобы
создать свой бизнес.
Высокая степень уверенности (вероятно, должно быть)
Для предположения
о действии, которое
относится к настоящему.
a)	утвердительные пред¬
ложения:
She must be about twenty.
Ей должно быть около 20.
b)	отрицательные
предложения:
Probably he doesn’t know
English, (с отрицанием)
Должно быть, он не
знает английского.
Для предположения
о действии,которое
относится к будущему
используются эквива¬
лентные выражения:
The weather is likely to
change.
Вероятно, погода
изменится.
Probably, he will speak
English.
Он, должно быть, будет
говорить по-английски.
Для предположения о дей¬
ствии, которое относится к
прошлому.
a)	утвердительные предло¬
жения:
She must have known about it.
Должно быть, она об этом
знала.
b)	отрицательные предло¬
жения:
Probably, she didn’t know my
address. Должно быть, она
не знала моего адресса.
Запрет (строгая форма запрета)/необходимость не совершать действие (нельзя;
запрещено)
You mustn’t walk on the grass.
По газону ходить запрещено.
You must not miss classes.
(Вам) нельзя пропускать уроки.
Не told me I mustn’t cry.
Он сказал, что я не должен
плакать.
Приказание/настоятельный совет (должен; обязан)
You must leave the room at once.
Немедленно покиньте комнату.
You must revise for your test.
Ты должен повторить весь материал к тесту.
Ex. 3. Analyse the following sentences and translate them into Russian.
1. As a postgraduate student you must obtain some new scientific results. 2. Must
we send them the results of our work immediately? - Yes, do please. 3. You must
46

pay more attention to details. 4. The door to the laboratory must not be left open.
5. The meeting is at 10 o’clock sharp and you mustn’t be late. 6. He must be at the
library now. 7. You must let him know about it. 8. You must be tired after your hard
work. 9. Must I type the document? - No, you needn’t. 10. People must not cross
the border without passports. 11. Everyone must go to school. 12. Must we measure
the perimeter? - Yes, you must. 13. Must I really help him with the translation?
'	) To have to (to have got to)
Present
have/has to
do/does not have to
Do you have to ... ?
Future
shall/will (not) have to
Will you have to ... ?
Past
had to
did not have to
Did you have to ... ?
Необходимость/обязанность (возникающая из-за обстоятельств)
(должен, нужно, надо, приходится, вынужден)
She has to look for a new job.
Ей приходится искать
новую работу.
I’ve got to go now.
Мне надо идти.
(have got to употребляется
только в настоящем
времени)
I shall have to speak to
them about this plan.
Мне придется
поговорить с ними об
этом плане.
He had to return home.
Ему пришлось
возвратиться домой.
Отсутствие необходимости (не нужно, не надо, не приходится)
We don’t have to attend
classes on Sunday.
Нам не нужно ходить на
занятия в воскресенье.
Не won’t have to come
back till April.
Ему не придется
возвращаться до апреля.
She didn’t have to take
a taxi.
Ей не пришлось брать
такси.
Ex. 4. Analyse the following sentences and translate them into Russian.
1. Do I have to present another schedule? 2. I had to explain the rule twice to
make it perfectly clear. 3. They didn’t have to change the date of the conference.
4. You ve got to study the relation between these two discoveries. 5. Does she have
to wear glasses? - Yes, she does. 6. They will have to arrange everything for the
meeting. 7. Did you have to walk all the distance to the station yesterday? 8. Do we
have to define prime numbers? - No, you needn’t. 9. Have I got to make another
drawing? - Yes, you have.
47

То be to
Present
am/is/are (not) to do
Future
am/is/are (not) to do
Past
was/were (not) to do
Необходимость (предусмотренная планом, договоренностью, расписанием)/
ожидаемое, запланированное действие/предопределенное событие (должен,
предстоит, суждено)
What am I to do?
Что я должен делать? / Что мне делать?
The train is to come at 7.
Поезд должен прибыть в 7 часов.
You are to make a report at the conference.
Ты должен сделать доклад на конференции.
They are to arrive in London tonight.
Они должны прибыть в Лондон сегодня
вечером.
You were to stay there.
Ты должен был остаться там.
(выполнено ли действие или
нет - неизвестно)
Не was never to see her again.
Ему не суждено было увидеть ее
снова.
Обозначает запланированное
действие в прошлом, которое не
произошло:
was/were to + Perfect Infinitive
We were to have met at the station.
Мы должны были встретиться
на вокзале.(но не встретились)
Строгий приказ/инструкции (должен, не должен, не делай, не смей делать)
You are to finish your article tomorrow.
Вы должны закончить статью завтра.
No-one is to leave the room.
Никому не покидать комнату!
Ex. 5. Analyse the following sentences and translate them into Russian.
1. The students are not to take the books out of the reading-room. 2. Were they to
meet at four o’clock at the University? 3. You are to write an abstract of your thesis.
4. Am I to follow you? 5. We are to have six lessons of English this week. 6. You are
to give the books back to the library at the end of the year. 7. The concert was to
take place yesterday. 8. Who is to meet Jack? 9. She was to finish school last year.
10. The lessons are to begin at eight o’clock. 11. Is she to make a speech? - Yes,
she is.
48

Need
Need как модальный глагол (нужно, надо) употребляется только в фор¬
ме Present Indefinite преимущественно в вопросительных и отрицательных
предложениях.
Present
need, needn't
Future
need, needn't
Past
Необходимость (нужно, надо)
В вопросительных предложениях выражает сомнение в целесообразности выпол¬
нения действия.
Need we go to the library tomorrow?
Нам завтра нужно идти в библиотеку?
- Need I do the whole exercise?
Мне надо сделать все упражнение?
a)	- Yes, you must.
b)	- No, you needn’t.
Отсутствие необходимости (не нужно, не надо, незачем; зря)
Не needn’t speak to her about it.
Ему не нужно говорить с ней об этом.
I needn’t get up early today.
Мне не надо сегодня рано вставать.
You needn’t wait any longer.
Тебе не надо больше ждать.
needn’t + Perfect Infinitive выра¬
жает отсутствие необходимости
совершения действия в прошлом,
но действие было совершено:
Не needn’t have performed the
operation of division first.
Зря он сначала выполнил деление.
Need как смысловой глагол (нуждаться, нужно) имеет те же временные
формы, что и обычные глаголы, употребляется в сочетаниях с существи¬
тельными; а при употреблении с глаголами выражает необходимость выпол¬
нения привычного, повторяющегося действия.
Present
need, needs, don't need,
doesn't need
Future
shall/will need
Past
needed, didn't need
I don’t need to work at the
lab every day.
Мне не надо каждый день
работать в лаборатории.
I’ll need your help.
Мне будет нужна
твоя помощь.
We needed the dictionary badly.
Нам был очень нужен словарь.
Не didn’t need to stay after classes.
Ему не надо было оставаться
после уроков.
49

Ex. 6. Analyse the following sentences and translate them into Russian.
1. Need we continue working by this plan? 2. You needn’t do all this in written
form, you know. 3. He didn’t need to read his paper at the seminar. 4. Do you need
to write sentences about numbers in all branches of mathematics? 5. Need you
worry about the man who always deceives you? 6. They needn’t hurry now. They
will have to wait for another train. 7. Need I come myself? - Yes, you must. 8. Need
we perform any other construction? - No, you needn’t.
Should, ought to
Present	Future
should (not), ought (not) to	should, ought (not) to
Past
Совет (следует, должен)
В этом значении чаще употребляется should, который показывает личную заинте¬
ресованность.
You should try again.
Тебе следует снова попытаться.
Не should be more careful about his health.
Ему следует более внимательно относиться к своему
здоровью.
Обязанность/моральный долг (следует, должен)
В этом значении чаще употребляется ought to.
You ought to tell your parents the truth.
Ты должен говорить своим родителям правду.
Большая степень уверенности (должно быть, вероятно, по-видимому)
Для предположений о настоящем и будущем, основанных на каких-то известных
говорящему или вытекающих из контекста фактах, используется should/ought to,
а не must.
Its 10 o’clock. Не should/ought to be at work.
Сейчас 10 часов. Он, должно быть, на работе.
Не should pass his exam easily. He is good at maths.
Он должен легко сдать экзамен. Он хорошо знает матема¬
тику.
Недоумение/удивление/несогласие (выраженное вопросом)
"Why should I go there?
Чего ради мне идти туда?
Why should I always wait for you?
Почему я всегда должен тебя ждать?
50

Критика (не следовало, не надо было)
Критика по поводу действия в прошлом, которое было или не было совершено,
(употребляется перфектный инфинитив)
You shouldn't have read the letter.
Вам не следовало читать письмо.
We should have done it long ago.
Нам давно следовало бы это сделать.
Ex. 7. Analyse the following sentences and translate them into Russian.
1. You shouldn’t miss lectures and seminars if you want to pass the exams. 2. If you
want to know mathematics, you should work hard at it. 3. Doctor, what should I do?
4.	You ought to go further with this investigation. 5. Ought you to tell everybody
what has happened? 6. You ought not to participate in their useless argument.
7.	You should switch off your mobile in class. 8. They ought to ban smoking in
public places.
Pre-Reading Activity
Guess the meaning of the following words.
Decimal (adj) [4 dsszzz VI], combination (n) [.kcir/zz'rzeznegative
(adj) [V.szszzv], symbol (n) pszrW], plus (n) [р1лн], minus (n) ['ziazzss],
indicate (v) [4z'dzkszz], sum (n) [sa~], product (n) ['pzczskz], numeration
(n) z^s'rsz.'srj, base (n) [zezs], arithmetic (n) [Vrz9~is:zk], system (n)
['szizzirj, represent (v) [.repzz'zs'r].
Read and learn the basic vocabulary terms:
number (n) ['	=]
numeral (n) ziszsl]
digit (n) [' ±z<z:]
division (n) [dz'vz
divide (v) [zz'vazz]
dividend (n) [' zivz W]
divisor (n) [zz'vazzs]
quotient (n) pkwszj'szz]
value (n) ['vs?z ]
derive (v) [zz'zazv]
introduce (v) [.zr.zrs' s]
число, количество; (v) перечислять
цифра, символ, число
цифра
деление
делить
делимое
делитель
частное
величина, значение; (v) ценить
происходить, получать
вводить, представлять, знакомить
51

invent (v) [zs'vesr]
denote (v) [di'r.s'J:]
determine (v) [з:ч:з 22:2]
enable (v) [:'	V.]
appear (v) [s\c:s]
sign (n) [sa:r_]
addition (n) [s'di'Vr.]
add (v) [ssz]
addend (n) [s4 dss2]
subtract (v) [ssb'trsk:]
minuend (n) [4m:r.>=s£]
difference (n) [4 difsrsr.s]
inverse (adj) [zr/vs 5]
factor (n) [4 fsskts]
multiplication (n) [, 22лк:2к4 kszj" Vr.]
multiply (v) [422s.lt:cla:]
multiplicand (n) [z22A?:pk'ks22]
multiplier (n) [' mAltzplais]
meaningless (adj) ['22:2:-l=s]
subtraction (n) [ssVtrssk.'V2]
subtrahend (n) pSAb::shs2.2]
изобретать, создавать, придумы¬
вать, открыть
обозначать, отмечать, означать
определять, устанавливать
давать возможность, делать воз¬
можным
появляться
знак; (v) подписывать
сложение
складывать
слагаемое
вычитать
уменьшаемое
разность
обратный, противоположный
сомножитель; (v) разложить
на множители
умножение
умножать
множимое
множитель
бессмысленный
вычитание
вычитаемое
Memorise the following word combinations:
the former/the latter - первый (из вышеупомянутых)/последний (из выше¬
упомянутых)
no matter how great - не важно, насколько большое
ten times as great - в десять раз больше
the number to be made smaller - число, которое должно быть уменьшено
since - так как
the number resulting from multiplication - число, полученное в результате
умножения
therefore - следовательно
what is left over - то, что осталось
whenever - всегда, когда бы ни
except - за исключением
52

Reading Activity
FOUR BASIC OPERATIONS OF ARITHMETIC
Many thousands of years ago this was a world without numbers. Today, using
the same numbers in many different ways, man can build bridges, skyscrapers,
fly off the Earth like a bird, even measure the distance to the Moon. So you see,
mathematics and numbers, are very important to life nowadays.
Until XVI century people in Europe used Roman numerals. The Roman system
of numbers is based upon the letters I, V, X, L, C, D, and M. These letters were
mixed together to form many different combinations. The arithmetic symbols now
in use were derived from the Arabs and the Hindus, the latter of whom introduced
the symbol 0. The invention of this symbol for zero was very important, because it
enabled the nine Hindu symbols 1,2, 3,4, 5,6,7,8 and 9 to represent any number,
no matter how great. Hie work of a zero is to keep the other nine symbols in their
proper place. The Hindu-Arabic numeration system is a decimal system: that is,
it is based on tens. In this system the value a digit represents is determined by the
place it has in the number; if a digit is moved to the left one place, the value it
represents becomes ten times as great.
Hie invention of our present notation for the decimal number system made
possible simple and systematic operations with numbers. Arithmetic is the
elementary branch of mathematics dealing with the properties of numbers and
their operations. We work with only positive numbers in arithmetic. Negative
numbers appear in algebra. An operation is a way of thinking of two numbers
and getting one number. The fundamental operations of arithmetic are addition,
subtraction, multiplication, division.
Hiere are special signs to indicate operations with numbers. Hiey are plus (+),
minus (-), multiplication (x) and division (:) signs.
Hie process of finding a simple expression for the sum of two or more
numbers is known as addition. In the arithmetic sentence 3 + 5 = 8 three and five
are addends and eight is the sum.
In the operation of subtraction the number to be made smaller is called the
minuend. Hie subtrahend is the number to be subtracted. The result of the process
is the difference. A sentence like 6-4 = 2 represents an operation of subtraction.
Here the difference is the number that when added to the subtrahend gives the
minuend. Thus, subtraction is the inverse operation of addition since 2 + 4 = 6 and
6-4 = 2.
In multiplication the number being multiplied is the multiplicand. Hie
number by which we are multiplying the multiplicand is the multiplier. When
two or more numbers are multiplied, each of them is called a factor. Hie number
resulting from the multiplication is known as the product. You must remember
53

that the product of any number multiplied by zero is zero. The product of any
number multiplied by one is the same number.
In division the number to be divided is called the dividend. The divisor is the
number by which the dividend is to be divided. When we are dividing the dividend
by the divisor we get the quotient. You may check division by using multiplication
since 4:2 = 2 and 2x2 = 4. Therefore, division and multiplication are inverse
operations. The remainder is what is left over after the dividend has been divided
into equal parts. If there is a remainder, it may be written over the divisor and
expressed as a fraction in the quotient. There are some important facts that must
be remembered about division. Hie quotient is 0 (zero) whenever the dividend is 0
and the divisor is not 0. That is, 0 : n = 0 for all values of n except n = 0. Division
by 0 is meaningless for all values of n.
Post-Reading Activity
Ex. 8. Answer the following questions.
1.	What numerals were used in Europe until XVI century? 2. Who introduced the
symbol 0? 3. Is the Hindu-Arabic system a decimal system or a binary one? 4. What
does the value which a digit represents depend on? 5. What are the signs most used
in arithmetic? 6. What are the fundamental operations of arithmetic? 7. Are division
and multiplication inverse operations? 8. Are subtraction and multiplication
inverse operations? 9. What must you remember about multiplication? 10. What
important facts about division must be remembered?
Ex. 9. Find the Russian equivalents for the following English word combinations:
1) to use the same numbers; 2) to represent numbers; 3) a present notation;
4) a numeration system; 5) a decimal system; 6) to be based on tens; 7) an inverse
operation; 8) capital letters; 9) an arithmetic sentence; 10) to measure the distance,
а)	основываться на десятках; b) обратная операция; с) заглавные буквы;
d) измерять расстояние; е) десятичная система; f) представить числа; g) ис¬
пользовать одни и те же числа; h) система счисления; i) арифметическое вы¬
ражение; j) современная система записи.
Ех. 10. Mark the following as true or false.
1.	People used numbers thousands of years ago. 2. The decimal number system was
invented by the Romans. 3. Hie Roman system of numbers is based upon the letters
А, В, C, D, E, F and G. 4. The symbol 0 was introduced by the Arabs. 5. The work
of a zero is to keep the other nine symbols in their proper place. 6. Our decimal
number system is positional. 7. We go from right to left in forming larger and
54

larger units of ten, hundred, thousand and so on. 8. We work both with positive
and negative numbers in arithmetic. 9. Hie product of any number multiplied by
one is the same number. 10. The quotient is 0 whenever the divisor is 0.
Ex. 11. Fill in the blanks with necessary words and word combinations from
the box:
the sum, the quotient, the multiplication sign, to divide,
the difference, subtraction, factors, addends, the subtrahend,
a remainder, to be added, a numeral, division.
1. We get ... as a result of addition. 2. The numbers to be added are called ... .
3.	You may check ... by multiplication. 4. Will there be ... if you divide 25 by 7?
5.	If you are ... two numbers, you must remember that division by 0 is meaningless.
6.	To find the minuend,... and the subtrahend must be known. 7. Addition and ...
are inverse operations. 8. The multiplicand and multiplier are the names for the ....
9. Hie result of division is known as .... 10. The difference is the number that when
added to ... gives the minuend. 11. The plus sign between two numbers means
that these numbers are ... . 12. A dot placed between two numbers is used as ... .
13. A symbol used to represent a number is called ....
Ex. 12. Make the sentences negative and interrogative.
1. We can add numbers in any order. 2. You should discuss the rules of
multiplication at the next lesson. 3. Mathematicians were able to discover another
fundamental law of nature. 4. Hie result must be checked immediately. 5. You have
to decide on the subject of your thesis. 6. They may misunderstand the theoretical
character of the problem. 7. We were allowed to work on our experiment out of
class. 8. Hiey will be able to use some symbols instead of words. 9. We need to use
special methods to obtain necessary results. 10. You ought to study carefully the
definitions given above.
Ex. 13. Ask special questions.
1. Scientists should develop this important branch of mathematics, (what) 2. In
antiquity people could count using positive integers, (when) 3. We are not allowed
to use zero as a divisor, (who) 4. Natural numbers may be divided into two classes:
even and odd. (how many) 5. She was able to obtain the solution by multiplying
two numbers, (how) 6. He had to speak English at the international conference,
(where) 7.1 must discuss the details of my thesis with the science adviser, (who(m))
8.	You need to follow your teachers instructions during the test, (whose) 9. You
ought to do your best and fulfil the task, (what) 10. First you are to perform the
operation of division and then multiply the quotients, (which)
55

Ex. 14. Choose the phrase closest in meaning to the given statement.
1.	Dan cant be a teacher.
a)	Гт sure Dan isn’t a teacher.
b)	I think Dan isn’t a teacher.
2.	Need I take the tablets every day?
a)	Is it a good idea to take the tablets every day?
b)	Is it necessary to take the tablets every day?
3.	If it is hot tomorrow, we may go to the beach.
a)	We will definitely go to the beach tomorrow.
b)	It is possible that we will go to the beach tomorrow.
4.	You mustn’t steal.
a)	It is against the law to steal.
b)	It isn’t necessary to steal.
5.	Alison has to work on Saturday. Her boss told her so.
a)	Alison wants to work on Saturday.
b)	Alison’s boss wants her to work on Saturday.
6.	Late-comers are to report to the Dean’s office.
a)	It’s a good idea.
b)	It’s the rule.
7.	Astronauts must feel afraid sometimes.
a)	They are supposed to.
b)	It’s only natural.
8.	You can’t come in here.
a)	It isn’t allowed.
b)	I don’t believe it.
9.	We should be there soon.
a)	I expect so.
b)	It’s absolutely certain.
10.	All motorcyclists have to wear crash helmets.
a)	It’s a good idea.
b)	It’s the rule.
Ex. 15. Choose the correct modal verb or its equivalent.
1. You (may/ought to/are to) take care of your parents. 2. My sight is getting worse.
Next year, I’m afraid I (cannot/may not/wont be able to) read without glasses.
3. Twelve delegates from several countries (can/have to/are to) meet at the end of
February. 4. Excuse me, (could/may/must) you tell me the way to the Houses of
Parliament? 5. The weather is getting worse. It (must/is likely/may) rain. 6. There
are no people in the hall, we (must/can/need) have a talk there. 7. Although he felt
56

ill, he (could/was able to/may) finish all the paperwork. 8. You (can/must/ought
to) go and see that movie. Its very interesting. 9. Don’t worry, you (dont have to/
mustnt/may not) pay now. 10 When we were at school, we (had to/ought to/must)
wear a uniform.
Ex. 16. Give the proper English equivalents for the Russian expressions.
1.	Нам пришлось perform the operation of addition to find the answer. 2. Ему
предстоит specify the conditions of the experiment. 3. Им разрешают use a
dictionary if necessary. 4. Я в состоянии solve this difficult problem myself.
5. Вам следует remember that multiplication is associative. 6. Ей не надо use this
theorem. 7. Они могут apply their theories in practice. 8. Вы обязаны remember
several rules about division. 9. Можно мне start the calculations now? 10. Вам
следует to accept everything your parents say as an axiom.
Ex. 17. Translate from English into Russian.
1. Everyone studying mathematics must have a good understanding of the
meaning of each symbol. 2. These expressions may contain numbers and letters.
3.	Multiplication can be distributed over addition, e.g. 14* 12 = (14 x 10)+ (14x2).
4.	A prime number is a number that can be divided evenly by only 1 and itself.
5.	No multiplication sign is to be used between two letters or a letter and a number
written side by side. 6. The plus sign should be understood when there is no sign
before the number. 7. Will you be able to name other factors of 18 except nine
and two? 8. In some cases, when we have to name a whole number in a factored
form, more than two factors can be used. 9. A positive number can be indicated
by placing a plus sign (+) before the number. 10. When the minus sign is placed
between any two numbers, it indicates that the difference of the two numbers is
to be found. 11. Division may be indicated also by writing the dividend above the
divisor with a line between them.
Ex. 18. Translate from Russian into English.
1. Числа, которые нужно сложить, называются слагаемыми, а результат
сложения называется суммой. 2. Эти числа предстоит перемножить. 3. Мы
должны записывать результат справа от знака равно. 4. Число можно поде¬
лить на два без остатка (точно), если это четное число. 5. Деление обратно
умножению. 6. Произведение любого числа, умноженного на ноль, равно
нулю. 7. Никто не может сказать, когда люди начали считать. 8. Они обозна¬
чили (указали) операцию сложения знаком плюс. 9. Ни один студент не мог
решить задачу, заданную профессором. 10. Им придется использовать дво¬
ичную систему счисления. 11. Данный элемент может быть обозначен тем
же символом.
57

UNIT 5
Времена группы Perfect
(Действительный залог)
Present
Past
Future
I
we
you
they
have
asked
I
we
you
they
he
she
it
had asked
I
we
shall will
have
asked
he
she
it
has
you
they
he/she/it
will
just, already, by now,
yet (еще, уже), before,
this morning/week/year,
today, never, ever (когда-
либо), recently (недавно),
lately (недавно), since,
for (в течение), how long
by 5 o'clock yesterday,
by that time, as soon as,
by Monday, up to that
year/week/day, after,
before he came
by 5 o’clock yesterday, by this
time, by the end of the year,
before he comes
Hie Present Perfect Tense употребляется:
1. Для выражения однократного действия, завершившегося к данному
моменту речи и связанного с настоящим временем через результат. Действие
могло совершиться как непосредственно перед моментом речи, так и в более
отдаленное время в прошлом.
I haven’t seen him today.
Its the best book I have ever read.
Не hasn’t written his thesis yet.
He has just shown me his course paper.
Я его сегодня (еще) не видел.
Это лучшая книга, которую я когда-
либо читал.
Он еще не написал диссертацию.
Он только что показал мне свою
курсовую работу.
Какую задачу он решил?
What kind of problem has he solved?
2.	Для выражения действия, которое началось до момента речи и все
еще продолжается в момент речи (с глаголами, которые не употребляются
во временах группы Continuous).
I’ve been in the laboratory since two
o’clock.
We’ve known him for three years.
Я (нахожусь) в лаборатории с двух
часов.
Мы знаем его три года.
58

3.	Для выражения завершенного будущего действия в придаточных
предложениях времени и условия после предлогов when, after, as soon as, until,
till, if.
After I have checked the results,	После того как я проверю резуль-
I	will discuss them with my science таты, я обсужу их с моим научным
adviser.	руководителем.
The Past Perfect Tense употребляется:
для выражения действия, которое
действия или момента в прошлом.
We had translated the article by five
o’clock.
He had passed his exam in physics when
I met him.
совершилось до какого-то другого
Мы (уже) перевели статью к пяти
часам.
Он (уже) сдал экзамен по физике,
когда я его встретил.
The Future Perfect Tense употребляется:
для выражения действия, которое завершится к определенному моменту
в будущем. Это редко употребляемая форма.
Не will have informed them about the
changes in the program before they start
the experiment.
They will have finished their research
by the end of the next year.
Он уведомит их об изменениях в
программе до того, как они начнут
эксперимент.
Они закончат свое исследование к
концу следующего года.
Ex. 1. Compare the Russian and English tense forms.
Я решил эти задачи ...
Я решу эти задачи ...
I solved the problems yesterday. (Past Ind.)
* I have already solved the problems. (Pr. Perf.)
I had solved the problems before he came. (Past Perf.)
" I shall solve the problems tomorrow. (Fut. Ind.)
I shall have solved the problems when he comes. (Fut.
< Perf.)
If I solve the problems, will we obtain the required
< result of the experiment? (Pr. Ind.)
Ex. 2. Read and compare the follow ing sentences. Explain the use of the English
tense forms (Present Perfect, Past Indefinite, Past Perfect, Future Perfect, Future
Indefinite, Future Continuous, Present Continuous).
1. a) He has enlarged his English vocabulary lately.
b)	He enlarged his English vocabulary when he was in Great Britain.
c)	Before he went to Great Britain he had enlarged his English vocabulary.
59

2.	a) How long have you been here?
b)	How long ago were you there?
c)	He wondered how long I had been there.
3.	a) Since when have you started working as a teacher?
b)	When did you start working as a teacher?
c)	By that time she had started working as a teacher.
4.	a) The scientists have just changed the order of the whole process.
b)	The scientists changed the order of the whole process just now (a moment
ago).
c)	He said that the scientists had just changed the order of the whole process.
5.	a) Perhaps man will improve the devices used in calculations.
b)	By the end of this century man will have radically improved the devices used
in calculations.
c)	Man is going to improve the devices used in calculations radically in the near
future.
6.	a) Our students will translate this article next week.
b)	Our students will be translating the article this time tomorrow.
c)	Our students will have translated this article by the time you come.
7.	a) I promise I shall find a proper solution of the problem soon.
b)	By the middle of the lesson I shall have found a proper solution of the problem.
c)	Don’t phone now. I shall be looking for a proper solution of the problem.
8.	a) In the future she will try to enter the faculty of Cybernetics.
b)	She is going to enter the faculty of Cybernetics after she finishes school.
c)	I am sure, she will have become a student of the faculty of Cybernetics, when
we meet again.
9.	a) They will take part in the international conference on mechanics in a year.
b)	They will already have taken part in the international conference on
mechanics by the time you come there.
c)	This time next week they will be taking part in the international conference
on mechanics.
Ex. 3. Make the following sentences interrogative and negative.
1. By the middle of the 21sl century we’ll have built a lot of space stations. 2. The
teacher has just spoken about rational and irrational numbers. 3. By that time
natural scientists had learnt to use the parallelogram as a means of addition.
4.	A young mathematician has found a better proof of the theorem recently. 5. By
the end of the week I’ll have written the second chapter of my dissertation. 6. He has
already taken his exam in differential equations. 7. They had done their laboratory
work by 2 o’clock. 8. We have just replaced the terms in the equation. 9. This week
60

the students have learnt to perform operations on complex fractions. 10. I had
simplified the fractions before multiplying them.
Ex. 4. Translate the following sentences into Russian.
1.	Mathematicians have used mathematical formulas in solving these problems.
2.	By the end of the lesson we’ll have been able to obtain the modified definition of
the function. 3. Scientific theories have often suggested directions for mathematical
investigations. 4. Physical objects and observed facts had often served as a source
of the postulates in Maths. 5. Einstein was able to achieve some of his results after
Maths had suggested new ways of thinking about space and time. 6. Abel had hardly
reached the age of 22 when he made two of his most famous discoveries. 7. Algebra
has become the science that can deal effectively with anything. 8. By the end of June
the students will have passed their exams and gone on holiday. 9. Throughout the
centuries people have improved their ability to record, process and communicate
information. 10. When you come, I will have solved these equations with fractions.
11. Ulis century scientists have made a lot of discoveries about the universe.
Pre-Reading Activity
Guess the meaning of the following words.
Fraction (n) [frssk.'V n], fractional (a) ['frsskkrd], equivalent (a) [f kwivslsr.:],
rational (a) [kss 'sr.l], process (n) ['prsnses], concept (n) ['kcssept], arithmetic
(n) [skiGmsrik], operation (n) [cpskeiki], horizontal (a) [.hcri'zcnrl], separate
(v) [kepsrei:], (a) [kep.-it], decimal (a) ['desiml], type (n) [kaip].
Read and learn the basic vocabulary terms:
part (n) [pa t]
proper (a) ['preps]
improper (a) [irr/preps]
change (n, v) [rjkizd^]
proportional (a) [prs'pc fsl]
equal (a) ['1 kw'sT]
reduce (v) [n'd> s]
relation (n) [”ke: ki]
ratio (n) [ks: ks”]
numerator (n) [к.1~ izisreirs]
denominator (n) [di'scminsirs]
slanting (a) ['slz s::~.]
часть
правильный
неправильный
замена, заменять
пропорциональный
равный
сокращать
отношение
пропорция, отношение
числитель (дроби)
знаменатель (дроби)
наклонный
61

define (v) [di'fain]
quantity (n) ['kwcsuri]
mixed (a) [mik-st]
imply (v) [irr/plai]
point (n) [pcin:]
repeat (v) [:f pi t]
repeating (a) (fraction) [rf pi :ir]
terminating (a) [.is irii'neiti"]
agreement (n) [s'sri ~iSi_i]
produce (v) [prs'd1" s]
consider (v) [ksr.'sizs]
the former (of)(adj) ['fc ms]
compare (v) [ksm'pss]
irreducible (a) [iri'r'J ssbl]
nought (n) [nc 1]
whole (a) ['hsirl]
power (n) ['pans]
определять
величина, количество
смешанный
значить
точка
повторять
периодическая (дробь)
конечная(дробь)
соглашение
производить
рассматривать, полагать, считать
первый (из)
сравнивать
несократимая (дробь)
ноль
весь, целый
степень
Memorise the following word combinations:
over and over again - многократно
may be changed to - может быть превращена
in full agreement with - в полном соответствии с
a part - to - whole relationship - отношение части к целому
by a slanting line - наклонной линией
the absolute value of the entire fraction - абсолютное значение всей дроби
in its lowest or simplest terms - в своем простейшем виде
to the left of (to the right of) - налево, направо
leaving out the decimal point - пропуская десятичную точку
to have in common - иметь что-то общее (сходное)
rather than - скорее чем, вернее чем
Reading Activity
RATIONAL NUMBERS
AND DECIMAL NUMERALS
In mathematics, a fraction is a concept of a proportional relation between
a part and a whole. In other words, it is an example of a specific type of ratio, in
which the two numbers are related in a part-to-whole relationship, rather than
a comparative relation between two separate quantities.
62

A fraction is a quotient of numbers, the quantity obtained when the numerator
is divided by the denominator. Thus, % represents three divided by four.
Hie denominator represents the number of equal parts that an object is divided
into, and the numerator tells the number of those parts indicated for the particular
fraction. The numerator and the denominator of a fraction may be separated by
( 1
a slanting line e.g. — , or may be written above and below a horizontal line.
■	i
Fractions are rational numbers and that means that the denominator and the
numerator are integers. Any rational number might be defined as a number named
by — where a and n ~t- 0.
n
Usually there are several ways of reading fractions. One may say ‘three quarters’
3	t	, , i
tor — and one sixth’ for — . In strictly mathematical contexts these fractions might
4	6
also be read as ‘three over four’, one over six’ or ‘three upon four’, one upon six’.
Especially more complex fractions may be expressed by using the word over’
( 317Л
e.g. 	 .
t	509 J
A common fraction is called a proper fraction if the absolute value of the
numerator is less than the absolute value of the denominator - that is, if the
absolute value of the entire fraction is less than one. An improper fraction names
the absolute value of the numerator greater than or equal to the absolute value of
(	9 9^
the denominator e.g. —, — .A mixed number is the sum of a whole number and
v 7 9 J	. ,	(	2A
a proper fraction. This sum is implied without the use of ‘+’ sign e.g. 1— .
к 3.7
12 3
Fractions which represent the same fractional number like and so on,
2	4 6
are called equivalent fractions.
You have already known that any fractional number multiplied by one has the
(	3 3 Л
same value as the original number e.g. lx— = - .
\	4 4,
As soon as we have multiplied the numerator and the denominator of a fraction
by the same (non-zero) number, the resulting fraction will be equivalent to the
original fraction. We simply produce another name for the fraction. Consider the
fraction - . When both the numerator and the denominator are both multiplied
2	2	1
by two, the result is — which has the same value as - .
4	2
63

Dividing the numerator and the denominator by the same non-zero number
we just reduce or simplify the fraction. A fraction in which the numerator and the
denominator have no factors in common other than 1 is called irreducible or in its
3	3
lowest or simplest terms. Consider the following fractions: — and - . The former
9	8
3
is not in the lowest terms because both 3 and 9 can be divided by 3. In contrast —
is in lowest terms - the only number that is a factor of both 3 and 8 is 1.
A decimal fraction is a special type of fraction written without a denominator
(which is 10 or a power of 10) but in which the number of figures on the right-hand
side of a dot, called the decimal point, indicates whether the denominator is 10
or a higher power of 10. All digits to the left of the decimal point represent whole
numbers, and all digits to the right of the decimal point represent fractional parts
of one.
Decimals like .111, .3535, .282828 are called repeating decimals and those,
which repeat zeros, - terminating decimals (e.g. 0.25000).
We have just studied different types of decimals. Its important to know how
decimals are read nowadays. Let’s take such numerals as 9.3 (nine point three),
21.65 (twenty-one point six five), 0.182 (nought point one eight two or zero point
one eight two).
Rational numerals can be named by decimal numerals. The arithmetic of
numbers in decimal form is in full agreement with the arithmetic of numbers in
fractional form.
Post-Reading Activity
Ex. 5. Answer the following questions.
1.	What’s a fraction in mathematics? 2. In what form is a common fraction
generally written? 3. What does the denominator (the numerator) represent?
4.	How can a rational number be defined? 5. What types of fractions do you knowr
in algebra? 6. What is an equivalent (mixed) fraction? 7. Is it possible to change a
mixed number to an improper fraction? 8. What happens to a common fraction
when wre multiply it by one? 9. In what way can wre reduce a fraction? 10. What is
a decimal fraction?
Ex. 6. Find the English equivalents for the Russian w ords and wrord combinations:
1) часть целого; 2) числитель дроби; 3) знаменатель дроби; 4) должны быть
превращены; 5) в десятичной форме; 6) наименьший общий знаменатель;
7) полученная дробь; 8) неправильная дробь; 9) сократить дробь; 10) пра-
64

вильная дробь; 11) члены дроби; 12) представлено дробью; 13) искомая
дробь; 14) иметь что-либо общее.
a) the fraction sought for; b) a proper fraction; c) to reduce a fraction; d) a part
of the whole; e) to have in common; f) in decimal form; g) the resulting fraction;
h) the terms of a fraction; i) the denominator of a fraction;)) an improper fraction;
k) the least common denominator (LCD); 1) the numerator of a fraction; m) must
be changed to; n) is represented by the fraction.
Ex. 7. Give the proper English equivalents for the Russian expressions:
the greatest common factor; to invert; the quotient; performed
operations; the numerator and denominator; decimal numerals;
proper fractions; the minuend, subtrahend and remainder;
to reduce a fraction; mixed numbers; improper fractions.
1.	A rational number is частное (divisor is not zero) of two integers. 2. Fractions
which represent values less than one are called правильными дробями. 3. If we
divide both числитель и знаменатель by the same number, not zero, or one we
leave the fractional number unchanged. 4. To bring a fractional number to lower
terms means сократить дробь. 5. Наибольший общий делитель is the largest
possible integer by which both numbers in the fraction are divisible. 6. We can
express rational numbers as десятичными числами. 7. In order to divide one
fraction by another it is necessary перевернуть the divisor fraction and then
multiply. 8. We have just выполнили операции on complex and rational expressions
as well. 9. When we have to subtract decimal fractions we write them so that the
decimal points of уменьшаемого, вычитаемого и остатка are below each other.
10. To multiply смешанные числа we reduce them to неправильные дроби.
Ex. 8. Mark the following as true or false.
1. Every fraction has a numerator and a denominator. 2. A rational number cant
be another name for a fraction. 3. In the proper fraction the denominator is less
than the numerator. 4. In the improper fraction the denominator is greater than
the numerator. 5. A mixed fraction contains an integer and a proper fraction.
6. We change a fraction if we multiply it by 1. 7. If we multiply the numerator and
the denominator by the same whole number, we produce another name for the
fractional number. 8. Principles of arithmetic are valid in the case of mathematics.
9. Its impossible to express rational numbers as decimal numerals. 10. The digits
to the right of the decimal point represent whole numbers. 11. The name given to a
decimal like 0.1313 is terminating. 12. We obtain a tenth by dividing 1 by 10.
65

Ex. 9. Fill in a suitable verb in the Present Perfect Tense:
to give, to become, to show, to draw, to learn, to find,
to be, to make, to take, to rewrite.
1.	They ... an attempt to overcome difficulties of this complicated procedure.
2.	Hie science adviser ... the whole situation under control.
3.	They ... not all the examination material yet.
4.	I... this article on Maths very helpful.
5.	In the given example he ... the validity of these principles.
6.	Hie post-graduate ... the previous chapter of his report.
7.	He ... a first-year student of the Mechanics and Mathematics faculty this year.
8.	My friend ... already the diagram which has certain advantages.
9.	Hie teacher ... an example of an algorithm.
10.... you ... able to name these rational numbers by decimal numerals?
Ex. 10. Choose the correct tense form.
1.	(Have you ever heard/Did you ever hear) about continued fractions?
2.	We (have started/started) learning fractions when we were at school.
3.	By the end of the lecture the students (had learnt/ has learnt) that in Higher
Mathematics a fraction is viewed as an element of a field-of fractions.
4.	I (have looked up/looked up) in a dictionary just now the meaning of the word
“a solidus”(a slanting line).
5.	She just (has written/wrote) this mixed number in another way; i.e. as an
improper fraction.
6.	She (will have studied/will study) carefully the chapter about complex fractions
by the end of the week.
7.	When you come back I (will have divided/have divided) these fractions using
the rule “invert and multiply”.
8.	I need to subtract two fractions but I (haven’t got/didn’t get) the least common
denominator yet.
9.	Before he solved the problem he (had had to/have had to) convert the repeating
decimals into fractions.
10.	You (have remembered/remembered) since school that a fraction is in its
lowest terms when its numerator and denominator have no common factor
other than one.
Ex. 11. Ask special questions.
1. By 5 o’clock the experiment will have been over, (by what time) 2. By the age of
41 Sophia Kovalevskaya had won recognition of mathematicians all over the world.
66

(whose) 3. Zero concept has got many new applications in modern science and
engineering recently, (what kind of) 4. Russian scientists had introduced thousands
of new concepts by the end of the 20-th century, (who) 5. Hie students haven’t
attended the course in the history of mathematics this month, (when) 6. They
had obtained some equations by using mathematical terms before the teacher
collected their papers, (how) 7. For a long time the major task of mathematicians
has been to express ancient algebra in modern symbols, (how long) 8. By the end
of the year the historian will have written four new books about the work of genii
of mathematics, (how many) 9. Mans technical progress has developed greatly,
(what) 10. Most people have come across the term “industrial robots”, (what, who)
11.	Some European mathematicians have tried to prove Fermat’s last theorem ever
since it became known, (since when) 12. Prof. Smirnov’s postgraduates will have
finished the experiment before he comes, (whose)
Ex. 12. Translate into English.
1. Дробь представляет часть целого. Она показывает, что разделили что-то
на несколько равных частей. 2. В дроби число, стоящее над чертой, назы¬
вается числителем, число, стоящее под чертой, называется знаменателем.
Числитель и знаменатель - члены дроби. 3. Мы уже узнали, что дробь, у
которой числитель меньше знаменателя, называется правильной дробью.
Правильная дробь - меньше единицы. 4. Неправильной дробью называется
дробь, числитель которой равен знаменателю или больше него. Таким обра¬
зом, неправильная дробь равна или больше единицы. 5. Числа, состоящие
из целого числа и дроби, называются смешанными. 6. Если вам нужно со¬
кратить дробь, то вы должны разделить числитель и знаменатель этой дро¬
би на одно и то же число. Это число называется общим делителем. 7. Пре¬
подаватель только что объяснил нам, что дроби, знаменателями которых
являются числа, выраженные единицей с последующими нулями (одним
или несколькими), называются десятичными числами. 8. Десятичные чис¬
ла, в которых одна или несколько цифр повторяются многократно, назы¬
ваются периодическими, а те, в которых повторяются нули, непериодиче¬
скими десятичными числами (или конечными десятичными числами). 9. Ты
уже перенес десятичную точку на один знак вправо, чтобы увеличить число
в 10 раз? 10. К концу семестра мы закончим изучение дробей и десятичных
чисел. 11. Он еще не сложил эти дроби, так как не знает, как привести их к
общему знаменателю.
67

UNIT 6
Степени сравнения.
Времена группы Perfect Continuous
Degrees of Comparison
Короткие слова:
-er, -est
hot - hotter - the hottest
cold - colder - the coldest
large - larger - the largest
Слова, оканчивающиеся на -у:
-ier, -iest
Но:
dirty - dirtier - the dirtiest
happy - happier - the happiest
gay - gayer - the gayest
Длинные слова:
more, the most
famous - more famous - the most famous
general - more general - the most general
Некоторые прилагательные имеют
два способа образования степеней
сравнения:
clever, stupid, gentle, friendly, cruel,
common, pleasant, quiet, narrow,
shallow
. ___—■ quieter - the quietest
quiet CL	.	?
л	- more quiet -the most quiet
^-shallower - the shallowest
s a ow •'^more shallow - the most shallow
real
right
wrong
real - more real - the most real
right - more right - the most right
wrong - more wrong - the most wrong
Усиление значения
a bit/а little + comparative (немного)
a bit longer - немного длиннее
a little cheaper - немного дешевле
Усиление значения much/far/
a lot + comparative (намного, гораздо)
Your car is much older than mine.
Твой автомобиль значительно старее
моего.
This way is far longer than that one.
Эта дорога гораздо длиннее, чем та.
Ex. 1. Analyse these sentences and compare the adjectives given there. Translate
them into Russian.
1.	He has a difficult test. I have a more difficult test. Her test is the most difficult of
all. 2. Your problem is easy. His problem is easier than yours. That students problem
is the easiest. 3. This definition is too simple. There is a more simple (simpler)
definition. That is the most simple (simplest) definition of all. 4. My explanation of
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this task is wrong. My friends explanation of this task is more wrong than mine.
I think that his explanation is the most wrong explanation I have ever heard. 5. Jill
is 25, Gary is 24 lA. Jill is a bit older than Gary. 6. France isn’t very big. Canada is
much bigger than France. 7. That film is interesting. I consider it is by far the most
interesting film I have ever seen. 8. This method is complicated. The new one is
much more complicated. It is the most complicated method that I remember.
Особые случаи образования степеней сравнения
good/well - хороший/хорошо
better- лучший/лучше
best - самый лучший
bad/badly - плохой/плохо
worse - худший/хуже
worst - самый худший
much/many - много
more - больше/более
most - наибольшее
количество
little - мало
less - меньше/менее
least - наименьшее
количество
far - далекий/далеко
farther - дальше
further - дальше,
дополнительный,
добавочный
farthest - самый дальний
furthest - самый дальний,
дальше всего
late - поздний/поздно
later - более поздний/
позже
latter - последний
(из упомянутых)
the latest (there may be
more to come) - самый
поздний, но не последний
the last (final, before this) -
последний, окончательный
old - старый ~~	"
older - старше
elder - старше в семье
the oldest - самый
старший (о возрасте)
the eldest - старший в
семье
near - близкий/близко
nearer - ближе
the nearest (о расстоянии)
the next (порядок)
Ex. 2. Analyse these sentences and compare the adjectives given there. Translate
them into Russian.
1.	Ulis example is not quite good. You ought to find a better one. I do not think it
is the best example that you can give. 2. The result of their exam is bad. It is much
worse than we expected. In fact, it is the worst in many years. 3. I have little free
time. Mary has less free time than me. Jane has the least free time. 4. My house is
far from the University. The hostel is farther from the University. I saw them in
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the farthest corner of the park. Please, send the books back without further delay.
5.	Peter has 5 notebooks. Mary has more notebooks. She has 10. John has the most
books. He has 15. 6. He came home later than usual. Have you heard the latest
news? The last train leaves in half an hour. 7. My elder brother is 5 years older than
me. My grandmother is the oldest in our family. Her eldest son is my father. 8. The
nearest cafe is in a five - minute walk from here. The next news bulletin comes in
10 minutes.
Сравнительные конструкции
Сравнительная степень +
than чем
New York is larger than Washington.
Нью-Йорк больше Вашингтона.
My friend is three years older than me/than I am.
Мой друг на три года старше меня.
the most наибольший, больше
всего
the least наименьший,
меньше всего
This is the most exciting place I have ever been to.
Это самое замечательное место, где я когда-либо
побывал.
Carol is the least experienced person in our team.
Кэрол является наименее опытным специалистом
в нашей команде.
As...as
такой же... , как
также... , как
Не is as handsome as his brother.
Он такой же красивый, как и его брат.
Their car is as expensive as ours.
Их автомобиль такой же дорогой, как и наш.
Not as ... as
не такой ... , как
Not so ... as
не так... , как
She is not as slim as her sister.
Она не такая стройная, как ее сестра.
Mrs. Green is not so friendly as she looks.
Миссис Грин не такая дружелюбная, как выглядит.
Twice as... as
Three times as ... as
Half as... as
в два, три раза,
наполовину..., чем
Petrol is twice as expensive as it was several years ago.
Сейчас бензин в два раза дороже, чем он был
несколько лет назад.
This room is half as small as the next one.
Эта комната наполовину меньше, чем соседняя.
The same... as
такой же..., как
Не has the same habits as his father.
У него такие же привычки, как у отца.
The more ..., the better
чем..., тем
The more we practice, the better results we get.
Чем больше мы занимаемся, тем лучше у нас
результаты.
The easier the text, the quicker you will be able to
translate it.
Чем легче текст, тем быстрее вы сможете его
перевести.
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Ex. 3. Follow the model and make the sentences in which comparison is
expressed.
Model 1: This problem is ... (difficult) ... the first problem.
This problem is as difficult as the first problem.
1.	Ulis text is ... (interesting) ... that one. 2. This sentence is ... (long) ... the
second one. 3. This definition is ... (exact)... the definition given in the text-book.
4. His answer is ... (good)... that girl’s answer. 5. English classes are... (important)
... lectures on mathematics.
Model 2: This theorem is... (famous) ... people may think.
This theorem is not so famous as people may think.
1.	This system is ... (reliable) ... the one we studied. 2. That dictation is ... (easy)
... we’ll write next lesson. 3. Hie proof is ... (valid) ... he supposed at first. 4. This
story is ... (boring) ... he thought about it before. 5. This solution is ... (good) ...
she suggested at the conference.
Model 3: (Big) the plan, (long) they will work.
The bigger the plan, the longer they will work.
1.	(Soon) the problem is solved, (good). 2. (Long) the student refuses to learn the
words, (bad) for him. 3. (Hard) they work, (good) is for their salary. 4. (Much) she
practices, (healthy) she becomes. 5. (Convincing) the lecturer speaks, (attentive)
the audience listens.
Ex. 4. Open the parentheses and use the correct form.
1.	Our sitting-room is (light) room in our flat. 2. There are (many) students in our
group than in yours. 3. Hie railway station is (far) from here than the airport. 4. My
suit is much (expensive) than yours. 5. Betty is a little (short) than her brother.
6.	Who was (late) person to leave the building yesterday? 7. Her (old) brother is
a well-known Belarusian mathematician. 8. Who is (old) in your group? 9. Her
translation is (bad) than his. 10. You will get (far) instructions in a few days. 11. His
equation is (difficult) than hers. 12. Silver is (heavy) than copper. 13. Hiere were
(few) problems than we expected. 14. It is (successful) experiment we have ever
made. 15. (Long) the days, (short) are the nights. 16. He has to work a lot (hard) in
his new job than he used to (early). 17. Hie (carefully) you do it, the (well) it will
be. 18. The (much) I get to know you, the (little) I understand you.
Ex. 5. Give the proper English equivalents for the Russian expressions:
the best, greater, as difficult as, less, longer ... that one, as interesting as,
not so important as, the youngest, better, not so famous as,
the sooner ... the better, the older ... the happier.
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1.	In the improper fraction the denominator is меньше than the numerator. 2. If
you want to say that six is больше than 5, we write: 6 > 5. 3. Чем скорее you answer
the text, тем лучше. 4. This film is такой же интересный, как the one we have
already seen. 5. This sentence is длиннее than то предложение. 6. She reads
English лучше than I do. 7. Чем старше I get, тем счастливее I am. 8. Ann is
самая молодая in our family. 9. The first equation is такое же трудное, как the
second one. 10. These problems are не такие важные, как those we have dealt
with. 11. Our laboratory is самая лучшая in the University. 12. This scientist is не
такой знаменитый, как Einstein.
Ex. 6. Translate into English.
1.	Эта статья - самая трудная из всех, которые мы когда-либо переводили.
2.	Сегодня намного холоднее, чем вчера. 3. Его книга гораздо интереснее ва¬
шей. 4. Эта аудитория меньше той. 5. Мое пальто такое же теплое, как его.
6. «Ваша сестра старше вас?» - «Нет, она немного моложе меня». 7. Этот сту¬
дент самый младший в своей группе. 8. Сегодня ветер не такой сильный, как
вчера. 9. Моя ручка гораздо хуже вашей. 10. Ее перевод значительно лучше
того, который она сделала вчера. 11. Какой язык труднее: немецкий или ан¬
глийский? 12. В пятой группе больше студентов, чем во второй. 13. У меня
меньше книг, чем у Кати. 14. Мария гораздо красивее своей сестры. 15. Я по¬
лучил дополнительную информацию по этому вопросу. 16. У мамы меньше
времени, чем у отца. 17. Этот текст намного больше, чем предыдущий. 18. Он
не так молод, как мой брат.
The Perfect Continuous Tenses
Present
Past
Future
I
you
we
they
have
been
working
I
you
he
she
it
we
they
had been
working
I
we
shall
will
have been
working
he
she
it
has
he/she
it/you
they
will
for a long time, for 5 years,
since 2 o’clock, all morning/
day/week
for 2 hours when he
came,
since 2 o’clock when
you came
for two hours by the time he comes,
next year for five years already,
for forty minutes when you ring us up
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The Present Perfect Continuous Tense употребляется:
1.	Для выражения действия, которое началось в прошлом, длилось опре¬
деленный период времени и еще не завершилось к моменту речи.
We have been applying these methods Мы применяем эти методы
of investigation for a long time.
исследования уже в течение долгого
времени.
Как долго они обсуждают новый
метод?
For how long have they been
considering the new method?
2.	Для выражения действия, которое началось в прошлом, длилось неко¬
торое время и закончилось непосредственно перед моментом речи.
I feel tired as I have been translating
the article for two hours.
Although the sun is shining, it is still
cold as it has been raining hard.
3.	Для выражения гнева, недовольства, раздражения.
Who has been using my cup?	Кто пользовался моей чашкой?
I suppose you have been telling lies Я полагаю, ты снова говоришь не-
again.	правду.
Я чувствую себя усталым, так как
два часа я переводил статью.
Хотя светит солнце, все еще холод¬
но, так как шел сильный дождь.
Они готовились к экзамену с 10 ча¬
сов, когда вы пришли.
The Past Perfect Continuous Tense употребляется:
1.	Для выражения действия, которое началось в прошлом, длилось опре¬
деленное время до другого момента либо действия в прошлом и еще не за¬
вершилось.
They had been revising for the
examination since 10 o’clock when you
came.
We had been trying to come to a
certain agreement for two hours when
the director came.
2.	Для выражения действия, которое началось в прошлом, длилось опре¬
деленный период времени и завершилось непосредственно перед наступле¬
нием другого прошедшего действия.
Не felt very tired when he came home
as he had been playing football.
Мы пытались достичь определен¬
ного соглашения уже два часа, когда
пришел директор.
They were wet because they had been
walking in the rain.
Он чувствовал себя очень усталым,
когда пришел домой, так как играл в
футбол.
Они промокли, так как шли под
дождем.
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Hie Future Perfect Continuous Tense употребляется:
для выражения длительного действия, которое будет протекать некото¬
рый период времени до другого момента в будущем и все еще будет совер¬
шаться в этот момент. Эта форма встречается очень редко.
By the end of the year he will have been К концу года он будет работать
working at his book for ten months.	над своей книгой уже десять месяцев.
Ех. 7. Analyse the following sentences and translate them. Compare the
predicates in these pairs of sentences.
a)	affirmative
1.	I am studying English now.
2.	They were discussing the definition
in class yesterday.
3.	He will be speaking at the
conference next month.
b)	negative
1.	He is not doing any experiments
right now.
2.	She wasn't watching TV at 8 p.m.
yesterday.
3.	Next year they will not be living
here as they are moving to a new
house.
c)	interrogative
1.	Are they working in the garden
now?
2.	Was he translating the text in
class yesterday?
3.	Will she be doing her research
next week?
1.	I have been studying it since September.
2.	They had been discussing the definition
for some time, when the teacher came.
3.	He will have been speaking at the
conference for half an hour, when his
scientific adviser comes.
1.	He has not been doing any experiments
since last year.
2.	She had not been watching TV for some
time when her mother came.
3.	Next year they will not have been living
here for 5 years but for 6 years already.
1.	Have they been working for a long time?
2.	Had he been translating this text for an
hour when his mother came?
3.	Will she have been doing her research
for five years by the end of this year?
Note the difference in translation between the Present Perfect Continuous and
the Present Perfect tenses.
1. - You look hot.	1. He’s run all the distance to the finish
-	I’ve been running all the way. fairly well.
-	Я бежала всю дорогу.	Он пробежал всю дистанцию до
финиша довольно хорошо.
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2.	I’ve been learning irregular verbs all
afternoon.
Я учил неправильные глаголы весь
день.
3.	Sorry about the mess. I’ve been
painting the house.
Извините за беспорядок. Я крашу
дом.
2.	I’ve learnt irregular verbs.
(= I know them.)
Я выучил неправильные глаголы
3.	I’ve painted two rooms since
lunchtime.
Я покрасил две комнаты с обеда.
Ех. 8. Choose the correct variant.
1.	For how long (have they discussed, have they been discussing) the situation?
2.	Why (have you repeated, have you been repeating) these English words over
and over again? 3. Hie students (have taken, have been taking) the examination for
more than 5 hours. 4. They (were discussing, have been discussing) the situation
for three hours. 5. She (has been answering, has answered) the lesson already.
6. She (has worn, has been wearing) glasses for two years. 7. Peters English is
getting much better. He (is practising, has practised, has been practising) a lot
this year. 8. I (have written, am writing, have been writing) my course paper for
three months, but I (am not finishing, haven’t been finishing, haven’t finished) it
yet. 9. ... you (are defending, have defended, have been defending) your course
paper? - No, I (haven’t done, am not doing, haven’t been doing) it yet. 10. Tom
(is having, has been having, has had) a toothache for nearly a week. He (is going,
has been going, has gone) to the doctor today and I’m waiting for him. 11. What
you (are doing, have been doing, have done) with my cassette-recorder? I can’t
find it anywhere. 12. You look tired! - Yes, I (am dancing, have danced, have been
dancing) and I (haven’t danced, am not dancing, haven’t been dancing) for years, so
I’m not used to it. 13. Everybody (is looking, has looked, has been looking) forward
to this holiday for months. 14. Recently this scientific theory (is being proved, has
been proved, has been proving) to be false.
Ex. 9. Match the beginnings and the ends of the sentences.
1.	Tom had been working for two hours
2.	You look tired.
3.	Aren’t you hungry?
4.	By the lsl of August, 2018
5.	He was out of breath
6.	Why are my books all over the floor?
7.	After I had been walking for an hour
8.	Her hair is white
a)	as if he had been running for several
hours without a rest.
b)	she will have been working at the
department for 35 years.
c)	I decided to have a cup of tea.
d)	because she has been painting the
ceiling.
e)	I have been writing my course paper
for more than a month.
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9.	Why is his coat wet?
10.	She looked very dirty
f)	because she had been cleaning the flat
the whole day.
g)	No, I’ve been eating all day.
h)	He has been walking in the rain.
i)	when his brother came
j)	Your little sister has been playing with
them.
Ex. 10. Translate into English.
1. Они учат эти правила больше года. 2. Как долго этот студент переводит эту
статью? 3. Весь день идет снег. 4. Она преподаватель английского языка. Она
преподает с тех пор, как закончила лингвистический университет. 5. Ты вы¬
глядишь усталой. - Я учила алгебру весь день. 6. Я вымыл свою машину. Раз¬
ве она не выглядит чудесно? 7. Сейчас она учит испанский язык, но она еще
не очень хорошо говорит на нем. 8. Он уже два часа наигрывает эту мелодию
на фортепияно. 9. Все студенты пишут этот тест уже 20 минут, и только один
студент написал его. 10. На этой неделе я послала несколько писем своим
друзьям. 11. Как долго вы будете писать контрольную работу перед тем, как
сдадите ее преподавателю? 12. Мой друг ждет вас уже с двух часов. Почему
вы не пришли вовремя?
Pre-Reading Activity
Guess the meaning of the following words.
Generalization (n) [z<szs:sla:4zs:' Yr.], arithmetic (n) [s'riGmetik], procedure
(n) [prs"si z<s], symbol (n) [' szmb si], formula (n) [fo characteristic (n)
[zksr:kts4risrik], coefficient (n) [zks”:4f:?sr.:], zero (n) [4z:=iz'z].
Read and learn the basic vocabulary terms:
compute (v) [ks~/p1i:]
deal (dealt) with (v) [di •]
apply (v) [s'pla:]
instead of (adv) [:z 4
particular (adj) [ps'iikrils]
concerning (prep) [ksr.'ss
replace (v) [iz'plszs]
hold (held) for (v) [hc~ld]
to be true pr" ]
вычислять
иметь дело с, рассматривать
использовать, применять
вместо
определенный
относительно, касательно
заменять, замещать
годится для
быть верным, справедливым,
удовлетворять
76

likewise (adv) [la:kwa:z]
raising to a power [reiz". ts s
term (n) [rs ~ ]
in terms of
multinomial (n) [тлк:'~с™гз1]
binomial (n)
trinomial (n) ["а.:' 2
monomial (n) [22c'2ccrrjsl]
подобно, так же, таким же образом
возведение в степень
член
через, посредством
многочлен, полином
двучлен, бином
трехчлен
одночлен
Memorise the following word combinations:
let the number 20 be replaced - давайте заменим число 20
then the statement is true - тогда утверждение справедливо
no matter what - независимо от того, какие
for convenience - для удобства
serve to distinguish - служат для того, чтобы различить
both phis and minus - как плюс, так и минус
is to be treated as - следует рассматривать как
Reading Activity
THE NATURE OF ALGEBRA
Algebra is a generalization of arithmetic. Each statement of arithmetic has
been dealing with particular numbers for years: the statement (20 + 4)2 = 202 + 2 x
x 20 x 4 + 42 = 576 explains how the square of the sum of the two numbers, 20
and 4, may be computed. It can be shown that the same procedure applies if the
numbers 20 and 4 are replaced by any two other numbers. In order to state the
general rule, we write symbols, ordinary letters, instead of particular numbers. Let
the number 20 be replaced by the symbol a, which may denote any number, and
the number 4 by the symbol b. Then the statement is true that the square of the sum
of any two numbers a and b can be computed by the rule (a + b)2 = a2 + 2a x b + b2.
Ulis is a general rule which remains true no matter what particular numbers
may replace the symbols a and b. A rule of this kind is often called a formula.
Algebra is the system of rules concerning the operations with numbers. These
rules can be most easily stated as formulas in terms of letters, like the rule given
above for squaring the sum of two numbers.
Hie outstanding characteristic of algebra is the use of letters to represent
numbers. Since the letters used represent numbers, all the laws of arithmetic hold
for operations with letters.
77

In the same way, all the signs which have been introduced to denote relations
between numbers and the operations with them are likewise used with letters.
For convenience the operation of multiplication is generally denoted by a dot
as well as by placing the letters adjacent to each other. For example, a x b is written
simply as ab.
The operations of addition, subtraction, multiplication, division, raising to
a power and extracting roots are called algebraic expressions.
Algebraic expressions may be given a simpler form by combining similar
terms. Two terms are called similar if they differ only in their numerical factor
(called a coefficient).
Algebraicexpressionsconsistingofmorethan one term are calledmultinomials.
In particular, an expression of two terms is a binomial, an expression of three
terms is a trinomial. In finding the product of multinomials we make use of the
distributive law.
In algebra the signs plus (+) and minus (-) have their ordinary meaning
indicating addition and subtraction and also serve to distinguish between
opposite kinds of numbers, positive (+) and negative (-). In such an operation as
+ 10 - 10 - 0, the minus sign means that the minus 10 is combined with the phis
10 to give a zero result or that 10 is subtracted from 10 to give a zero remainder.
The so-called “double sign” (±), which is read “plus-or-minus”, is sometimes
used. It means that the number or symbol which it precedes may be “either plus or
minus” or “both plus and minus”.
As in arithmetic, the equality sign (=) means “equals” or “is equal to”.
The multiplication sign (•) has the same meaning as in arithmetic. In many
cases, however, it is omitted.
The division sign (+) has the same meaning as in arithmetic. It is frequently
replaced by the fraction line; thus - means the same as 6 + 3 and in both cases the
3
result or quotient is 2. The two dots above and below the line in the division sign
(+) indicate the position of the numerator and the denominator in a fraction, or
the dividend and the divisor in division.
Parentheses ( ), brackets [ ], braces { }, and other enclosing signs are used
to indicate that everything between the two signs is to be treated as a single
quantity.
Another sign which is sometimes useful is the sign of strict inequality which
means “greater than” or “less than” Hie sign (>) means “greater than” and the sign
(<) means “less than”. Thus, a> b means that “a is greater than b” and 3 < 5 means
“3 is less than 5”.
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Post-Reading Activity
Ex. 11. Answer the following questions.
1. What is the relationship between arithmetic and algebra? 2. In what arithmetic
operations do we use numbers? 3. What do we use in algebra to represent numbers?
4.	What examples of the close relationship between arithmetic and algebra can
you give? 5. What is algebra? 6. What is the outstanding characteristic of algebra?
7.	Name algebraic expressions you know. 8. When are two terms called similar?
9.	What signs of strict inequalities are used in algebra? 10. What is the meaning of
the multiplication sign, the equality sign and the division sign? 11. What does the
expression (a + b) mean? 12. What enclosing signs do you know and what do they
indicate?
Ex. 12. Find the English equivalents for the following Russian words and word
combinations:
1) утверждение; 2) иметь дело; 3) через, посредством; 4) вычислять; 5) подоб¬
но, таким же образом; 6) алгебраические выражения; 7) многочлен; 8) для
удобства; 9) член; 10) трехчлен; 11) возведение в степень; 12) тогда утверж¬
дение справедливо; 13) представлять.
a) then the statement is true; b) a trinomial; c) raising to a power; d) to deal with;
e) for convenience; f) a term; g) to compute; h) likewise; i) in terms of; j) algebraic
expressions; k) a statement; 1) to represent; m) a polynomial.
Ex. 13. Give the proper English equivalents for the Russian expressions:
computed, instead of, simpler, ordinary letters, replace, hold,
a generalization, relations, concerning, multinomials.
1. Algebra is обобщение of arithmetic. 2. In order to state the general rule we write
symbols, обычные буквы, instead of particular numbers. 3. These signs have been
introduced to denote отношения between numbers. 4. Algebra is the system of
rules относительно the operations with numbers. 5. Particular numbers may
замещать the symbols a and b. 6. All the laws of arithmetic верны for operations
with letters. 7. We write symbols вместо particular numbers. 8. The square of the
sum of any two numbers c and d can be вычислен by the rule (c + d)2 = c2 + 2cd + d2.
9. Algebraic expressions may be given а более простая form by combining similar
terms. 10. Algebraic expressions consisting of more than one term are called
многочленами.
79

Ex. 14. Mark the following as true or false.
1. Algebra is a generalization of geometry. 2. In order to state the general rule,
we write numbers instead of particular letters. 3. Algebra is the system of rules
concerning the operations with numbers. 4. Since the letters used represent
numbers, all the laws of arithmetic fail to hold in operations with letters. 5. The
operations of addition, subtraction, multiplication, division, raising to a power and
extracting roots are called algebraic expressions. 6. An expression of two terms is
a trinomial. 7. As in arithmetic, the equality sign means “not equal to”. 8. In finding
the product of multinomials we make use of commutative law. 9. These rules
cannot be easily stated as formulas in terms of letters, like the rule given above for
squaring the product of two numbers. 10. The outstanding characteristic of algebra
is the use of numbers to represent letters.
Ex. 15. Ask special questions.
1. A polynomial is an algebraic expression composed of one or more terms (what,
how many) 2. Algebraic expressions are divided into two groups, (how many) 3. An
expression бх5 + 4X3 + 8 is of the fifth degree in x. (what) 4. If a polynomial contains
but one term, it is called a monomial, (when) 5. The fundamental operations with
polynomials are addition, subtraction, multiplication and division, (what) 6. If the
remainder is zero, the division is exact, (when) 7. The so-called “double sign” (±)
is sometimes used, (what) 8. The equality sign (=) means “equals” or “is equal to”,
(what) 9. In the operation + 10 - 10 = 0, the minus sign means that 10 is subtracted
from 10 to give a zero remainder, (what) 10. We use the signs phis (+) and minus
(-) to indicate addition and subtraction, (who, what for) 11. There are three
requirements for an equation, (how many)
Ex. 16. Translate into Russian.
Monomials and Polynomials
1. Algebraic expressions are divided into two groups according to the last operation
indicated. 2. A monomial is an algebraic expression whose last operation is neither
addition not subtraction. 3. So, a monomial is either a separate number represented
by a letter or by a figure, for example -x, +9, or a product, for example ab, (x + y),
or a quotient, for example -—or a power x3, but must never be either a sum
c
or a difference. 4. An algebraic expression which consists of several monomials
connected by the phis and minus signs, is known as a polynomial. 5. Such is, for
instance, the expression x+yz + c. 6. Terms of a polynomial are separate expressions
which form the polynomial by the aid of the + and - signs. 7. Usually the terms of a
polynomial are taken with the signs preceding them; for instance, we say: term
term +b* and so on. 8. When there is no sign before the first term it is xy or + xy.
80

Ex. 17. Translate into English.
1.	Алгебра - это система правил, касающихся действий с числами. 2. В алге¬
бре числа обозначаются буквами, а не цифрами. 3. Поскольку буквы обозна¬
чают числа, все законы арифметики годны для действий с буквами. 4. Зна¬
ки, которые обозначают действия с цифрами, также употребляются для
букв. 5. Операции сложения, вычитания, умножения, деления, возведения
в степень и извлечения корней называются алгебраическими операциями.
6.	В алгебре мы применяем следующие знаки: плюс, минус, знак равенства,
знак умножения, знак деления, скобки круглые, квадратные и фигурные,
знак «больше, чем», знак «меньше, чем» и другие. 7. Алгебраическому выра¬
жению можно придать более простую форму путем приведения подобных
членов. 8. Алгебраическая сумма нескольких одночленов называется мно¬
гочленом. 9. Двучлен - это алгебраическое выражение, состоящее из двух
членов, трехчлен - алгебраическое выражение, состоящее из трех членов.
UNIT 7
Времена группы Indefinite
в страдательном залоге
В русском и английском языках существуют два залога: действительный
и страдательный. В действительном залоге подлежащее выполняет дей¬
ствие, выраженное сказуемым. В страдательном залоге действие направ¬
лено на подлежащее, а носителем действия является дополнение. Соответ¬
ственно, когда говорящего в первую очередь интересует объект какого-то
действия, а лицо или предмет, совершающий действие, не известен или не
важен, тогда употребляется страдательный залог. Таким образом, поставив
объект действия на первое место в предложении (вместо подлежащего), мы
акцентируем на нем больше внимания. В тех случаях, когда также необхо¬
димо акцентировать внимание и на носителе действия, употребляется пред¬
лог by.
Времена страдательного залога образуются при помощи вспомогатель¬
ного глагола to be (в соответствующем времени, лице и числе) и Participle II,
которое остается неизменным, и употребляются согласно тем же правилам,
что и соответствующие им формы действительного залога.
to be + Participle II
81

Present Indefinite Passive
Past Indefinite Passive
Future Indefinite Passive
I	am
he/she/it	is	asked
we/you/they are
I/he/she/it was (	. .
...	{ asked
we/you/they were (
I/we	shall/will be
he/she/it will be	< asked
you/they will be
She is often asked at
the lessons. Ее часто
спрашивают на уроках.
Computers are used
everywhere.
Компьютеры
используются везде.
He was asked at the lesson
yesterday. Его спросили на
уроке вчера.
A computer was used to
solve a difficult problem
last week. Компьютер был
использован для решения
трудной задачи на
прошлой неделе.
You will be asked at the
lesson tomorrow. Вас
спросят на уроке завтра.
This new computer will be
used in our laboratory next
week Этим новым компью¬
тером воспользуются в
нашей лаборатории на
следующей неделе.
Ex. 1. Read the sentences and compare the translation of predicates in them.
Present
I attend all lectures at the University.
We often use the Internet.
He applies a new method in his
research.
Scientists know all fundamental
discoveries.
All lectures at the University are
attended by me.
The Internet is often used by us.
A new method is applied in his research.
All fundamental discoveries are known
to our scientists.
сл
сЗ
(X
He finished the article yesterday.
We studied the rule at the lesson last
week.
I developed a new system of notation.
The article was finished by him
yesterday.
The rule was studied at the lesson last
week.
A new system of notation was developed
by me.
Future
We shall discuss a new program at the
lecture tomorrow.
He will answer all your questions after
the lecture.
The writer will publish his book next
year.
A new program will be discussed at the
lecture tomorrow.
All your questions will be answered after
the lecture.
The book will be published by the writer
next year.
Interrogative
The operations with symbols are
performed in algebra.
The first coefficient is represented by the
letter.
Some interesting facts were found in
that book
The report will be prepared in time.
Are the operations with symbols
performed in algebra?
Is the first coefficient represented by the
letter?
Were any interesting facts found in that
book?
Will the report be prepared in time?
82

Negative
This combination is used in the new
system.
Such members are easily multiplied.
I was told to solve another equation.
The order of the operations will be
discussed later.
This combination is not used in the new
system.
Such members are not easily multiplied.
I was not told to solve another equation.
The order of the operations will not be
discussed later.
Ex. 2. State the voice of the verb in the following sentences. Translate these
sentences.
1.	The students left the experiment unfinished. 2. The algebraic language is used to
express mathematical ideas. 3. The members of the equality are connected by the
equality sign. 4. The result will be checked immediately. 5. We shall study higher
mathematics next term. 6. This property was discussed in the previous chapter.
7.	All the facts are summarized in this statement. 8. Will the test be written on
Monday? 9. The student showed me his graduation paper a few days ago. 10. She
will be told about their recent investigations in the field of algebra. 11. They told
the foreign scientists about their studies in the theory of programming. 12. Their
calculations will not be used in his work.
Pre-Reading Activity
Guess the meaning of the following words.
Expression (n) [iks'prej'V'l, identical (adj) [a.:4dsr.tikVl], conditional (adj)
[ksr.4d:/'s^nl], accuracy (n) p ssk;“ssi], classify (v) pklssifa:], transformation
(n)	reduce (v)	d1" sJ, linear (adj) [4kr.:s].
Read and learn the basic vocabulary terms:
equation (n) [1'kwsi 'srJ
statement (n) [' steitmsr.:]
finite (adj) pfair.ait]
variable (adj) [4 vssnsdl]
identical (adj) [a:'derziksl]
briefly (adv) ['k:i fl:]
root (n) [:" t]
aid (n) [s:d]
illustrate (v) [':lss:re:t]
restriction (n) t”k 'sr.]
substitute (v) ['sAdstir^:]
satisfy (v) [4ss::s/a:]
уравнение
утверждение, формулировка
конечный
переменная
аналогичный
кратко
корень
помощь
иллюстрировать
ограничение
замещать, заменять
удовлетворять
83

linear (adj) [4kr.:s]
quadratic (adj) [kws'd:s::k]
cubic (adj) [4k1~ b:k]
integral (n) [':r.::grsl]
fractional (adj) ['::skj'snsl]
rational (adj) [4rs 'snl]
irrational (n) [1'гзе 'sr.l]
original (adj) [с' г:2312. V1]
extraneous (ad) [eki4t:e:r/ss]
линейный
квадратичный
кубический
интеграл, целое число
дробный
рациональный, целесообразный
иррациональное число
первоначальный
посторонний, чуждый
Memorise the following word combinations:
other than - кроме
in question - рассматриваемый
to check the accuracy - проверить точность
for convenience - для удобства
regardless of the form - независимо от формы
when applied to an equation - в применении к уравнению
is said to be equivalent with respect to - считается эквивалентным относи¬
тельно
can readily be recognized and discarded - можно легко распознать и отбросить
Reading Activity
EQUATIONS AND IDENTITIES
An equation is a statement of equality between two algebraic expressions. The
two expressions are called members or sides of the equation. If the two members of
an equation are finite and are exactly the same, or become the same, for every value
of the symbols or variables involved, the equation is called an identical equation or
an identity, for example
(x - 2)2 = x2 - 4x + 4,	(x + 3) (x - 2) = x2 + x - 6.
If the two members of an equation are equal for certain particular values of
the symbols involved, but not for all values, the equation is called a conditional
equation, or briefly, an equation. An equation in one unknown, say x, is a way of
describing one or more numbers by stating a condition the numbers must satisfy.
To solve an equation is to find values of the unknowns that make the two members
equal. Such values of the unknowns are called roots or solutions of the equation.
The following rules aid in finding the root:
1.	Hie roots of an equation remain unchanged if the same expression is added
to or subtracted from both sides of the equation.
84

2.	The roots of an equation remain the same if both sides of the equation are
multiplied or divided by the same expression other than zero and not involving the
letter whose value is in question.
Hie equation 2x = 4, where x is the unknown, is true for x - 2. To illustrate
the first of the above two rules, add 5x to both sides of the equation 2x = 4. We get
2x + 5x = 4 + 5x which, like equation 2x = 4, is true for only x = 2. To illustrate the
importance of the restriction in the second of the above two laws, multiply both
sides of the equation by x and get (2x) x = (4x) x which is true not only for x = 2,
but also for x = 0.
It is always a good plan to check the accuracy of one’s work by substituting the
result in the original equation to see whether the equation is true for this value.
These numbers or values of the unknown x actually satisfy the equation upon
substitution.
For convenience equations are classified in various ways; according to degree
they may be linear, quadratic, cubic, etc.; in form, integral or fractional, rational or
irrational. Regardless of the form the equation is in at first, the process of solving
will involve transformations which will finally put it in the form:
the unknown = one or more definite values.
Those transformations when applied to an equation will give a new or derived
equation. A derived equation is said to be equivalent with respect to an original
equation if it contains all the roots of that equation and no others. The following
operations will always lead to equivalent equations, i.e.:
1.	Adding to or subtracting the same finite quantity from both members.
2.	Multiplying or dividing both members by the same quantity provided this
quantity is not zero and does not contain the unknown.
If the equation is fractional it may be changed into an integral equation by
multiplying both sides by the Least Common Denominator. This process is called
clearing the equation of fractions. Hie integral equation will have all the roots of
the original fractional equation but sometimes will include additional roots. Hiese
extraneous roots will be values of the unknown for which the Least Common
Denominator is zero and they can readily be recognized and discarded.
An equation in which the variable is raised to the first power only is usually
called a linear, or first degree, equation.
To solve an equation containing fractions, first reduce each fraction to its
lowest terms. Hien multiply each side of the equation by the Least Common
Denominator of all the denominators. This process is called clearing of fractions.
A quadratic equation is one which can be reduced to the form ax1 + bx + c = 0
(a * 0) where a, b and c are known and x is unknown.
85

Post-Reading Activity
Ex. 3. Answer the following questions.
1. What is an equation? 2. What are the members or expressions on the either side
of the sign of equality called? 3. What must we do to solve the equation? 4. What
do we call the solution of the equation? 5. In what way are the equations classified?
6. How do we check the equation? 7. How are equations classified? 8. In what way
can one solve an equation containing fractions? 9. What operations will lead to
equivalent equations? 10. What is a linear equation?
Ex. 4. Find the Russian equivalents for the following English words and word
combinations:
1) may be true for; 2) in finding; 3) a linear equation; 4) the Least Common
Denominator; 5) for every value of the variables involved; 6) to be equal for certain
particular values; 7) the expression in question; 8) to satisfy the equation upon
substitution; 9) regardless of the form; 10)by substituting.
а) независимо от формы; b) линейное уравнение; с) равный для некоторых
определенных величин; d) рассматриваемое выражение; е) путем подстанов¬
ки; f) наименьший общий знаменатель; g) для каждого значения включенной
переменной; 11) при нахождении; i) может быть верным для;)) удовлетворять
уравнению при подстановке.
Ех. 5. Give the proper English equivalents for the Russian expressions:
is checked, clearing, to satisfy, to solve, the unknown quantity,
have been transposed, the Least Common Denominator,
substituted into the expression, the value, roots.
1. No rule can be given чтобы решить the given problem. 2. We must write
out a definite description of what the letter selected for неизвестная величина
represents. 3. There is an indefinite number of pairs that can be подставлены в это
выражение. 4. Check the величину obtained in the solution of the equation. 5. Hie
derived equation may have fewer number of корней than the original one. 6. All
the terms containing the unknowns были перенесены to the left member. 7. Each
member of the equation is multiplied by the наименьший общий знаменатель.
8.	Hie result проверяется by the given formula. 9. The process of multiplying each
member by the Least Common Denominator is called устранение the equation of
any fractions. 10. Hie statement of condition may be such that no number can be
found to удовлетворить it.
86

Ex. 6. Make the following sentences interrogative and negative.
1. This relation is expressed symbolically. 2. Such numbers are easily multiplied.
3.	These problems are discussed at the seminars. 4. Various examples were given
at the last lesson. 5. I was told to solve another equation. 6. All the facts were
summarized in that expression. 7. The line will be divided into several parts. 8. Hie
conference will be held next week. 9. This exercise was done in the classroom
yesterday. 10. The common solution will be examined tomorrow.
Ex. 7. Ask special questions.
1. All the students will be examined next week, (when) 2. This equation is called
linear, (what) 3. The Latin alphabet is used in algebra, (where) 4. They were told
about the scientific conference, (who) 5. Some new rules were given at the last
lesson, (what, when) 6. The necessary equation will be written on the blackboard,
(what) 7. This algebraic expression was discussed in the previous chapter, (where)
8. Terms are usually written with the signs before them, (how) 9. The concept of
an equation was explained yesterday, (when) 10. The values of the unknowns are
called the roots, (what)
Ex. 8. Translate from English into Russian. Mind the use of Modal Verbs.
1. This equation can be solved easily. 2. This algebraic expression can be evaluated.
3.	The result must be obtained today. 4. One or more numbers can be described by
stating a condition. 5. The following condition must be satisfied. 6. The result must
be checked by division. 7. A fractional equation may be changed into an integral
equation. 8. Like terms must always be combined. 9. The results of the research can
be sent tomorrow. 10. Some interesting information on the system of equations can
be given at the next lecture.
Ex. 9. Translate the sentences from English into Russian.
1.	In solving problems by means of algebraic equations, the first and the most
difficult step which must be done is to translate the words into the algebraic language.
2.	Definite rules cannot be given to enable the student to solve mechanically the
given problem. 3. The following suggestions may be helpful for the students of
mathematics. 4. The difficult problem must be read over and over again until it
is clearly understood. 5. A full and definite description of the unknown quantity
represented by the letter is written out. 6. The expressions for all quantities in the
example involving this unknown will be given afterwards. 7. Expressions, which,
according to the statement of the problem, represent the same number, are found
and set equal to each other. 8. An equation in one unknown has a finite number
of solutions, or values of this unknown, which will satisfy the equation. 9. Any
87

term on one side of an equation may be transposed to the other side if its sign is
changed. 10. An equation which can be reduced to the form ax + b = 0 is called a
linear equation in x.
Ex. 10. Translate the sentences from Russian into English.
1. Уравнение - это утверждение, выражающее равенство двух алгебраиче¬
ских выражений. 2. Корень уравнения остается прежним, если к обеим ча¬
стям уравнения прибавить или от обеих частей уравнения вычесть одно и
то же выражение. 3. Корень уравнения остается прежним, если обе части
уравнения умножить или разделить на одно и то же выражение. 4. Решить
уравнение - значит найти те значения неизвестного, при которых обе части
уравнения равны одному и тому же числу (другими словами, все те значе¬
ния неизвестного, при которых равенство будет верным). 5. После подста¬
новки эти значения неизвестного удовлетворяют уравнению. 6. Значения
неизвестного, которые удовлетворяют уравнению, называют корнями или
решениями уравнения. 7. Величина, обозначенная через х, является неиз¬
вестной. 8. Такие задачи обычно решаются алгебраически, и используются
определенные правила. 9. Выражение, написанное слева от знака равенства,
называется левым членом уравнения. 10. Результат обычно проверяется по
данной формуле. 11. В этом случае будет получено новое уравнение. 12. Оба
члена умножаются на выражение, содержащее неизвестную. 13. Как называ¬
ется система линейных уравнений?
UNIT 8
Времена группы Continuous и Perfect
в страдательном залоге
Present Continuous Passive
Past Continuous Passive
Future
Continuous
Passive
I	am \
he/she/it	is	>	being	asked
we/you/they are J
I/he/she/it was \
> being asked
we/you/they were J
	*
now, at this moment, while (в mo
время как, пока), at present
at 6 o'clock, when she came, from 6 till
7 o'clock, the whole evening
	*
88

This problem is being discussed
at the moment.
Эту задачу обсуждают в
данный момент.
These natural numbers are being
multiplied now.
Эти натуральные числа
умножают сейчас.
The problem w’as being discussed in
class yesterday when the bell rang.
Эту задачу обсуждали вчера в
классе, когда прозвенел звонок.
These natural numbers were being
multiplied at the very end of the
lesson yesterday.
Эти натуральные числа умножали
вчера в самом конце урока.
* Вместо отсутствующей формы Future Continuous употребляется форма Future
Simple.
Present Perfect Passive
Past Perfect Passive
Future Perfect Passive
he j
she r has
it ’
I 1
"e I have
you 1
they J	j
> been asked
I >
he
she
it
we
you
they >
> had been asked
I 1
We 1
He 1
She
It
you
they >
shall/will
► will
>
>. have been
asked
The house has already
been built.
Дом уже построили.
The fraction has just been
reduced.
Дробь только что
сократили.
The house had been
built before I arrived.
Дом уже был построен
до того, как я приехал.
The whole chapter had
been studied by the end
of the semester.
Вся глава была изучена
к концу семестра.
The house will have been built
by January.
Дом построят к январю.
A new method will have been
introduced by the end of the
month.
Новый метод будет
представлен к концу месяца.
Ex.l. Analyse these pairs of sentences and compare the predicates given there.
They are solving the equation at the
moment.
He was dividing these numerals at
2 o’clock yesterday.
We have already reduced the fraction.
When I came back you had already
replaced the terms in the equation.
They will have discussed the definition
by 3 o’clock.
The equation is being solved at the
moment.
These numerals were being divided at
2 o’clock yesterday.
The fraction has already been reduced.
When I came back the terms in the
equation had already been replaced.
The definition will have been discussed
by 3 o’clock.
89

Are you changing the improper
fraction to the whole number now?
Was she writing the decimal fractions
at that moment?
Has he omitted the plus sign in this
sentence?
Will you have translated the article by
tomorrow?
Is the improper fraction being changed
to the whole number now?
Were the decimal fractions being
written at that moment?
Has the plus sign been omitted in this
sentence?
Will the article have been translated by
tomorrow?
The students are not multiplying these
integers right now.
We were not subtracting the fractions
when the teacher came in.
You have not divided the numerator
yet.
The student had not proved the
theorem by the end of the class
yesterday.
They will not have checked the result of
the calculation by 5 o’clock tomorrow.
These integers are not being multiplied
by the students right now;
The fractions were not being
subtracted when the teacher came in.
The numerator has not been divided
yet.
The theorem had not been proved by
the end of the class yesterday.
The result of the calculation will
not have been checked by 5 o’clock
tomorrow;
Ex. 2. State the voice and the tense-form of the verbs in the following sentences
and translate them into Russian.
1.	The students are being given a lecture now. 2. The students were being asked about
mathematical sentences the whole lesson. 3. The given quantity hasn't been divided
yet. 4. All the data had been obtained by that time. 5. The algorithm will have been
carefully worked out by tomorrow. 6. Have any of these articles on mathematics
been translated recently? 7. All the digits have already been aligned as appropriate.
8. The conference is being held at the moment. 9. Are the numbers being added
without a calculator right now? 10. His graduation paper hasn't been presented yet.
Ex. 3. Open the parentheses and give the correct form of the verb in the Passive
Voice.
1. Don’t enter the room! A student (to examine) there just now. 2. The letter (to
type) by the typist when I came in. 3.1 am sure that his work (to complete) by the
end of the month. 4. A lot of new words (to learn) already by the students. 5. All
the dinner (to eat) before they finished the conversation. 6. The question (not to
answer) yet. 7. The proposal (to consider) by 9 o’clock yesterday. 8. The papers (to
sign) just by the dean of the faculty. 9. The results of the test (to discuss) by the
students at the moment. 10. The article (to translate) by the time you return.
90

Pre-Reading Activity
Guess the meaning of the following words.
Algebraic (adj) [,si:4bre::k], polynomial (n) [zpcl:42s~iri:sl], integral (adj)
'=;■], constant (n) ['kc'stsr.t], coefficient (n) [zks'2:'f: 's2:]. exponent (n)
[sks'ps'22sr.:], fundamental (adj) [длг.с=чт=г.:1], process (n) ['prs^se-s].
Read and learn the basic vocabulary terms:
term (n) [ts 22]
upper (adj) [члрэ]
degree (n) [d:'cn]
appear (v) [s'p:s]
above (prep) [s'bAv]
monomial (n) [22с '2s2~i:sl]
binomial (n) [ba:' 2S'22s:sl]
trinomial (n) [:ra:' 2S~~i:9a]
power (n) [4pa~s]
place (v) [pie *5 J
obtain (v) [sb'te:?.]
arrange (v) [s're:22j]
arrangement (n) [s'rer^sisr.:]
ascend (v) [s4 5=22]
descend (v) [d:4 5522]
state (v) [s:s::]
concern (v) [ksr.'se 2]
precede (v) [p"'s: 2]
член, термин
верхний, высший
степень
показываться, появляться
над, выше, сверх
одночлен, adj. - одночленный
двухчлен, adj. - двухчленный
трехчлен, adj. - трехчленный
показатель степени
размещать, ставить
получать, приобретать
размещать, располагать, расставлять
размещение, расположение
подниматься, восходить
спускаться, снижаться
сообщать, формулировать
касаться, иметь отношение
предшествовать, стоять перед чем-либо
Memorise the following w ord combinations:
such as - такой как
in other words - другими словами
is known as a polynomial - известен как многочлен, называется многочленом
thus - таким образом
either ... or - либо ... либо, или ... или
in order to - для того, чтобы
by the aid of - с помощью (чего-либо)
91

Reading Activity
POLYNOMIALS
A number represented by algebraic symbols is referred to as an algebraic
expression.
An algebraic expression whose parts are not separated by + or - is called
a term; as 2 x 3, -5 x yz, and xylz.
In the expression 2x3- xyz - xylz there are three terms. The expression
c x (a - b) is a term.
An algebraic expression of one term is known as a monomial or a simple
expression, (xy and 3ab are monomials).
An algebraic expression of more than one term is called a polynomial. Such is,
for instance, the expression ab - a + b - 10 + (a - b)!c.
In other words we can say that algebraic expressions which consist of several
monomials connected by the + and - signs are known as polynomials.
Terms of a polynomial are separate expressions which form the polynomial by
the aid of addition and subtraction. Usually, the terms of a polynomial are taken
with the signs preceding them; for instance, we say: term -a, term +b2, and so on.
A polynomial of two terms is called a binomial, e.g. 3a + 2b and x2 - y2 are
binomials. Similarly a + b + c is a trinomial.
Thus, all algebraic expressions are divided into two groups according to the
last algebraic operation indicated: monomials and polynomials.
An expression, any term of which is a fraction, is referred to as a fractional
expression, as -3x + «/x; all the other expressions are called integral ones.
An algebraic expression such as Бх3 - 7X2 + 9x + 6 is a polynomial or an integral
expression in the letter x. It is composed of one or more terms, each of which is
either an integral power of x multiplied by a constant or a constant which is free
of x. The constant multipliers 5, 7, 9 are called coefficients-, the upper numbers: 3,
2 are exponents-, 6 is the constant term. The polynomial is of the third degree in x
since 3 is the highest exponent appearing in the expression.
An expression such as 2x2y + 5x*yz3 - 9xyz + 2x + 7 is a polynomial in x, у
and z. The degree of a polynomial in several letters is the highest degree that any
single term has in those letters. Thus, the above expression is of the seventh degree
in x,y and z since the sum of the exponents of the second term is seven.
Lets consider four fundamental operations of polynomials.
The first operation is addition. In order to add polynomials, you should place
them in such a way that like terms fall under each other, and add the coefficients in
each column to find the final coefficient of that term.
92

Hie second one is subtraction. To subtract one polynomial from another place
the terms of the subtrahend under like terms of the minuend, change the signs of
the terms of the subtrahend and add.
Hie third operation is multiplication. Suppose, you have been given two
polynomials and have been asked to multiply one of them by the other. In order to
do it, you are to multiply each term of one by every term of the other and to add
the products thus obtained.
And the last one is division. To divide one polynomial by another, arrange
both the dividend and the divisor in ascending or descending powers of some
letter common to both and write the quotient as a fraction.
Hie rule concerning the operation of division may be stated in the following way:
1.	Divide the leading term of the dividend by the leading term of the divisor,
obtaining the first term of the quotient.
2.	Multiply each term of the divisor by this term of the quotient and subtract
the product from the dividend.
Hie remainder found by this subtraction is used as the dividend and the
process is repeated. The work is continued until a remainder is reached which is of
lower degree than the divisor. In any case of division if the remainder is zero, the
division is exact.
Post-Reading Activity
Ex. 4. Answer the following questions.
1. What is an algebraic expression? 2. What algebraic expression is called polynomial
(monomial, binomial)? 3. What are the terms of a polynomial? 4. What numbers
in a polynomial are called coefficients (exponents, the constant term)? 5. How do
we define the degree of a polynomial? 6. What are the fundamental operations of
polynomials? 7. How is the sum of two polynomials obtained? 8. How is subtraction
of polynomials performed? 9. How is the product of two polynomials obtained?
10. What is the rule of polynomial division?
Ex. 5. Find the English equivalents for the following Russian word combinations:
1) состоять из нескольких одночленов; 2) алгебраическое выражение; 3) об¬
разовывать многочлен; 4) знаки, предшествующие им; 5) состоять из одного
или нескольких членов; 6) разместить таким образом; 7) сложить коэффи¬
циенты; 8) разместить делимое по возрастающим или убывающим показа¬
телям степени; 9) касающееся операции деления; 10) сложить произведения,
a) to form a polynomial; b) to be composed of one or more terms; c) to consist of
several monomials; d) to place in such a way; e) an algebraic expression; f) to add
93

the products; g) signs preceding them; h) to arrange the dividend in ascending
or descending powers; i) to add the coefficients; j) concerning the operation of
division.
Ex. 6. Give the proper English equivalents for the Russian expressions:
a trinomial, descending, subtraction, obtain, a fraction,
coefficients, terms, exponents, sum, remainder, fractional.
1.	Each of these polynomials is composed of two членов. 2. In the algebraic
expression 3x3 + 2x? + 5 the constant multipliers 3,2, 5 are called коэффициенты.
3.	In the polynomial 2x3 + 5л2 + 9 the upper numbers 3 and 2 are called показате¬
лями степени. 4. A polynomial consisting of three terms is called трехчлен. 5. One
of the fundamental operations that had been applied to those polynomials before
other operations was вычитанием. 6. The algebraic expression 2y3 - Зу2 + 2y is
arranged in убывающим powers of the letter y. 7. Multiplying two polynomials
we получаем a product. 8. If the остаток of division is zero, it is exact. 9. An
expression, any term of which is дробь, is called а дробным expression. 10. Adding
two polynomials we obtain а сумму.
Ex. 7. Make the following sentences negative and interrogative.
1. The result of subtraction is being checked now. 2. Polynomials and their
fundamental operations were being studied by the students the whole day
yesterday. 3. These polynomials are being multiplied at the moment. 4. Each step
of the process has already been carefully studied. 5. Hie necessary information has
just been obtained. 6. All the material about polynomials has been learned by the
student recently. 7. Those algebraic expressions have been carefully arranged in
descending powers. 8. Four operations with polynomials were being discussed at
the seminar. 9. The remainder in this expression will have been found by the end of
the lesson. 10. Hie experiment was being carried out when you came in.
Ex. 8. Mark the following as true or false.
1. A polynomial is composed of one term only. 2. A number represented by
algebraic symbols is called a fractional expression. 3. Each term of a polynomial
is either an integral power of x multiplied by a constant or a constant free of x.
4.	The division isn’t exact if the remainder is zero. 5. Division of one polynomial
by another is rather a long process. 6. A monomial consists of several terms. 7. All
algebraic expressions are divided into different groups. 8. When subtracting we
change the signs of the terms of the subtrahend. 9. The polynomial Зх2^2 + 2xy + 5
is of the fifth degree. 10. A polynomial of three terms is called a binomial.
94

Ex. 9. Ask special questions.
1. A polynomial is an algebraic expression composed of one or more terms,
(what) 2. In the expression 2x3- xyz - xylz there are three terms, (how many)
3. A polynomial of two terms is called a binomial, (how) 4. Hie polynomial Зх3 +
+ 4л2 + 5 is of the third degree in x. (what) 5. Hie whole material has already been
learned by the students, (by whom) 6. All the trinomials were being subtracted
when we came, (what) 7. If the remainder of division is zero, it is exact, (when)
8. You should divide the leading term of the dividend by the leading term of the
divisor, (what, who) 9. Hie remainder found in the result of subtraction is used as
the dividend, (how) 10. Hie translation of the text hadn’t been completed by the
end of the class yesterday, (what)
Ex. 10. Translate these sentences from English into Russian.
1.	An algebraic expression of one term is called a monomial or simple expression.
2.	An algebraic expression of more than one term is called a polynomial. 3. Hie
terms of a polynomial are taken with the signs preceding them. 4. Hie polynomial is
of the third degree in x since 3 is the highest exponent appearing in the expression.
5.	You have been given two polynomials and have been asked to multiply one of
them by the other. 6. We place the terms of the subtrahend under like terms of
the minuend. 7. Hie fractional numerals are being written as the corresponding
decimal numerals by the students right now. 8. In dividing polynomials both the
dividend and the divisor must be arranged in ascending or descending power of
the letter common to both. 9. To add or to subtract polynomials we must place
them so that like terms fall under each other. 10. Hie remainder is of lower degree
than the divisor.
Ex. 11. Translate these sentences from Russian into English.
1. Многочлен состоит из двух и более членов. 2. Алгебраическое выраже¬
ние, которое содержит только действия умножения, деления и возведения в
степень, называется одночленом. 3. Алгебраическая сумма нескольких одно¬
членов называется многочленом. 4. Трехчлен - алгебраическое выражение,
состоящее из трех членов. 5. Числа при неизвестных х, у называются коэф¬
фициентами многочлена. 6. Многочлены можно складывать, вычитать, ум¬
ножать и делить. 7. Чтобы разделить многочлен на одночлен, нужно делимое
и делитель разместить в убывающем или возрастающем порядке общего не¬
известного. 8. Правило, касающееся деления, может быть сформулировано
определенным образом. 9. Деление продолжается до тех пор, пока не будет
найден остаток с числовым значением, меньшим,чем делитель. 10. Если оста¬
ток при делении равен нулю, то деление называют точным или без остатка.
95

SUPPLEMENTARY READING
IMAGINARY NUMBERS
When imaginary numbers were introduced, mathematicians spoke of them as
“imaginary” simply because they didn’t exist in the system of numbers to which
they were accustomed. Actually, they are no more imaginary than the ordinary
real numbers. The so-called imaginary numbers have carefully defined properties
and can be manipulated as easily as the other numbers. And yet because the new
numbers were considered “imaginary”, the symbol i was used. We can speak of
positive imaginary numbers (+/) and negative imaginary numbers (-/), whereas
(+1) is a positive real number and (-1) is a negative real number. Thus, we can say
n/—1 — ± i.
The system of real numbers can be exactly matched in the system of imaginary
numbers. If we have +5, -17.34, +3/10, we can also have +5/, -17.32/, +3/710. You
can even picture the imaginary system of numbers.
Suppose you represent the real number system on a straight line with 0 (zero)
in the center. The positive numbers are on one side of the zero and the negative
numbers are on the other.
You can then represent the imaginary system of numbers along another line,
crossing the first at right angles at the zero point, with the positive imaginaries on
one side of the zero and the negative imaginaries on the other.
BASE-TWO SYSTEM
Any number at all can be expressed by some combination of Is ans Os. The
first civilized mathematician to work it out systematically, however, was Gottfried
Wilhelm Leibniz, about three centuries ago.
It turned out that this base-two system of numbers (also called the binary
system) is ideal for electronic computers.
After all, the two different digits, 1 and 0, can be matched in the computer by
the two different positions of a particular switch: “on” and “off”.
However, since we are only human beings, the question is, can we handle the
base-two system? For instance, can we convert back and forth between base-two
numbers and ordinary numbers? If we are shown 110001 in the base-two system,
what does it mean in ordinary numbers?
Actually, this is not difficult. Hie based-two system uses powers of 2, starting at
the extreme right with 2° and moving up a power at a time as we move leftward. So
we can write 110001 with little numbers to represent the exponents, thus 110001 =
= 1 x 25 + 1 x 24 + 0 x 23 + 0 x 22 ~ 0 x 21 + 1 x 2°.
96

Only the exponents with the Is are used, so 110001 represents 25 plus 24 plus
2° or 32 plus 16 plus 1. In other words, 110001 in the base-two system is 49 in
ordinary numbers.
Working the other way is even simpler.
Suppose you wish to convert an ordinary number into the base-two system.
You divide it by 2 and set the remainder to one side. Working only with the whole¬
number portion of the quotient, you divide that by 2 again, and again set the
remainder to one side and work only with the whole-number portion of the new
quotient. When the whole-number portion of the quotient is reduced to 0 as a
result of the repeated divisions by 2, you stop. The remainders, read backward, give
the original number in the base-two system.
THE ORDER OF OPERATIONS
Hie order of operations is a mathematical and algebraic set of rules. It is used
to evaluate (solve) and simplify expressions and equations. The order of operations
is the order in which different mathematical operations are done. The standard
mathematical operations are addition (+), subtraction (-), multiplication (• or x),
division (/), brackets (which are grouping symbols, like parentheses () or [] and
exponentiation (also called orders or indices).
Mathematicians have agreed on a correct order to use operations, and it is very
important that they know these rules. When people are solving a problem with
more than one operation, they will need to know the correct order to solve the
problem correctly. Otherwise the answer will be wrong.
One should follow all the rules in this order from left to right in the equation,
Brackets and indices. Use operations inside brackets and solve any indices. You
should always solve brackets first when solving an equation. Example: (2 + 3) x
x (4- 0 + 23 = 5x 3 + 23.
Multiplication and division. Solve any multiplication and division in the
problem. Note that multiplication does not precede division; this is a common
mistake. Both are solved from left to right as they occur. Example: 5x4- 9/3 =
= 20 - 9/3 = 20 - 3.
Addition and subtraction. Finally, solve any addition or subtraction. Two
examples of all rules.
Example one: (1 + 8) x (4 - 1) + 16/8 = 9x 3+ 16/8 = 27 + 2 = 29.
Example two: (7 + 3) x (6 - 3) + 216/27 = 10 x (6 - 3) + 216/27 = 10 x 3 +
+ 216/27 = 30 + 216/27 = 30 + 8 = 38.
97

RULES OF ALGEBRA
In algebra, there are a few rules that can be used for further understanding
of equations. These are called the rules of algebra. While these rules may seem
senseless or obvious, it is wise to understand that these properties do not hold
throughout all branches of mathematics. Therefore, it will be useful to know how
these axiomatic rules are declared, before taking them for granted. Before going on
to the rules, reflect on two definitions that will be given:
1.	Opposite - the opposite of a is -a;
2.	Reciprocal - the reciprocal of a is Ma.
Rules: a) Commutative property of addition. “Commutative” means that a
function has the same result if the numbers are swapped around. In other words,
the order of the terms in an equation does not matter. When the operator of two
terms is addition, the “commutative property of addition” is applicable. In algebraic
terms, this gives a + b = b + a. Note that this does not apply for subtraction!
b)	Commutative property of multiplication. When the operator of two terms
is multiplication, the “commutative property of multiplication” is applicable. In
algebraic terms, this gives a x b = b x a. Note that this does not apply for division!
c)	Associative properly of addition. “Associative” refers to grouping the
numbers. The associative property of addition implies that, when adding three or
more terms, it doesn’t matter how these terms are grouped. Algebraically, this gives
a + (b + c) = (a + b) + c. Note that this does not hold for subtraction!
d)	Associative property of multiplication. The associative property of
multiplication implies that, when multiplying three or more terms, it doesn’t matter
how these terms are grouped. Algebraically, this gives a x (b x c) = (a x b) x c. Note
that this does not hold for division!
e)	Distributive property. Hie distributive properly states that the multiplication
of a number by another term can be distributed. For instance, a x (b + c) = ab + ac.
f)	Additive identity property. “Identity” refers to the property of a number that
it is equal to itself. In other words, there exists an operation of two numbers so that
it equals the variable of the sum. The additive identity property states that the sum
of any number and 0 is that number: a + 0 = a. This also holds for subtraction:
a - 0 = a.
g)	Multiplicative identity property. The multiplicative identity property states
that the product of any number and 1 is that number: a x 1 = a. This also holds for
division: all - a.
h)	Additive inverse property. The additive inverse property is somewhat like
the opposite of the additive identity property. When an operation is the sum of a
number and its opposite, and it equals 0, that operation is a valid algebraic operation.
Algebraically, it states the following: a - a = 0. Additive inverse of 1 is (-1).
98

i)	Multiplicative inverse property. Hie multiplicative inverse property entails
that when an operation is the product of a number and its reciprocal, and it
equals 1, that operation is a valid algebraic operation. Algebraically, it states the
following: at a = 1. Multiplicative inverse of 2 is Vi.
PROGRESSIONS
A sequence is a succession of quantities formed according to some fixed law.
The simplest illustration of a sequence is that formed by the numbers used in
counting, i.e. by natural numbers 1,2, 3,4, 5 ....
Hie sequence of natural numbers may be divided into two classes: even and
odd.
Hiis gives 2,4,6,8,10 as a sequence of even numbers, 1,3,5,7,9 as a sequence
of odd numbers.
Hie sequence formed by the squares of the natural numbers I2, 22, 32,42, 52 or
1, 4, 9, 16, 25 is another illustration of a succession of quantities in which there is
an evident law of formation.
A progression in which the difference between two consecutive terms is
always the same is called an arithmetic progression. The common difference may be
positive or negative. The progression is increasing if the difference is positive, and
decreasing if the difference is negative.
Let us call a the first term, d - the common difference, n - the number of term
considered, / - the n-th or last term, and S - the sum of n terms of the progression.
Hiese five numbers a, d, n, I, S are known as elements of the arithmetic progression.
A sequence of numbers is said to form a harmonic progression if the reciprocals
of the numbers form an arithmetic progression. Thus, if a, b, c are in arithmetic
progression, 1/a, 1/b, 1/c are in harmonic progression.
A progression in which any term after the first can be obtained by multiplying
the preceding term by a given constant is called a geometric progression. Hie
constant is called the common ratio and may be found by dividing any term by the
one that precedes it.
Problem. An Indian Prince, Sirahm, is said to have offered to the inventor of
chess any reward he might desire.
Hie inventor asked to be given 1 wheat seed for the first square on the chess¬
board, 2 seeds for the second square, 4 for the third and so on, doubling the number
of seeds for every subsequent square. Hie prince agreed. But when the quantity of
wheat to be given for all the 64 squares of the chess-board was calculated, it was
found that the reward could not be granted at this rate because of the lack of wheat
on hand. How much wheat had to be given to the inventor?
99

The number of seeds due for all the 64 squares is equal to the sum S of the
following series of numbers:
S = 1 + 2 + 22 + 23 + 24+ ... + 262 + 263.
As a result, we obtain the following formula:
S = 2M- 1.
The final number of seeds will be:
S = 264- 1 = 18,446,774,073,709,551,615.
We may find by calculation that if this number of seeds were scattered evenly
over the surface of the earth, the layer of wheat formed would be about 9 mm thick.
In this problem we have dealt with a series of numbers each term of which,
beginning with the second, is derived from the preceding one by multiplying it by
a constant number. Such a series is known as a geometric progression.
PRACTICE THE WAY
OF PUTTING QUESTIONS IN ENGLISH
Ex. 1. Complete the questions.
1.1 want to go out.
Where do you want to go?
2. Ann and Paul aren’t going to the party.
Why aren’t they going?
3. I’m reading.
What... ?
4. Sue went to bed early.
What time... ?
5. My parents are going on holiday.
When... ?
6.1 met Torn a few days ago.
Where... ?
7. Tina has gone away.
Where... ?
8.1 can’t come to the party.
Why... ?
9.1 need some money.
How much... ?
10. Angela doesn’t like me.
Why... ?
11. If rains sometimes.
How often...
12.1 did the shopping.
When... ?
Ex. 2. Make questions with who or what (subject or object).
1.1	bought something.
2. Somebody lives in this house.
3.1	phoned somebody.
4.	Something happened last night.
5.	Somebody knows the answer.
What did you buy?
Who lives in this house?
100

6.	Somebody did the washing-up.
7.	Jill did something.
8.	Something woke me up.
9.	Somebody saw the accident.
10.1 saw somebody.
11.	Somebody has got my pen.
12.	This word means something.
Ex. 3. Fill in the blanks with the auxiliary verb where it is necessary,
a) do	b) does	c) -
1.	Who of you ... speaks English fluently?
2.	... you like playing chess?
3.	Where ... she live?
4.	... your mother like cooking?
5.	Who ... you always walk your dog with?
6.	Where ... your parents spend their vacation?
7.	What... your father do in his spare time?
8.	Why ... they learn poems by heart?
9.	How often ... Dan come to the club?
10.	Who ... his friend like to play chess with?
11.	When ... you clean your teeth?
12.	What... you think of me?
13.	How often ... you go swimming?
14.	What time ... your sister come back home?
15.	How much ... your trousers cost?
16.	When ... your father go to work?
17.	Why ... your sister go shopping on Saturdays?
18.	Who ... knows the way out?
19.	What time ... you get up?
20.	How much ... your sweater cost?
21.	Whose parents ... want to help us?
22.	Which of your sisters ... lives in the country?
Ex. 4. Chose the corresponding auxiliary verb for making questions.
a) do	b) does	c) is	d) are
1.	What subject... you like best?
2.	Where... the capital of your country?
3.	... you know what time it is ?
4.	How far ... London from Liverpool?
101

5.	... you speak English?
6.	Where ... your son study?
7.	... it snowing now?
8.	When ... first spring flowers appear on the ground?
9.	... it often rain in autumn?
10.	When ... it get light in January?
11.	... it still dark?
12.	What circle ... you going to join?
13.	How long... it take you to get to the Institute?
Ex. 5. Chose the corresponding auxiliary verb to complete questions.
a) do	b) does	c) is
d) are	e) have	f) has
1.	What subjects ... she good at?
2..	.. your brother got a camera?
3..	.. your mother like cooking?
4.	What floor ... your bedroom on?
5..	.. your parents in France now?
6.	Where ... the nearest book-store?
7..	.. your friend have any money?
8.	Where ... your uncle work?
9.	What sports ... they fond of?
10.	What... the weather like today?
11.	What languages ... you speak?
12.... you like science fiction?
13.	What... your favourite pop group?
14.	What bike ... she got?
15.	How many apples ... you got?
Ex. 6. Which auxiliary verb would you use for making the following statements
interrogative.
a) do	b) did	c) does
d) have	e) has	f) will
1.	We have to change the whole method.
2.	If rained hard yesterday.
3.	They have to speak to their teacher.
4.	He has to work in the laboratory every Friday.
5.	She had to translate only one text.
6.	I shall have to tell him about the incident.
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7.	They will have to go there tomorrow.
8.	She had to meet him at the airport yesterday.
9.	She often goes on business to the USA.
10.	They had to discuss the matter at once.
11.	Some students combine studies and work.
12.	We had to send the fax to the office.
13.	We shall have to inform his family about our arrival.
14.	She has to take an exam on geometry today.
Ex. 7. Make the following sentences interrogative and negative.
1.	I had to put the baby to bed at once.
2.	Someone will have to stay and do this job.
3.	They were to finish school that year.
4.	We had to take our children to the Zoo.
5.	Hie train is to come at 5 o’clock.
6.	As a rule she has dinner at home.
7.	They have to speak to your teacher.
8.	We are to meet after classes.
9.	On Sunday I had breakfast at 9 o’clock.
10.	She was to take her exam that morning.
11.	You will have to help the boy.
12.	They had to check the results of the test.
Ex. 8. Ask special questions.
1.	I am to give you only the general idea of our work, (what)
2.	She has to translate only one text, (how many)
3.	He was to speak to Mike after the seminar, (when)
4.	They will have to answer all the questions, (what)
5.	You had to tell your father about it. (who)
6.	Tomorrow my sister will have dinner at 3 o’clock, (when)
7.	I’ll have to speak to her tomorrow; (to whom)
8.	I have to make a report at the conference. (wrhere)
9.	We are to leave the laboratory at 8 o’clock. (wrhen)
10.	Hie lecture wras to begin in the afternoon, (what)
11.	Usually he has supper at 7 o’clock, (when)
12.	My friend had to go there at once, (w'here)
Ex. 9. Chose the correct item for completing questions.
1.	When ... the 350th anniversary of Rembrandt’s birth celebrated?
a) does	b) was	c) did
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2.	... any exhibitions devoted to this anniversary organized at the university?
a)	were	b) was	c) did
3.	... the Belarussian Government do its best to improve the living conditions of
the population?
a) do	b) does	c) is
4.	When ... you finish writing your report?
a) have	b) did	c) are
5.	What magazine ... you looking through when I came in?
a) did	b) were	c) have
6.	... it still raining? — No, the rain has already stopped.
a) is	b) will	c) does
7.	... he meet us at the station tomorrow?
a. will	b) does	c) is
8.	... you finished writig your article yet?
a) were	b) did	c) have
9.	... your sister want to buy a new radio-set?
a) has	b) is	c) does
10.	When ... you install a new modem?
a) did	b) were	c) are
11.	Where ... you going when I met you last night?
a) did	b) were	c) are
12.	... your friend like to go jogging in the evening?
a) do	b) does	c) is
Ex. 10. Put questions to the underlined words.
1.	My mother runs the house perfectly.
2.	My students are very nice.
3.	John and Dick are playing tennis.
4.	They run for the bus every morning.
5.	We like English tea.
6.	Luisa has got a very interesting job.
7.	I have dinner at 7 o’clock in the evening.
8.	Babies have five meals a day.
9.	There is a round table and five chairs in the middle of the room.
10.	These are my postcards.
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11.	Jill goes to work by bus.
12.	The Normans invaded Britain in 1066.
13.	Columbus discovered America in 1492.
14.	She does her room every day.
15.	They were at the disco last night.
16.	He is crying as he has cut his finger.
17.	He had to spend a lot of money on education.
18.	He had ironed all the linen by 5 o’clock yesterday.
19.	This pair of trousers cost seven dollars.
20.	The fax will be received in an hour.
21.	The contract has been signed.
22.	A new supermarket is being built in our district.
23.	His father wants him to become a student.
24.	Г11 buy a new toy-car for my son when I get the money.
Ex. 11. Make up all possible types of questions on the basis of the given sentences.
a) questions to the subject b) alternative	c) disjunctive
d) special questions	e) general
1.	The children are swimming in the river.
2.	The work can be done in two weeks.
3.	You have to send e-mails to your friends.
4.	There will be a new service station here.
5.	They lay in the sun for half an hour.
6.	I have some pets at home.
7.	She did the work nicely last week.
8.	You’ll get a nice present for your birthday.
9.	He is driving a new Ford today.
10.	The telegram was brought by a stranger.
11.	The lecture will be delivered by a visiting professor.
12.	This dish must be served hot.
13.	There are three foreign students in this group.
14.	She is a careless driver.
15.	That man has been to Australia five times.
16.	I have been learning English all my life.
17.	You must call your elderly parents every week.
18.	There are all modern conveniences in the cottage.
19.	She was fixing breakfast at five o’clock yesterday.
20.	It often rains in autumn in this country.
21.	He always has a swim before breakfast.
22.	The manager expects the secretary to arrive at 9.
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СШи1 □□
UNIT 9
Согласование времен (Sequence of Tenses)
Косвенная речь (ReportedSpeech)
Согласование времен
1.	Если сказуемое главного предложения выражено глаголом в настоя¬
щем времени (Present Simple, Present Perfect) или в будущем времени (Future
Simple), то глагол в придаточном предложении употребляется в любом вре¬
мени, которое требуется по смыслу:
I know
he works there
he is working there
работает
he worked there
he was working there
работал
he will work there
he will be working there
будет работать
2.	Если сказуемое главного предложения выражено глаголом в прошед¬
шем времени (обычно Past Simple), то глагол придаточного предложения упо¬
требляется в одной из форм прошедшего времени или будущего в прошедшем
(Future-in-the Past).
I knew
he worked there
he was working there
работает
he had worked there
he had been working there
работал
he would work there
he would be working there
будет работать
3.	Изменение модальных глаголов при согласовании времен,
can -> could	shall should (совет)
may might	must had to (пришлось, был вынужден)
will -» would
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Она напомнила мне, что я должен
быть осторожнее.
Я не понимал, почему именно я
должен это делать.
Не изменяются при согласовании времен модальные глаголы: would,
could, might, ought to, must (обязан)/mustn’t.
She reminded me that I ought to be
more careful.
I didn’t understand why it was I who
should do it.
Косвенная речь
Правило согласования времен действует и при обращении предложения
в косвенную речь.
1.	Повествовательные предложения в косвенной речи вводятся глаго¬
лами to say smth to smb, to tell smb (he said that..., he said to me that..., he
told me that...).
Запомните: to tell a lie» to tell the truth» to tell a storv and to tell the time.
Глагол в главном предложении,
ящем времени.
Прямая речь
Jack says, “She knows the answer.”
Anna says, “We are leaving tonight.”
Bob says, “I have read the story.”
Sue says, “They told the truth.”
Jim says, “I was thinking about it.”
Greg says, “Dad will speak to you.”
Paul says, “He can swim here.”
вводящий прямую речь, стоит в насто-
Косвенная речь
Jack says (that) she knows the answer.
Anna says (that) they are leaving tonight.
Bob says (that) he has read the story.
Sue says (that) they told the truth.
Jim says (that) he was thinking about it.
Greg says (that) dad will speak to me.
Paul says (that) he can swim here.
Глагол, вводящий прямую речь, стоит в прошедшем времени.
Прямая речь
Present Simple
Jack said, “She knows the answer.’
Present Continuous
Anna said, “We are leaving soon.’
Present Perfect
Bob said, “I have read the story.”
Past Simple
Sue said, “They told the truth.”
Косвенная речь
Past Simple
Jack said (that) she knew the answer.
Past Continuous
Anna said (that) they were leaving soon.
Past Perfect
Bob said (that) he had read the story.
Past Simple/Past Perfect
Sue said (that) they (had) told the truth.
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Past Continuous
I said, “I was thinking about it.”
Past Perfect
Matt said, “I had read this book.”
Past Perfect Continuous
Ted said, “I had been doing it all day.’
Future Simple
Greg said, “Dad will speak to you.”
Past Continuous/Past Perfect
Continuous
I said (that) I was /had been thinking
about it.
Past Perfect (no change!)
Matt said (that) he had read this book.
Past Perfect Continuous (no change!)
Ted said (that) he had been doing it all
day.
Future-in-the Past
Greg said (that) dad would speak to me.
Для передачи косвенной речи используется ряд других глаголов:
a)	для сообщения информации: remark (отмечать), explain (объяв¬
лять), mention (упоминать), insist (настаивать), declare (объявлять),
announce (объявлять), state (заявлять), promise (обещать).
b)	следующие глаголы показывают, что далее последует ответ на уже
высказанную реплику: answer (отвечать), confirm (убеждать), reply (от¬
вечать), object (возражать), agree (соглашаться), deny (отрицать), assert
(утверждать), protest (выражать протест).
2.	Общие вопросы вводятся в косвенной речи союзами if или whether,
которые помещаются перед косвенным общим вопросом. Сам косвенный
вопрос приобретает структуру утвердительного предложения.
Прямая речь	Косвенная речь
Hie dean asked Lucie, “Do you live far The dean asked Lucie if she lived far
from the university?”	from the university.
My brother asked, “Can you answer the My brother asked me if I could answer
phone?”	the phone.
Вместо глагола ask могут употребляться другие вводящие глаголы: wrant
to knowr (хотеть знать), inquire (спрашивать), wonder (интересоваться).
3.	Специальные вопросы в косвенной речи вводятся вопросительными
местоимениями, которые становятся союзными словами.
Прямая речь	Косвенная речь
Не asks me, “What book has she read He asks me what book she has read
since Monday?	since Monday.
He asked me, “What places have you He asked me what places I had visited,
visited?”
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4.	Повелительные предложения. Если прямая речь выражает приказание,
то глагол to say заменяется глаголом to tell (велеть, сказать) или to or¬
der (приказывать). Если прямая речь выражает просьбу, то глагол to say
заменяется глаголом to ask (просить).
Следует иметь в виду, что после глаголов to ask, to tell, to order в
английском языке всегда следует косвенное дополнение, обозначающее лицо,
к которому обращена просьба или приказание, а само содержание просьбы
или приказа передается глаголом в форме инфинитива с частицей to.
Здесь также возможно употребление таких глаголов, как to invite (при¬
глашать), to advise (советовать), to recommend (рекомендовать), to warn
(предупреждать).
Прямая речь
She said to him, “Come at five o’clock.”
The teacher said to me, “Don’t sit down.’
I said to her, “Please, bring me a glass
of water.”
Косвенная речь
She told him to come at five o’clock.
Hie teacher told me not to sit down.
I asked her to bring me a glass of water.
5.	Отступления от правил согласования времен.
A.	Правило согласования времен не соблюдается, если придаточное
предложение содержит абсолютную истину.
They didn’t even know that life has	Они не знали даже того, что жизнь
been developing on our planet for	на нашей планете развивается
billions of years.	миллиарды лет.
B.	Если придаточное предложение содержит пословицы или поговорки,
которые говорящий считает истинными, безотносительно ко времени.
When he returned from France he said Когда он вернулся из Франции, он
that East or West home is best.	сказал, что в гостях хорошо, а дома
лучше.
C.	Если предложение в косвенной речи употребляется во II или III типе
условных предложений.
Не said that if he were me, he would Он сказал, что если бы он был на
attend that conference.	моем месте, он принял бы участие в
той конференции.
6.	Дополнительные изменения в косвенной речи. Личные, притяжатель¬
ные и указательные местоимения, а также наречия места и времени изме¬
няются следующим образом при преобразовании предложений из прямой
в косвенную речь.
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Прямая речь
Косвенная речь
I, you, he, she
we, you, they
he or she
they
my, your, his, her
our, your, their
his or her
their
this, these
that, those
now
today
tonight
yesterday
tomorrow
the day after tomorrow
in an hour
last week
next week
next Friday
two days ago
then, at that time
that day
that night
the day before, the previous day
the next day, the following day
two days later
an hour later
the week before, the previous week
the week after, the following week
the following Friday
two days before
Ex. 1. Answer the questions according to the model. Don’t forget to make the
necessary changes in the reported speech.
a)	A. What does he say? (I am a first-year student).
B. He says he is a first-year student.
1.	What does he say? (I live in the students’ hall of residence).
2.	What does he promise? (As soon as my exams are over I shall go to Brest for a
short vacation).
3.	What do the students say? (We had two tests last week).
4.	What does mother know? (My son is afraid of dogs).
5.	What does the reporter mention? (There have been two accidents on the road).
6.	What has the Prime Minister declared? (I am going to London next week).
7.	What does the child say? (We have been reading this book for three days, Mom).
8.	What will you tell her? (We need some help).
b)	A. What did he say? (My sister learnt French).
B. He said his sister had learnt French.
1.	What didn’t you know? (She can speak Polish).
2.	What did you decide last week? (We all will go to the Canary Isles).
3.	What did she know? (Her boyfriend has already come back to Minsk).
4.	What did the administrator announce? (The press conference is taking place in
the main hall now).
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5.	What did the student say? (I have been writing my term paper since Friday).
6.	What did Bill say? (I took my dog out for a walk in the morning).
7.	What did mother remind you about? (I ought to be more careful when doing
my homework).
c)	A. What did she ask you about? (Has anybody read the book?)
B. She asked me if somebody had read the book.
1.	What did your friend ask you about? (Do you know the password for the
computer?)
2.	What did the policeman ask you about? (Does the car belong to you?)
3.	What did the interviewer want to know? (Do you watch TV every evening,
Chris?)
4.	What did your father ask you about? (Do you know what you have done?)
5.	What did your sister want to know? (Are they getting married this weekend?)
6.	What did you say to her? (Can a correct solution be found?)
7.	What did he want to know? (Has the situation changed recently?)
d)	A. What did he say? (What field of maths are you concerned with?)
B. He asked what field of maths I was concerned with.
1.	What did the scientist say? (When will it be possible to introduce a new method?)
2.	What did the teacher want to know? (How many English books have you read
since September?)
3.	What did he say to him? (Who has provided you with this material?)
4.	What did the sales manager want to know? (Why did you apply for this job?)
5.	What did the examiner say? (How long have you been learning English?)
6.	What did the customer want to know? (What are the advantages of a netbook?)
7.	What did the secretary say? (When are you leaving: today or tomorrow?)
e)	A. Read (Do not read) the book, (a teacher)
B. The teacher told me to read (not to read) the book.
1.	Don’t forget to put your name at the top of the page, (an examiner)
2.	Be careful when crossing the road, Bob. (mother)
3.	Please, write to me as often as you can. (a friend)
4.	Don’t drive too close to the car in front, (a driving instructor)
5.	Take this medicine three times a day. (a doctor)
6.	Fasten your seat belts! (a flight attendant)
7.	Don’t go near the house, it is dangerous, (a firefighter)
111

Ex. 2. Rewrite each sentence as direct speech.
1.	I am not sure if they will discover the truth.
2.	He asked me whether that scientist was popular.
3.	She didn’t know if they had covered all the problems.
4.	The inspector wanted to know what the average number of students in an
academic group was.
5.	Hie science adviser asked his post-graduate if he would be included in the
experimental group.
6.	Helen says she has chosen the topic of her graduation paper.
7.	Hie librarian reminded that I had to fill in those forms.
8.	Father told me his favourite team had lost the game two days before.
9.	Peter has just said he has found all information on his site recently.
10.	Hie teacher advised his students to read each question twice.
Ex. 3. Choose the correct variant.
1.	She asked if I planned to join them ... week.
a)	next	b) the last
c)	following	d) the following
2.	Hiey have said that their new computer ... tomorrow.
a) would	deliver	b) will deliver
c) would	be delivered	d) will be delivered
3.	Hie driver said that he ... be there at 8.40.
a) has to	b) had to
c) ought	d) will have to
4.	He ... why I had been standing at the bus stop the night before.
a) asked to me	b) asked me
c) told me	d) had told me
5.	She asked me if I... time to help him two hours later.
a) have	b)	will have
c) would	have	d)	am having
6.	Hie manager wondered when ... them the goods they had ordered.
a) would	they send	b)	will they send
c) they would send	d)	they will send
7.	Hiey complained that the coffee machine they had bought in that store ....
a) didn’t work	b) don’t work
c) hasn’t worked	d) isn’t working
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8.	He said that actions ... louder than words.
a)speaks	b)speak
c) will speak	d) had spoken
9.	Jane ... there was nothing she could do.
a)	said me	b) told me
c) told to me	d) say to me
10.	The customer wanted to know ....
a) how much that book is	b) how much is this book
c) how much that book was	d) how much was this book
Pre-reading Activity
Guess the meaning of the following words.
Geometry [zsz'csszrzz], process ['presses], geometric [,chzs'me:rzk]> axiom
['sskizsrrj, postulate ['pcstyjkr], theorem ['Gzsz’sm], vertical ['vs tzksl], figure
['fzss], form [fc rrj.
Read and learn the basic vocabulary terms:
reasoning (n) [Tz zsrzzz}]
principal (adj) ['pzzr.sspslj
discover (v) [dis'k-ws]
acquaint (v) [s'kweznt]
establish (v) [zs'zssblzj
definition (n) [,zefz'r.z’sr.]
proposition (n) [,p:cps'zz sr.]
meaning (n) ['22: r_:z~ ]
attribute (v) [s'zrzbyj.:] (v)
accept (v) [sk'sspr] (v)
proof (n) f]
unique (adj) [yx 'r_z k]
plane (n) ['plszr.]
congruent (a) ['kc~.cz”srzr]
arc (n) [a:k]
remaining (adj) [zi'meinizr ]
corollary (n) [ks'zclsrz]
distinguish (v) [dzs'ti-pwz']
hypothesis (n) [haz'pc9:szs]
рассуждение, обоснование
главный
обнаружить, открыть, находить
познакомить
устанавливать
определение
предложение, утверждение, выска¬
зывание, суждение
значение, смысл
приписывать, относить, придавать
принимать, допускать
доказательство
единственный, однозначный
плоскость
конгруэнтный, совмещающийся
дуга, арка
остающийся
следствие
различать, выделять, распознавать
гипотеза
113

conclusion (n) [ksr/k?z jsr.]
consider (v) [ksr/s:d=]
require (v) [rfkwazs]
rephrase (v) ["':re:z]
converse (n) ['kcr.vs s]
bisector (n) [ba:'sek:s]
assumption (n)	,’sr.]
deduce (v) [d:' d1"s]
вывод
считать, полагать, рассматривать
требовать, нуждаться
перефразировать
обратная теорема
биссектриса
предположение, допущение
выводить (заключение, формулу)
Memorise the following word combinations:
properties of geometric figures - свойства геометрических фигур
it would be instructive - было бы поучительно
one of the four angles turns out to be right - один из четырех углов оказывается
прямым
at most - самое большее, максимум
it is useful to notice - полезно отметить
the theorems are converse to each other - теоремы обратны друг другу
This is not always the case. - Это не всегда так.
to be true/false - быть истинным, ложным
to acquaint yourself with the forms of reasoning - ознакомиться с моделями
рассуждения
it follows from the theorem - из теоремы следует
one can distinguish two parts - можно выделить две части
to take for (as) granted - считать доказанным, принимать без доказательства
Reading Activity
MATHEMATICAL PROPOSITIONS
In geometry, the process of reasoning is a principal way to discover properties
of geometric figures. It would be instructive therefore to acquaint yourself with the
forms of reasoning usual in geometry.
All facts established in geometry are expressed in the form of propositions.
Hie propositions that we take for granted without proof are called assumptions.
With regard to a different set of assumptions the same proposition may, or may not
be true. The assumptions themselves are neither true nor false. They may be said to
be “true” only in the sense that their truth has been assumed.
Definitions are propositions which explain what meaning one attributes to
a name or expression.
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Axioms (some axioms are traditionally called postulates) are those facts which
are accepted without proof. Th is includes, for example, some propositions: through
any two points there is a unique line; if two points of a line lie in a given plane then
all points of this line lie in the same plane.
Propositions that can be logically deduced from the assumptions are often
called theorems. For example, if one of the four angles formed by two intersecting
lines turns out to be right, then the remaining three angles are right as well.
Corollaries are those propositions which follow directly from an axiom or
a theorem. For instance, it follows from the axiom “there is only one line passing
through two points” that “two lines can intersect at one point at most.”
In any theorem one can distinguish two parts: the hypothesis and the
conclusion. The hypothesis expresses what is considered given, the conclusion what
is required to prove. For example, in the theorem “if central angles are congruent,
then the corresponding arcs are congruent” the hypothesis is the first part of the
theorem: “if central angles are congruent,” and the conclusion is the second part:
“then the corresponding arcs are congruent;” in other words, it is given (known to
us) that the central angles are congruent, and it is required to prove that under this
hypothesis the corresponding arcs are congruent.
It is useful to notice that any theorem can be rephrased in such a way that the
hypothesis will begin with the word “it? and the conclusion with the word “then.”
For example, the theorem “vertical angles are congruent” can be rephrased this
way: “if two angles are vertical, then they are congruent.”
The theorem converse to a given theorem is obtained by replacing the hypothesis
of the given theorem with the conclusion (or some part of the conclusion), and
the conclusion with the hypothesis (or some part of the hypothesis) of the given
theorem. For instance, the following two theorems are converse to each other:
If central angles are congruent, If arcs are congruent, then the
then the corresponding arcs are	corresponding central angles are
congruent.	congruent.
If we call one of these theorems direct, then the other one should be called
converse.
In this example both theorems, the direct and the converse one, turn out to
be true. This is not always the case. For example the theorem: “if two angles are
vertical, then they are congruent” is true, but the converse statement: “if two angles
are congruent, then they are vertical” is false.
Indeed, suppose that in some angle the bisector is drawn. It divides the angle
into two smaller ones. These smaller angles are congruent to each other, but they
are not vertical.
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Reading Activity
Ex. 4. Answer the following questions.
1.	What is a principal way to discover properties of geometric figures? 2. Dwell
on the types of propositions. 3. What is a definition? 4. Axioms are statements
that must be proved, aren’t they? 5. What is particular about theorems? 6. Does a
corollary follow directly from a definition or from a theorem? 7. How many parts
can one distinguish in any theorem? 8. Can the hypothesis of one theorem become
the conclusion of the other? 9. Give your own examples of two theorems which are
converse to each other. 10. What is the difference between an assumption and an
axiom?
Ex. 5. Match the English words and word combinations with their Russian
equivalents:
1)	the process of reasoning
2)	to discover properties of figures
3)	what meaning one attributes
4)	to accept without proof
5)	there is a unique line
6)	to lie in the same plane
7)	congruent arcs
8)	the remaining angles
9)	to intersect at one point
10)	under this hypothesis
11)	to be converse to the given theorem
13)	to begin with the words
14)	to draw a bisector
a)	пересекаться в одной точке
b)	по этой гипотезе
c)	существует единственная линия
d)	какое значение придают
e)	процесс рассуждения
f)	начинать со слов
g)	конгруэнтные дуги
h)	обнаружить свойства фигур
i)	принимать без доказательства
j)	остающиеся углы
k)	провести биссектрису
l)	лежать на одной плоскости
т) быть обратным данной теореме
Ех. 6. Find out whether the statements are true or false according to the
information in the text. Use the introductory phrases.
I think, it is right.	I ant afraid, it is wrong.
Quite so. Absolutely correct.	I don't quite agree to it.
I quite agree to it.	On the contrary. Far front it.
1.	In geometry all facts are expressed in the form of formulas.
2.	Two parts are distinguished in any theorem: the proposition and the conclusion.
3.	Scientists discover properties of geometric figures by means of reasoning.
4.	Corollaries follow directly from definitions.
5.	We obtain a converse theorem by replacing the hypothesis of the given theorem
with the conclusion.
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6.	The direct and the converse theorems always turn out to be true.
7.	Axioms are postulates which should be proved.
8.	There are two types of propositions: congruent and central.
9.	In any theorem the hypothesis can begin with the word “if”, and the conclusion
with the word “then”.
Ex. 7. Fill in the blanks with the words from the box. Mind there are two extra
words:
a)	hypothesis
b)	further
c)	theorems
d)	reasoning
e)	accepted
f)	meaning
g)	to discover
h)	conclusion
i)	deduced
j)	remaining
k)	established
l)	propositions
1.	All that is necessary is that the words and phrases used shall have the same ...
for everybody.
2.	The ... that we take for granted without proof are called assumptions.
3.	That which is given is sometimes called the..., and that which is to be proved is
sometimes called the ....
4.	Propositions that can be logically deduced from the assumptions are often
called....
5.	All facts ... in geometry are expressed in the form of propositions.
6.	The answer to a problem in actual life can often be obtained by ... investigation
of the actual facts, while in geometry it can always be obtained by ... alone.
7.	The scientist has two problems - one,... new scientific propositions; the other, to
devise a set of assumptions from which all his propositions can be logically ....
Ex. 8. Match the left and the right parts
1.	If the same quantity is added or
subtracted from equal quantities,
then
2.	If two points of a line lie in a given
plane, then
3.	If each of two quantities is equal to a
third quantity, then
4.	If a number is divisible by 2 and by
3, then
5.	If central angles are congruent, then
6.	If one of the four angles formed by
two intersecting lines is right, then
of the following statements.
a)	these two quantities are equal to
each other.
b)	it is divisible by 6.
c)	the equality remains true.
d)	the remaining three angles are right
as well.
e)	all points of this line lie in the same
plane.
f)	the corresponding arcs are
congruent.
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Ex. 9. Translate the sentences according to the models.
Model 1: There are various ways of evaluating formulae.
Существуют различные способы вычисления формул.
1.	There are a lot of important theorems in this book.
2.	There are sets containing no elements.
3.	There has been recently developed a new method of proving the theorem.
4.	There are many measurements which must be made.
5.	There weren’t any problems with my term paper last year.
6.	There will be enough work for everybody at the next conference.
Model 2: There exist a lot of equivalent relations.
Существует много эквивалентных отношений.
1.	There exists no difference between these two expressions.
2.	There exists at least one element in a non-empty set.
3.	There exist some important statements in the article.
4.	There exist many different ways of defining a circle.
5.	There exist no solutions to the problem presented.
Model 3: To a pair of numbers there corresponds a point in the plane.
Паре чисел соответствует точка на плоскости.
1.	То a linear equation there corresponds a straight line in the Euclidean space.
2.	To a point in three dimensional space there correspond its three coordinates.
3.	To each number in x there corresponds a unique element in y.
4.	To any two objects a, b there corresponds a new object.
5.	If to each member x of a set there corresponds one value of a variable/, then у
is a function of x.
Ex. 10. Revision of Tenses (Active & Passive Voice).
a)	ask general questions with the following auxiliary verbs:
do, did, does, had, have, was.
1.	A lot of students combine work and studies.
2.	Some additional information was required.
3.	I used to read a lot of books in history.
4.	Last Friday I was late for classes because of the heavy rain.
5.	They got married last year.
6.	In Britain most shops close at 5.30 p.m.
7.	I usually have breakfast before I go to work.
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8.	We see each other only occasionally.
9.	Because of his bad answer the student had to solve another problem.
10.	Harry looks very untidy in these dirty jeans.
b)	choose the best English equivalent for the words given in Russian.
1.	He thought that you (занимаетесь) in for swimming.
a)	went	b) go	c) will go
2.	We are sure that they (решили) all the problems yesterday.
a)	were solved	b) are solved	c) solved
3.	In day-to-day life mathematics (используется) in every sphere, from telling the
time to hobbies.
a) are used	b) is used	c) used
4.	The train (отправляется из) London next Friday at 8 a.m. and (прибывает)
in Leeds at 11 a.m.
a) will leave, arrives b) leaves, will arrive c) leaves, arrives
5.	As soon as the classes (окончатся), we shall hurry to the canteen.
a) will be over	b) are over	c) is over
6.	The Dean said that a lot of interesting subjects (изучаются) by the students.
a) were studied	b) will be studied	c) studied
7.	This equation essentially (отличается) from the one which we (решали) last
time.
a) differed, solved b) differs, solved	c) differed, solves
8.	The main thing geometry (дает) us is the idea of a logical system and precise
thinking.
a) is given	b) gave	c) gives
Ex. 11. Rewrite the following passages in the Passive Voice.
A.	Charles Babbage, an English professor of mathematics, built the first computer in
1827. They called it a “Difference Engine”. Babbage also devised the basic principles
of the modern computer. He spent much of his own money on his inventions. In
1834 Babbage designed a more complex “Analytical Machine” - the worlds first
digital computer with a memory and programming, but couldn’t get the finance to
build it. People forgot about Babbages machine till 1937 when they rediscovered
his papers.
B.	The school provides the Internet for students to conduct research and
communicate with others in relation to schoolwork. They give the access to
network service to those students who agree to act in a responsible manner. Hie
staff thinks that access is a privilege, not a right. They expect that the user will
follow the certain rules of behaviour.
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Ex. 12. Ask special questions.
1.	Some properties are established by means of reasoning (how).
2.	Geometry is concerned with the properties and relationships of figures in
space (what... with).
3.	Some figures such as cubes and spheres have three dimensions (how many).
4.	Many discoveries were made in the nineteenth century (when).
5.	Hie truth of non-mathematical propositions in real life is much less certain
(where).
6.	The given proposition and its converse can be stated as follows (in what way).
7.	Pure mathematics deals with the development of knowledge for its own
purpose and need (what... with).
8.	Carl Gauss proved that every algebraic equation had at least one root (who).
9.	There are three words having the same meaning (how many).
10.	Hie given definition corresponds to the idea of uniqueness (what).
Ex. 13. Write the converses of the follow ing propositions. Decide in each case
if it is true or false.
1.	If all three sides of a triangle are equal, the three angles of the triangle are also
equal.
2.	If all four sides of a quadrilateral are equal, the quadrilateral is a parallelogram.
3.	If two triangles are equal, the angles of the two triangles are respectively equal.
4.	If two rectangles are equal, the diagonals of one rectangle are equal to the
diagonals of the other.
5.	If the line segment includes endpoints D and E then it includes all the points
between them.
6.	If the opposite sides of a quadrilateral are parallel and congruent, then the figure
is a rhombus.
7.	If the milkman has come, there are three bottles of milk on the back porch.
8.	If Aunt Marian is coming, we shall have waffles for supper.
9.	A squirrel (белка) is an animal having a thick bushy tail.
Ex. 14. Turn direct speech into reported speech.
1.	Plato advised, “The principal men of our state must go and learn arithmetic,
not as amateurs, but they must carry on the study until they see the nature of
numbers with the mind only”.
2.	Descartes, father of modernism, said, “All nature is a vast geometrical system.
Thus all the phenomena of nature are explained and some demonstration of
them can be given”.
3.	In Descartes’s words, “You give me extension and motion, then I’ll construct the
universe”.
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4.	The often repeated motto on the entrance to Plato’s Academy said, “None
ignorant of geometry enter here”.
5.	J. Kepler affirmed: “The reality of the world consists of its maths relations. Maths
laws are true cause of phenomena”.
6.	I. Newton said, “I don’t know what I may appear to the world; but to myself I
seem to have been only like a boy playing on the seashore, and diverting myself
now and then by finding a smoother pebble or a prettier shell than usual; whist
the great ocean of truth lay all undiscovered before me. If I saw a little farther
than others, it is because I stood on the shoulders of giants”.
Ex. 15. Choose the correct variant of translation.
1.	We thought that you were going to enter an institute.
a)	Мы думали, что вы собираетесь поступить в институт.
b)	Мы думали, что вы собирались поступить в институт.
c)	Мы думали, что вы собирались войти в институт.
2.	Scientists use mathematical formulas to express their findings precisely.
a)	Ученые используют математические формулы, чтобы аккуратно опи¬
сать свои находки.
b)	Ученые используют математические формулы для точного выражения
своих находок.
c)	Ученые используют математические формулы, чтобы точно выразить
полученные данные.
3.	Where there is a choice of two expressions, we should always choose the more
accurate one.
a)	Там, где существует выбор из двух выражений, нам всегда следует вы¬
бирать более точное выражение.
b)	Там, где есть выбор из двух выражений, мы всегда выберем более точ¬
ное выражение.
c)	Там, где есть выбор из двух выражений, мы бы всегда выбирали более
точное выражение.
4.	Assumptions are related to theorems in the same way as undefined terms are
related to definitions.
a)	Допущения были связаны с теоремами таким же образом, как неопре¬
деленные термины связаны с определениями.
b)	Допущения соотносятся с теоремами таким же образом, как неопреде¬
ленные термины соотносятся с определениями.
c)	Допущения зависят от теорем, а также от неопределенных терминов и
определений.
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5.	Very often a proposition is so worded that it requires thought to state the
converse proposition correctly.
a)	Очень часто утверждение формулируется таким образом, что нужно
как следует подумать, чтобы сформулировать обратное утверждение
правильно.
b)	Зачастую утверждение составляется так, что требуется поразмыслить,
чтобы правильно заявить об обратном утверждении.
c)	Очень часто утверждение выражается так, что оно требует размышле¬
ния над правильной формулировкой обратного утверждения.
Ех. 16. Translate the following sentences into English.
1. В данном случае обе теоремы - как прямая, так и обратная - оказывают¬
ся справедливыми. 2. Пять аксиом Евклида - это предложения, вводящие
отношения равенства или неравенства величин. 3. Учебник Евклида по ге¬
ометрии «Начала» читали, читают и будут читать математики. 4. Предложе¬
ние, которое следует непосредственно из аксиомы, называется следствием.
5.	Следующие две теоремы обратны друг другу. 6. Одно и то же утверждение
может быть или не быть истинным относительно другого множества допу¬
щений. 7. В любой теореме есть две части: гипотеза и вывод. 8. Вас просят
записать кратко предположения, которые вы сделали. 9. Аксиома - это ис¬
тинное, исходное положение теории. 10. Постулат - это утверждение, при¬
нимаемое в какой-либо научной теории как истинное, хотя и недоказуемое
ее средствами, и поэтому он играет в ней роль аксиомы.
Ех. 17. Read the text and find the answers to the following questions.
1.	What is logical deduction?
2.	Do we proceed from the general to the particular or from the particular to the
general in induction?
3.	Which method of thinking is more useful: deductive or inductive?
Deduction and Induction
The scientists have proved a chain of theorems and have come to recognize
the entire structure of undefined terms, definitions, assumptions, and theorems
as constituting an abstract logical system. In such a system we say that each
proposition is derived from its predecessor by the process of logical deduction. Hi is
process of logical deduction is scientific reasoning.
Ulis scientific reasoning must not be confused with the mode of thinking
employed by the scientist when he is feeling his way toward a new discovery. At
such times the scientist, curious about the sum of the angles of a triangle, proceeds
to measure the angles of a great many triangles very carefully. In every instance he
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notices that the sum of the three angles is very close to 180°; so he puts forward a
guess that this will be true of every triangle he might draw. Th is method of deriving
a general principle from a limited number of special instances is called induction.
Hie method of induction always leaves the possibility that further measurement
and experimentation may necessitate some modification of the general principle.
The method of deduction is not subject to upsets of this sort.
When the mathematician is groping for (ищет) new mathematical ideas, he
uses induction. On the other hand, when he wishes to link his ideas together into
a logical system, he uses deduction. The laboratory scientist also uses deduction
when he wishes to order and classify the results of his observations and his inspired
guesses and to arrange them all in a logical system. While building this logical
system he must have a pattern (модель) to guide him, an ideal of what a logical
system ought to be. The simplest exposition (изложение) of this ideal is to be
found in the abstract logical system of demonstrative geometry.
It is clear that both deductive and inductive thinking are very useful to the
scientist.
Ex. 18. Writing. Put the sentences into the right order to make a complete
paragraph. Hie first sentence is given to you.
What is Mathematics?
1.	Maths, as science, viewed as a whole, is a collection of branches.
A.	Hie largest branch is that which builds on ordinary whole numbers, fractions,
and irrational numbers, or what is called collectively the real number system.
B.	Hence, from the standpoint of structure, the concepts, axioms and theorems are
the essential components of any compartment of maths.
C.	Hiese concepts must verify explicitly stated axioms. Some of the axioms of the
maths of numbers are the associative, commutative, and distributive properties
and the axioms about equalities.
D.	Arithmetic, algebra, the study of functions, the calculus differential equations
and other various subjects which follow the calculus, in logical order are all
developments of the real number system. This part of maths is termed the
maths of numbers.
E.	Some of the axioms of geometry are that two points determine a line, all right
angles are equal, etc. From these concepts and axioms, theorems are deduced.
F.	A second branch is geometry consisting of several geometries. Maths contains
many more divisions. Each branch has the same logical structure: it begins
with certain concepts, such as the whole numbers or integers in the maths of
numbers or such as points, lines, triangles in geometry.
123

UNIT 10
Причастие (The Participle)
Причастие - неличная форма глагола, имеет признаки как прилагатель-
ного, так и глагола.
формы причастия
Active
Passive
Participle I
doing
being done
Выражает действие, одновременное с
действием глагола-сказуемого
Participle II
—
done
Выражает действие, одновременное
с действием глагола-сказуемого или
предшествующее ему
Perfect Participle
having
done
having been
done
Выражает действие, предшествующее
действию глагола-сказуемого
функции причастия
Participle I Active - doing - в предложении выполняет функцию:
1)	определения (an Attribute)
Hie writing student will be ...	Пишущий студент будет ...
Hie student writing a new programme ... Студент, пишущий новую
программу...
2)	обстоятельства (an Adverbial Modifier)
Solving these problems we must use	Решая эти задачи, мы должны
a new rule.	использовать новое правило.
While/When solving a problem use	Решая (при решении) задачу,
a computer.	используйте компьютер.
3)	части сказуемого (времена группы Continuous и Perfect Continuous)
(a part of Predicate)
Students are considering the properties Студенты рассматривают
of sets.	свойства множеств.
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Participle I Passive - being done - в предложении выполняет функцию:
1)	определения (an Attribute)
The computers being developed now Разрабатываемые сейчас компьютеры
will be extensively used.	будут широко использоваться.
2)	обстоятельства (an Adverbial Modifier)
Being w ritten on time, the article	Будучи написанной вовремя, статья
was published in the journal.	была опубликована в журнале.
3)	части сказуемого (a part of Predicate)
The system which is being tested Система, которую сейчас испыты-
now seems very complicated.	вают, кажется очень сложной.
Предложенный метод использовался в
наших вычислениях.
Метод, предложенный математиком,
использовался в наших вычислениях.
Метод, на который только что сосла¬
лись, представляет большой интерес.
Participle II - done, translated - выполняет функцию:
1)	определения (an Attribute)
The proposed method was used in
our calculations.
The method proposed by the
mathematician was used in our
calculations.
The method just referred to is of
great interest.
2)	обстоятельства (an Adverbial Modifier)
Translated from the language of
mathematics into everyday language
the relation became easier to
understand.
As seen from the results the
information was carefully collected.
When given enough time he will
write his paper.
Unless properly constructed the
device will not be reliable.
Будучи переведенным с языка мате¬
матики на обычный язык, это
соотношение стало легче для
понимания.
Как видно из результатов, инфор¬
мация была тщательно собрана.
Если ему дадут достаточно времени,
он напишет свою статью.
Если прибор неправильно сконст¬
руирован, он не будет надежным.
3)	части сказуемого (времена группы Perfect и Passive Voice) (a part of
Predicate)
He was told about some new
developments in this field of
mathematics.
Ему сказали о новых разработках в
этой области математики.
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Perfect Participle - having done, having been done - выполняет функцию:
1) обстоятельства (an Adverbial Modifier)
Having answered the teachers Ответив на вопросы учителя, сту-
questions the student left.	дент ушел (После того как студент
ответил на вопросы учителя, он
ушел).
Having been given the problem we После того как нам дали задачу, мы
began to analyse it.	начали ее анализировать.
Ex. 1. Read these groups of words and note the function and the form of the
Participle.
a)	1. The moving point is .... 2. The drawing man is .... 3. The line segment joining
points A and В .... 4. The scientist measuring distance....
b)	1. Hie student proving the correctness of the statement.... 2. Mathematicians
using symbols instead of words.... 3. A post-graduate collecting statistical data ....
4.	The researcher testing the new method ....
c)	1. Hie divided angle remained ... . 2. Hie named geometric objects ... . 3. Hie
extended line was .... 4. Hie expected information will be derived ....
d)	1. Hie points referred to as ... . 2. Hie two lines drawn parallel will never ... .
3.	Hie work continued the following day showed.... 4. This object taken as a model
served....
e)	1. The statement made is consistent with ... . 2. Hie problem dealt with seems
important ... . 3. Hie calculations made were accurate ... . 4. Hie number added
equals....
f)	1. Drawing a geometric figure one must.... 2. Finding the measure of an angle
you can ... . 3. When realizing this plan we .... 4. While considering the example
he ... . 5. When applying these rules one must remember ... . 6. While checking
these operations he....
g)	1. When asked about the date he ... . 2. When applied carefully this method
may .... 3. If changed a little, the problem .... 4. If continued further, the work....
h)	1. Being drawn carefully the figure will be ... . 2. Being multiplied the fraction
will not.... 3. Being published the article was .... 4. Being given the dimensions of
an object one can ....
i)	1. Having reduced the fraction the student... .2. Having obtained the expected
results the scientist.... 3. Having compared the results he could .... 4. Having been
asked to find the solution to the problem they....
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Ex. 4. Translate from Russian into English, using the Participles.
1.	Переходя улицу, будьте внимательны.
2.	Будучи очень усталыми, мы отказались идти на прогулку.
3.	Большое дерево, сломанное ветром, лежало поперек дороги.
4.	На собрании, проходящем сейчас в соседней комнате, обсуждается ряд
важных вопросов.
5.	Я покажу тебе статью, написанную моим научным руководителем.
6.	Получив хороший учебник, он смог быстро повторить сложную тему.
7.	Покажите мне список студентов, выполняющих эту лабораторную
работу.
8.	Составляя телеграмму, мы должны употреблять как можно меньше слов.
9.	Книги, прочитанные в детстве, кажутся старыми друзьями.
10.	Услышав об изменении погоды, они надели теплые куртки.
Pre-Reading Activity
Guess the meaning of the following words.
Capital [ksepztsl], subject ['sA2Z<zkt], fundamental [/Arzds'mestl], discuss
[скн'клн], mechanical [znz'ksszzzkl], direction [dz'rekki], perpendicular
[zp9 p=r.'2ikyjzls], interval [Zz2t=vs;].
Read and learn the basic vocabulary
location (n) [Lcz/kekz]
point (v, n) [pczr.z]
dot (n) [det]
dimension (n) [dz'mer/ss]
space (n) [spszs]
exact (adj) [zp'zskt]
exactly(adv) [zp'zssktk]
refer (v) (to) [rz'fs ]
mark (n) [222 k]
undefined (adj)
description (n) [dzs'kzzp Ч2]
length (n) [IszrG]
thickness (n) ['Gzkzzes]
depth (n) [dep 9 ]
extend (v) [zks'ter.d]
flat (adj) [fist]
surface (n) ['ss fzs]
terms:
определение местонахождения, место
указывать, точка
точка
измерение, размеры (pl.)
пространство, космос
точный
точно
ссылаться (на), иметь отношение
отметка, знак
неопределенный
описание
длина
толщина
глубина
простираться
плоский
поверхность
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infinitely (adv)
characteristic (adj) [zksr:kts'r:st:k]
intersect (v) [,:r.ts'sekt]
right (adj) [rait]
angle (n) ['sr.gl]
coincide (v) [zksczr/saiz]
vertex (n) ['vs tsks]
sine (n) [sais]
interior (n) [:r/::sr:s]
measure (n,v) ['irises]
degree (n) [di'cri ]
acute (adj) [s'k1":]
obtuse (adj) [sb't1" s]
straight (adj) ]
bisect (v) [bai'ssk:]
align (v) [s".a:r.]
protractor (n) [crs'tzsskts]
бесконечно
особенность
пересекаться
правильный, правый, прямой
угол
совпадать
вершина
синус
внутренняя часть
мера, измерять
степень, порядок, градус
острый
тупой (угол)
прямой, прямолинейный
делить пополам
ставить в ряд, выравнивать
транспортир
Memorise the following word combinations:
to be referred to as - называться
to extend forever - продлевать бесконечно
coplanar lines - компланарные линии
skew lines - ассиметричные линии
adjacent angles - смежные углы
complementary angles - взаимодополняемые углы (до 90°)
Reading Activity
POINTS, LINES, PLANES AND ANGLES
Hie most fundamental idea in the study of geometry is the idea of a point.
Think of a point as an exact location in space, it has no dimensions. When writing
about points, you represent the points by dots. Remember, the dot is only a picture
of a point, and not the point itself. Points are commonly referred to by using capital
letters. The dots mark points and are referred to as point A, point В and point C.
A line is one of the undefined terms in geometry. A description of a line is that
it has length but no thickness or depth. In theory, a line may be extended infinitely
in each direction.
A plane is a flat surface that extends infinitely in all directions. Imagine
extending the length and width of a table top forever.
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Lines that lie in the same plane are called coplanar lines. Any two coplanar
lines must have one and only one of the characteristics listed below.
Intersecting Perpendicular
Lines	Lines
Parallel Lines
Lines Which
Coincide
•	The lines may intersect. If they intersect and
form right angles, they are perpendicular lines.
•	The lines may be parallel. Parallel lines will
never meet.
•	Hie lines may coincide. Lines that coincide are
actually the same lines.
Lines that lie in different planes and do not
intersect are called noncoplanar lines or skew
lines.
•	If two planes do not intersect, the planes are
parallel.
• If two planes intersect, their intersection is a line.
An angle is formed by two rays that have the same endpoint, which is called
the vertex of the angle. The rays are called the sides of the angle. (A ray is a part of
a line drawn from a given point called the endpoint. Hie ray continues forever in
the other direction.) A point between the sides of the angle is in the interior of the
angle. “	” is the symbol for angle. To name an angle use three letters. The center
letter corresponds to the vertex. The other two letters are points on each ray.
Hie angle can be named ABC or CBA. It can be read as “angle ABC or angle CBA.
An angle is measured in degrees with an instrument called a protractor.
Hiere are five types of angles that are
essential to the study of geometry.
Acute angle - an angle whose measure
is less than 90°.
Right angle - an angle whose measure
equals 90°. A box in the vertex denotes
a right angle.
Obtuse angle - an angle whose
measure is greater than 90° and less than
180°.
Straight angle - an angle whose
measure equals 180°.
Reflex angle - an angle whose measure
is greater than 180° and less than 360°.
Acute Angle
Right Angle
Obtuse Angle
Straight Angle
Reflex Angle
•	Equal angles are angles that have the
same number of degrees.
•	A ray that bisects an angle divides it into 2 equal parts. Hie line is called the
angle bisector.
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•	Congruent angles have the same measure.
•	Perpendiculars are lines that form right angles.
•	All right angles are congruent.
•	The sides of a straight angle lie on a straight line.
•	All straight angles are congruent.
•	A perpendicular bisector of a line bisects the line and is perpendicular to
the line.
Post-Reading Activity
Ex. 5. Answer the following questions.
1.	What does this text deal with? 2. What is the most fundamental idea in the
study of geometry? 3. What is a point? 4. What do we usually use the letters of the
alphabet for? 5. What is a description of a line? 6. Does a plane extend infinitely in
all directions? 7. What lines are called coplanar? 8. What characteristics can you
list for coplanar lines? 9. What lines are called noncoplanar? 10. How is an angle
formed? 11. What is a ray? 12. Is there any special symbol to denote an angle?
13. What are the ways of naming an angle? 14. How is an angle measured? 15. What
kinds of angles do you know, and what are their degree measures?
Ex. 6. Match the English words and word combinations with their Russian
equivalents:
1)	the undefined term
2)	to extend indefinitely
3)	the vertex of the angle
4)	the interior of the angle
5)	distinguishing features
6)	the exterior part
7)	unless stated otherwise
8)	reflex angle
9)	perpendicular bisector
10)	adjacent angles
a)	вершина угла
b)	отличительные черты
c)	если не указано иное
d)	неопределенный термин
e)	внутренняя часть угла
f)	продлеваться бесконечно
g)	внешняя часть
h)	смежные углы
i)	угол между 180 °C и 360 °C
j)	перпендикулярная биссектриса
Ех. 7. Find out whether the statements are true or false according to the
information in the text. Use the introductory phrases.
I think, it is right.
Quite so. Absolutely correct.
I quite agree to it.
I ant afraid, it is wrong.
I don't quite agree to it.
On the contrary. Far front it.
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1.	A point has length, width or thickness.
2.	A line is limited and extends infinitely in one direction.
3.	A line unless stated otherwise is understood to be straight.
4.	A line is the shortest distance between two points.
5.	A surface has length and width, it doesn’t have thickness.
6.	Equal angles are angles that have the same number of degrees.
7.	Right angles are not congruent.
8.	A perpendicular bisector of a line bisects the line and is perpendicular to the
line.
9.	If two planes intersect, their intersection is a line.
10.	A point is a location and it has size.
11.	The size of the angle depends on the lengths of the rays forming it.
Ex. 8. Use the correct form of the Participle.
1.	(to name) geometric ideas we usually use letters of the alphabet.
2.	We insisted on the (to follow) notation of the geometric object.
3.	(to divide) both the numerator and the denominator by x you will get the
following expression.
4.	When (to speak) with my science adviser I got better understanding of the
latest development in my special field.
5.	The properties of the material (to use) in the experiment now are given in the
latest article.
6.	The advantages of the new system (to prove) by many tests are very important.
7.	Two angles (to have) the same vertex and a common side are refered to as
adjacent angles.
8.	The concepts (to introduce) at the seminar should be considered in detail.
9.	The (to obtain) difference must be checked carefully.
10.	The (to expect) result must prove that this law holds for similar cases.
Ex. 9. Fill in the blanks with the necessary words.
1.	To (измерить) an angle, compare its sides to the corner of this page.
2.	The corner represents (прямой угол), whose measurement is 90°.
3.	If the angle is smaller than the corner, the angle is (острый угол).
4.	If the opening is larger than the corner of the page, the angle is (тупой). Its
measure is more than 90°.
5.	Locate the point of your (транспортир) which represents the (вершина) and
align the vertex with the point.
6.	Rotate the protractor keeping the vertex aligned until one (сторона) of the
angle is on the 0°-180° line of the protractor.
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7.	The angle measure (определяется) by the side of the angle that is not on the
0°-180° line of the protractor.
8.	You may have to (продлить) one side of the angle so that it crosses the scale.
9.	Use the proper (обозначение), tn is the symbol for “measure of”.
Ex. 10. Match the left and the right parts of the following statements.
1.	A group of two angles is known a) two angles whose measures add up
2.	Adjacent angles are
3.	Vertical angles
4.	Complementary angles are
5.	One angle
6.	Supplementary angles are
7.	If an angle is cut into two adjacent
angles
8.	If the exterior sides of a pair of
adjacent angles are perpendicular
9.	Vertical angles are
to 180°.
b)	two nonadjacent angles formed by
two intersecting lines.
c)	is the complement of the other.
d)	two angles whose measures add up
to 90°.
e)	as a pair of angles.
f)	two angles that have the same vertex
and a common side.
g)	are congruent.
h)	then the sum of the measures of the
adjacent angles equals the measure
of the original angle.
i)	then each angle is a right angle.
Ex. 11. Mind the use of the Continuous and Perfect Continuous Tenses.
1.	I (to look for) a photograph my brother sent to me.
2.	They (to have) a meeting now.
3.	The phone always (to ring) when I (to have) a bath.
4.	Friends always (to talk) to me when I (to try) to concentrate.
5.	He (to watch) television when the door bell (to ring).
6.	He (to build up) his business all his life.
7.	They (to stay) with us for a couple of weeks.
8.	By 1992 he (to live) there for ten years.
9.	The video industry (to develop) rapidly.
10.	He (to work) nights next week.
11.	She (to spend) this summer in Europe.
12.	Why are you so late? I (to wait) for you since morning.
13.	The boys must be tired. They (to play) football in the garden all afternoon.
14.	The old town theatre is currently (to rebuild).
15.	I usually (to go) to work by car, but I (to go) on the bus this week while my car
(to repair).
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Ex. 12. Complete each of the sentences below by choosing one of the pronouns
in brackets.
1.	... arrived in good time and the meeting started promptly at 3.30. (anybody/
nobody/everybody)
2.	... in the village went to the party but... enjoyed it very much, (everybody/no
one/some one), (anybody/somebody/nobody)
3.	... heard anything, (everyone/ nobody/ somebody)
4.	“Who shall I give this one to? - You can give it to ... . It doesn’t matter.”
(everyone/nobody/anybody)
5.	That’s a very easy job.... can do it. (everyone/nobody/somebody).
6.	Would you like ... to drink? (anything/something/nothing)
7.	I thought I’d seen you .... (anywhere/somewhere/nowhere)
8.	There was ... to hide, (anywhere/somewhere/nowhere)
9.	You still haven’t told me .... (anything/something/nothing)
10.	Does ... agree with me? (anybody/somebody/nobody)
11.	I want to introduce you to .... (no one/ someone/ any one)
12.	Hie box was completely empty. There was ... in it. (nothing/anything)
13.	“Excuse me, you’ve dropped ... .Yes, look. It’s passport.” (something/anything/
everything)
14.	It’s all finished. I am afraid there’s ... left, (nothing/anything/something)
15.	I heard a noise, but I didn’t see .... (any one/no one)
16.	It’s too late. We can’t do ... to help, (anything/nothing)
17.	I agree with most of what he said, but I don’t agree with ... . (something/
everything/ anything)
18.	... offered to help. They probably didn’t have the time, (anybody/nobody/
everybody)
19.	If... asks, you can tell them I’ll be back soon, (somebody/anybody/everybody)
Ex. 13. Ask special questions to which the sentences below are the answers.
1.	A statement satisfying certain conditions is true, (what)
2.	Like terms being arranged in the following way will be enclosed in the
parentheses, (where)
3.	Reference is made to the commonly accepted system, (what... to)
4.	The force keeping all material bodies including people on the Earth is called
gravitation, (what kind)
5.	Having used the classification suggested by my science adviser I found it very
convenient, (when)
6.	Having been given little information they couldn’t continue the research, (why)
7.	Having followed the procedure they obtained the required results, (how)
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8.	Any fraction represents the quotient of its numerator divided by its
denominator, (what)
9.	Having obtained a proper interpretation of this fact they realized the
importance of the problem, (when)
10.	The created method has no advantages over the old one. (what)
Ex. 14. Find the corresponding Russian sentence.
1.	Geometry is a branch of mathematics concerned with questions of shape, size,
relative position of figures, and the properties of space.
a)	Геометрия - это область математики, которая рассматривала форму,
размер, относительное расположение фигур и свойства пространства.
b)	Геометрия - это область математики, имеющая отношение к вопро¬
сам формы, размера, расположения фигур относительно друг друга и
свойств пространства.
c)	Геометрия - это раздел математики, в котором рассматривали форму,
размер, относительное расположение фигур и свойства пространства.
2.	From what you already know you may deduce that drawing two rays originating
from the same end point forms an angle.
a)	Из того, что вы уже знаете, вы можете сделать вывод, что, рисуя два
луча, исходящих из одной конечной точки, вы получаете угол.
b)	Из того, что вам известно, вы можете сделать вывод, что изображение
двух лучей, берущих начало в одной и той же конечной точке, образует
угол.
c)	Из того, что вы уже узнали, вы, возможно, сделали вывод, что рисунок
двух лучей, берущих начало в одной конечной точке, образует угол.
3.	The approach to the problem being considered remained traditional.
a)	Рассматривался оставшийся подход к традиционной проблеме.
b)	Подход к оставшейся проблеме рассматривался традиционно.
c)	Подход к рассматриваемой проблеме оставался традиционным.
4.	Physical facts expressed in terms of mathematics do not seem unusual nowadays.
a)	Выраженные математические факты казались необычными в физиче¬
ских терминах в настоящее время.
b)	Физические факты, выраженные в математических терминах, не ка¬
жутся необычными сегодня.
c)	То, что физические факты в настоящее время выражаются математиче¬
скими терминами, не кажется сегодня необычным.
5.	Having made a number of experiments Faraday discovered electromagnetic
induction.
а)	При проведении ряда экспериментов Фарадей открыл электромагнит¬
ную индукцию.
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b)	Проведя ряд экспериментов, Фарадей открыл электромагнитную ин¬
дукцию.
c)	Так как Фарадей проделал ряд экспериментов, он сумел открыть элек¬
тромагнитную индукцию.
Ех. 15. Translate into English.
1.	Первая линия, с которой мы знакомимся, изучая математику, - это пря¬
мая линия.
2.	Дать строгое определение этого понятия совсем непросто.
3.	В работах Евклида (Euclid) линия определялась как длина без толщины.
4.	Угол - самая простая геометрическая фигура после точки, прямой, луча
и отрезка.
5.	Если на плоскости из точки О провести два различных луча ОА и ОВ,
то они разделят плоскость на две части, каждая из которых называется
углом с вершиной О и сторонами ОА и ОВ.
6.	Луч, делящий угол пополам и берущий начало в вершине угла, называется
его биссектрисой.
7.	Биссектриса развернутого угла делит его на два смежных угла, назы¬
ваемых прямыми углами.
8.	Большое значение для теории и практики имеет определение величины
или меры угла.
9.	Основное свойство меры угла должно заключаться в том, чтобы равные
углы имели одинаковую меру.
10.	Градусная мера используется в элементарной геометрии для измерения
углов.
11.	Каждый, наверное, знаком с транспортиром - измерителем углов на чер¬
тежах.
12.	Углы меньше прямого называются острыми, а углы больше прямого, но
меньше развернутого называются тупыми.
13.	Первая книга Евклида начинается с 23 «определений», среди них такие:
точка есть то, что не имеет частей; линия есть длина без ширины; линия
ограничена точками; наконец, две прямые, лежащие в одной плоскости,
называются параллельными, если они, сколь угодно продолженные, не
встречаются.
14.	Изложение геометрии в «Началах» Евклида считалось образцом, которо¬
му стремились следовать ученые и за пределами математики.
Ех. 16. Read the text and answer the follow ing questions.
1. How did Euclid define a point? 2. How is a point represented in two dimensional
space and in three dimensional space? 3. Is Euclids postulate about points
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confirmed under modern developments of Euclidean geometry? 4. Can one say
that Euclids postulation of points was complete and definite?
Points in Euclidean Geometry
Points are most often considered within the framework of Euclidean geometry,
where they are one of the fundamental objects. Euclid originally defined the
point vaguely, as “that which has no part”. In two dimensional Euclidean space, a
point is represented by an ordered pair (x, y) of numbers, where the first number
conventionally represents the horizontal and is often denoted by x, and the second
number conventionally represents the vertical and is often denoted by y. This
idea is easily generalized to three dimensional Euclidean space, where a point
is represented by an ordered triplet, (x, y, z) with the additional third number
representing depth and often denoted by z.
In addition to defining points and constructs (построения) related to points,
Euclid also postulated idea about points; he claimed that any two points can be
connected by a straight line. This is easily confirmed under modern developments
of Euclidean geometry, and had lasting consequences at its introduction, allowing
the construction of almost all the geometric concepts of the time. However, Euclids
postulation of points was neither complete nor definite, as he occasionally assumed
facts about points that didn’t follow directly from his axioms, such as the ordering
of points on the line or the existence of specific points. In spite of this, modern
developments of the system serve to remove these assumptions.
UNIT 11
Герундий (The Gerund)
формы герундия
Active
Passive
Indefinite
writing
being written
Выражает действие,
одновременное с действием
глагола-сказуемого
Perfect
having written
having been written
Выражает действие,
предшествующее действию
глагола-сказуемого
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I am surprised at
Меня удивляет то,
a)	his solving problems so quickly.
как быстро он решает задачи.
b)	his having solved the problem so quickly.
как быстро он решил эту задачу.
Герундий употребляется после
предлогов:
глаголов:
remember
excuse
thank
forgive j
+V-ing
(Gerund)
On/upon >
In
After
Before
Without 1
► +V-ing
(Gerund)
On obtaining the data the scientist went
on working.
Получив данные, ученый продолжил
работу.
After developing this
system we were
Разработав
эту систему, мы смог-
able to get good results.
ли получить хорошие результаты.
I remember learning this important
Я помню, что учил это важное ма¬
mathematical property.
тематическое свойство.
Excuse me for not attending your
Извините меня за то, что я не
lectures. I was ill.
ходил на ваши лекции. Я был болен.
Действительные формы герундия используются после глаголов: need
(нуждаться), want (нуждаться), require (требовать), deserve (заслуживать)
and an adjective worth (стоящий).
Your suggestion is worth paying attention Твоему предложению стоит уде-
to.	лить внимание.
His English requires improving.	Его английский нужно улучшить.
функции герундия
Подлежащее
(Subject)
Speaking English is easier than
writing.
Its no use waiting.
There is no knowing what may
happen.
Говорить по-английски легче,
чем писать.
Ждать бесполезно.
Неизвестно, что может
произойти.
Прямое дополнение
(Direct Object)
I remember preventing him
from making such a mistake.
He denied having been there.
Я помню, что помешал ему
совершить такую ошибку.
Он отрицал то, что был там.
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Предложное
дополнение
(Prepositional Object)
I look forward to meeting my
science adviser.
The students are accustomed
to using capital letters
to name geometric objects.
Я с нетерпением жду
встречи с моим научным
руководителем.
Студенты привыкли
использовать заглавные
буквы для обозначения
геометрических объектов.
Определение
(Attribute)
There are different ways of
solving this problem.
She hasn’t got any difficulty in
understanding the basic
geometric theorems.
Существуют различные
способы решения этой
задачи.
У нее нет никаких проблем
с пониманием основных
теорем геометрии.
Обстоятельство
(Adverbial Modifier)
In spite of being very tired he
continued his work.
After passing their exams
the students went on holiday.
Несмотря на то что он
был очень уставшим, он
продолжил работу.
Сдав экзамены, студенты
отправились на каникулы.
Именная часть
сказуемого
(Predicative)
Seeing is believing.
Our task is proving the
correctness of the given
statement.
Увидеть - значит поверить.
Наша задача - доказать
корректность данного
утверждения.
Часть составного
глагольного
сказуемого (Part of
a compound verbal
predicate)
When he entered the room I
couldn’t help smiling.
He has finished dictating
a letter.
Когда он вошел в комнату, я
не могла не улыбнуться.
Он окончил диктовать
письмо.
Герундий в функции прямого дополнения
Verb + -ing
to excuse - прощать, извинять
to finish - заканчивать
to forgive - прощать
to give up - бросать
to go on
to keep (on) > продолжать
to continue ,
to avoid - избегать, уклоняться,
стараться не делать
to enjoy - получать удовольствие,
to deny - отрицать
to risk - рисковать
to stop - прекращать
to suggest - предлагать
to postpone - откладывать
to mind - возражать, иметь
что-либо против
to imagine - воображать,
представлять
to dislike - не любить, испытывать
нравиться
неприязнь
139

Translate into Russian.
He tried to avoid answering the question.
On holiday, I enjoy not having to get up early.
Don’t keep interrupting me. I’m speaking.
She suggests going to the sea.
Герундий в функции дополнения с предлогом
a) verb + preposition + -ing
to accuse of - обвинять в
to aim at - стремиться к
to approve of - одобрять что-либо
to charge with - обвинять в
to complain of - жаловаться на
to depend on - зависеть от
to feel like - испытывать желание,
хотеть
to give up - бросать, прекращать
to insist on - настаивать на
to object to - возражать, быть против
чего-то
to rely on - полагаться на, доверять
to thank for - благодарить за
to suspect of - подозревать в
to think of - думать о
to agree to - соглашаться c
to succeed in - преуспевать в,
удаваться
to persist in - упорствовать,
упорно продолжать
to prevent from - мешать,
препятствовать
to look forward to - ждать
с нетерпением
to look like - похоже, что
Translate into Russian.
Do you feel like going out this evening?
Dave insisted on helping me.
He apologized to Sue for being rude to her.
I don’t approve of their watching a lot of TV.
b) to be + adjective (or Participle II) + preposition + ing
to be accustomed to - быть привыкшим к, приученным к
to be aware of - знать, сознавать, отдавать себе полный отчет в
to be capable of - быть способным на
to be engaged in - быть вовлеченным в, заниматься чем-либо
to be fond of - любить что-либо, нравиться
to be good at - быть способным, умелым, искусным в
to be guilty of - быть виновным, виноватым
to be indignant at - возмущаться, негодовать
to be interested in - интересоваться чем-либо
to be pleased at - радоваться, испытывать удовольствие от
140

to be proud of - гордиться чем-либо
to be responsible for - отвечать за, быть ответственным за
to be suitable for - быть подходящим для, годным для
to be sure of - быть уверенным в
to be surprised at - удивляться чему-либо
to be tired of - быть уставшим от
to be used to - привыкнуть к
Translate into Russian.
She is not capable of doing the work.
Are you interested in collecting coins?
We are all tired of listening to his complaints.
I am not accustomed to sleeping during the day.
Устойчивые выражения с герундием
a)	It’s no use, its no good - бесполезно, не стоит
There’s no point in - не имеет смысла
It’s (not) worth - (не) стоит, (не) заслуживает
to have difficulty (in) - столкнуться с трудностями, испытывать проблемы
I can’t (couldn’t) help - не могу не (не мог не)
can’t stand - не выносить кого-либо, не выдерживать что-либо
can’t bear (inf. also possible) - не переносить что-либо
b)	a waste of money - пустая трата денег
a waste of time - пустая трата времени
to be busy - быть занятым
to go swimming/go fishing (after go for activities) - заниматься плаванием,
ходить на рыбалку
Translate into Russian.
There is no point in persuading him.
I couldn’t help being late. My train was delayed by fog.
I can’t stand being fooled down.
It’s a waste of money buying this dress.
She is busy translating the text.
Следующие глаголы используются как с инфинитивом,
так и с герундием
а) без разницы в переводе
to attempt - пытаться, пробовать
to begin - начинать
to intend - намереваться, иметь в виду
to bother - надоедать, беспокоить
to propose - предлагать
to start - начинать
to continue - продолжать
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It began raining, or It began to rain.
Но такие глаголы, как знать, понимать, употребляются в форме инфинитива.
I began to understand.
b)	употребляются и герундий, и инфинитив, но со значительной разницей в
переводе
go on + -ing	go on + to
(продолжить делать что-то)	(начать делать что-то новое)
She went on talking about her illness for She told me about her son and then
hours.	she went on to talk about her other
problems.
stop + -ing	stop + to
(бросить, прекратить делать	(остановиться, чтобы)
что-то)
I’ve stopped smoking.	I stopped for a few minutes to rest.
c)	герундий используется для выражения действий в прошлом и привычек;
инфинитив - для будущих действий и в условных предложениях
to like - нравиться	to remember - помнить
to love - любить, нравиться	to regret - сожалеть
to hate - ненавидеть, не любить	to prefer - предпочитать
Translate into Russian.
I like climbing (habit). I would like to climb the top of this mountain.
I hate getting up early. I would hate to spend the night alone in the woods.
I remember seeing it on the notice-board. I must remember to post the letter.
Ex. 1. Choose the correct form.
1.	She had the feeling of....
a) being deceived	b) deceiving	c) having deceived
2.	Its a waste of time ... over trifles.
a) having argued	b) having been argued c) arguing
3.	My watch doesn’t keep good time. It needs ....
a) having been repaired b) being repaired c) repairing
4.	He mentioned ... it in the paper.
a) being read	b) reading	c) having read
5.	Is it worth while your ... to convince him of being wrong?
a) being tried	b) trying	c) having tried
142

6.	He insisted on ... with a certain respect.
a) having been treated	b) treating	c) being treated
7.	Father didn’t approve of my ... the offer.
a) having rejected	b) having been rejected c) rejecting
8.	Many apologies for not... to your letter.
a) having replied	b) replying	c) being replied
9.	She remembers ... him the message.
a) having been given	b) giving	c) being given
10.	I’m really looking forward to ... all your news.
a) being heard	b) having heard c) hearing
Ex. 2. Join the two sentences to make one sentence, beginning with a gerund.
Model: Shes a teacher. Its hard work.
Being a teacher is hard work. / Teaching is hard work.
1.	Capital letters are used to name geometrical objects. It is very convenient.
2.	You are to classify these quadrilaterals. It requires the knowledge of some
properties.
3.	We are going to locate this point on they axis. It will give us the first point on
the line.
4.	The student intends to divide a circle into a certain number of congruent parts.
It will help him to obtain a regular polygon.
5.	The base and the altitude of a rectangle are to be multiplied. It will give the
product of its dimensions or the area of the rectangle.
6.	Don’t argue! It’s no use. In a crossed quadrilateral, the interior angles on either
side of the crossing add up to 720°.
7.	Don’t deny this fact! It is useless. A square is a quadrilateral, a parallelogram,
a rectangle and a rhombus.
8.	You are going to divide a heptagon (a 7-sided polygon) into five triangles. Is it
any good?
Ex. 3. Choose the right preposition. .Make sensible sentences.
1.	Are you interested
2.	She is very good
3.	He insisted
4.	I apologize
5.	The teacher is fed up
6.	She succeeded
7.	My friend is keen
on
of
to
at
in
with
for
a)	disturbing you.
b)	looking after the children.
c)	learning foreign languages.
d)	having more time for doing things
he wants to.
e)	understanding this - its too
difficult.
143

8.	Professor is looking forward
9.	Tliis student is not capable
10. His sister is tired
in
on
of
f)	answering our stupid questions.
g)	studying.
h)	considering his solution of the
problem.
i)	doing sums.
j)	getting good education.
Ex. 4. Complete the sentences using a gerund as an attribute.
1.	I didn’t very much like the idea of....
2.	What is the purpose of... ?
3.	She had no difficulty (in)....
4.	You have made great progress in ....
5.	He was late, and he was afraid of....
6.	Can you imagine the pleasure of....
7.	He always produces the impression of....
8.	I am afraid you do not realize the importance of....
Ex. 5. Complete the second sentence so that it has a similar meaning to the first
one. Use the word in bold and other words to complete each sentence.
1.	I’ll be happy when I can have a rest after exams,
forward to
I’m looking ... a rest after exams.
2.	Learning new geometric theorems is something I like doing,
interested in
I’m always ... new geometric theorems.
3.	If I study a lot at night, it keeps me awake.
prevents from
... a lot at night... sleeping.
4.	I often operate the computer at university.
am used to
I... the computer at university.
5.	He didn’t want to take the books back to the library,
feel like
He didn’t... the books back to the library.
6.	He hates it if he has to do a lot of boring exercises.
can’t stand
He ... a lot of boring exercises.
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7.	Tm sorry. I’ve broken the speed limit’, said Sue.
apologized for
Sue ... the speed limit.
8.	Let us write a new program.
suggest
I ... a new program.
Ex. 6. Find and correct the mistakes in the sentences. Some of them are right
sentences.
1.	I’m looking forward to go on holiday.
2.	To cheat in examination is not allowed.
3.	It was kind of you inviting me joining you.
4.	It’s a waste of time watching TV.
5.	She said she was too busy to do this.
6.	Do you think that drawing a polygon is easier than drawing a circle?
7.	Please stop to make that noise, it’s driving me mad.
Pre-Reading Activity
Guess the meaning of the following words.
Classify [zklss:fa:], positive [Zpcz:::v], exterior [eksz::sr:s], polygon ['pckssr.],
parallel ['pssrs's'.J, parallelogram	rhombus [rcmbss],
diagonal [zaz'sssr. si], perimeter [p s'rzmirs ], positive [Zpcz:::vj.
Read and learn the basic vocabulary terms:
many-sided (adj) [rrisr.fsaidid]
closed (adj) [kkzzz]
according to (prep) [s'kcz:-]
quadrilateral (n) [,kwcz7ikss:=rsl]
pentagon (n) ['psczszsz]
hexagon (n) ['heksscszj
heptagon (n)
octagon (n) ['ckrsszsr.J
nonagon (n) ['zz'zzszsr.J
decagon (n) ['zskszsz]
regular (adj) [Z7sz"zls]
certain (adj) [zsa tsz]
connect (v) [kszr.ek:]
inscribe (v) [:r/skra:b]
многосторонний
закрытый, замкнутый
согласно
четырехугол ьн и к
пятиугольник
шестиугольник
семиугольник
восьмиугольник
девятиугольник
десятиугольник
правильный, регулярный
определенный, некий
связывать, соединять
вписывать
145

tangent (n) ['tszchsz:]
intersection (n) [:r.:sziek's]
circumscribe (v) [zss ksmsk:aib]
stand for (v) [iTssr.d]
emphasize (v) [Zs~iissa.iz]
separate (v) ['sepsrei:]
rectangle (n) ['rek/s-.cl]
square (n) [skwss]
trapezium (trapezoid) (n) [trs'p: z’sm]
convex (adj) ['kcz'veks]
concave (adj) [zkcr/ke:v]
adjacent (sides) (adj) [s' b e:s]
deltoid (adj, n) ['dsltcizj
kite (n) [ka::]
supplementary (adj) [zSApl:ziriSz:sr:]
consecutive (adj) [ksr/sek'“t:v]
касательная, тангенс
пересечение
описывать
символизировать, означать
подчеркивать, делать ударение
отделять
прямоугольник
квадрат
трапеция
выпуклый
вогнутый
смежный, примыкающий
дельтовидный, дельтоид
гладкий ромб
дополнительный
последовательный
Memorise the following word combinations:
to be mostly concerned with - главным образом интересоваться
the process of constructing - процесс построения
to have no difficulty listing - не иметь затруднений с составлением списка
it should be pointed out - следует отметить
lets dwell on - давайте остановимся на
to have in common - иметь общее
may prove helpful - может оказаться полезной
to be closely related to - быть тесно связанным с
it is worth remembering - стоит запомнить
crossed quadrilaterals - пересекающиеся четырехугольники
Reading Activity
REGULAR POLYGONS. SPECIAL QUADRILATERALS
In this chapter we’ll be mostly concerned with studying plane figures called
polygons. Polygons are many-sided figures, with sides that are line segments. These
simple closed figures are named according to the number of sides and angles they
have, and may be classified by the measures of the angles or the measures of the
sides. The simplest polygon is a triangle, a geometric plane figure having three
sides. We have no difficulty listing all the polygons having up to ten sides. In the
picture below you can see some of them.
146

A polygon is called regular if all of its sides and all of its interior angles are
congruent. For instance, a square is a regular quadrilateral having four right
angles and four equal sides.
It should be pointed out that the process of constructing a regular polygon is
closely related to division of a circle into congruent parts. Students of mathematics
will remember two major theorems concerning this problem.
Theorem. If a circle is divided into a certain number (greater than 2) of
congruent parts, then:
(1)	connecting every two consecutive division points by chords, we obtain
a regular polygon inscribed into the circle.
(2)	Drawing tangents to the circle at all the division points and extending
each of them up to the intersection points with the tangents at the nearest division
points we obtain a regular polygon circumscribed about the circle.
To find the sum of the interior angles of any polygon one can use the formula
sum of the angles = (n-2) 180° (where n stands for the number of sides or angles).
For a triangle the sum is 180°. By drawing all diagonals from one single vertex of
a polygon we can separate it into triangles. If you look back at the formula, you
will see that n - 2 gives the number of triangles in the polygon, and that number
is multiplied by 180°, which is the sum of the measures of all the interior angles in
a triangle.
To find the perimeter of a regular polygon you should multiply the length of
the sides by the number of sides.
Now let us dwell on polygons called quadrilaterals. In Euclidean plane
geometry a quadrilateral is a polygon with four sides and four vertices or corners.
Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting),
147

also called crossed. Simple quadrilaterals are either convex or concave. A kite is
a convex trapezium that has two congruent pairs of adjacent sides. A deltoid is
a concave trapezium.
The five most common types of quadrilaterals are the parallelogram, the
rectangle, the square, the trapezoid and the rhombus. All quadrilaterals have
some things in common. All of them 1) have four sides, 2) are coplanar, 3) have
two diagonals and 4) the sum of their four interior angles equals 360°.
In addition, some of quadrilaterals possess special properties. As an example
lets take a parallelogram. Its special properties are as follows:
•	Opposite sides are parallel.	• Opposite angles are congruent.
•	Consecutive pairs of angles are	• Opposite sides are congruent,
supplementary.	• Diagonals bisect each other.
Below is a summary of the types of quadrilaterals. You might think of a
quadrilateral like this: every square is a rectangle, but not every rectangle is a square.
A rectangle is also a parallelogram, but a parallelogram may not be a rectangle.
Such a classification may prove helpful. It is worth remembering.
Post-Reading Activity
Ex. 7. Answer the following questions.
1.	What is a polygon? 2. In what way do we classify polygons? 3. What polygon
is called regular? 4. How is the process of constructing a regular polygon related
to division of a circle into congruent parts? 5. Is it possible to obtain a regular
polygon inscribed into the circle? (a regular polygon circumscribed about the
148

circle) 6. Which formula is used for finding the sum of the interior angles of any
polygon? 7. How can we find the perimeter of a regular polygon? 8. What are the
most common types of quadrilaterals? 9. How many things do all quadrilaterals
have in common? 10. What special properties does a parallelogram possess? 11. Do
you know any other classifications of quadrilaterals? Are they worth remembering?
Ex. 8. Match the English words and word combinations writh the Russian
equivalents:
1)	to draw diagonals
2)	from one single vertex
3)	in addition to
4)	to circumscribe about the circle
5)	adjacent sides
6)	to draw tangents to the circle
7)	a convex or a concave trapezium
8)	stands for
9)	special quadrilaterals
10)	regular polygons
11)	supplementary angles
12)	bisect each other
13)	the measures of the sides
14)	a plane figure
15)	possess properties
16)	a consecutive pair
17)	to inscribe into the circle
18)	complex or crossed quadrilaterals
a)	делят друг друга пополам
b)	специальные четырехугольники
c)	дополнительные углы
d)	плоская фигура
e)	вписать в окружность
f)	описать вокруг окружности
g)	означает
h)	выпуклая или вогнутая трапеция
i)	последовательная пара
j)	начертить диагонали
k)	провести касательные к окруж¬
ности
l)	обладают свойствами
т) кроме того
п)	из одной вершины
о)	смежные стороны
р)	сложные или пересекающиеся
четырехугольники
q)	величины сторон
г) правильные многоугольники
Ех. 9. Fill in the blanks with the words from the box. Mind there are twTo extra
wrords:
a)	rectangle
b)	regular
c)	vertices
d)	line segments
e)	corners
f)	vertex
g)	convex
h)	concave
i)	special
j)	product
k)	rhombus
l)	dimensions
m)	trapezoid
n)	congruent
o)	interior
p)	quadrilateral
q)	diagonals
r)	number
s)	inscribed
t)	circumscribed
u)	kite
v)	length
149

1.	A simple closed figure formed by ... is called a polygon.
2.	In Euclidean plane geometry, a ... is a polygon with four sides and four ... or ....
3.	The area of a rectangle figure is the ... of its ....
4.	A shape that is both a ... and a ... is a square (four equal sides and four equal
angles).
5.	A polygon is called ... if all of its sides and all of its interior angles are ....
6.	Applying these geometric theorems we can obtain both a regular polygon ...
into the circle and a regular polygon ... about the circle.
7.	Finding the sum of the ... angles of a polygon is not difficult.
8.	If you wish to find the perimeter of a regular polygon you should multiply ...
of the sides by the ... of the sides.
9.	By drawing all ... from one single ... of a polygon we can separate it into
triangles.
10.	Simple quadrilaterals are either ... or ....
11.	A parallelogram possesses ... properties.
Ex. 10. Guess what figure possesses the following properties and memorise
them (square, trapezoid, kite, rectangle, parallelogram, rhombus).
1.	A ... has two parallel pairs of opposite sides.
2.	A ... has two pairs of opposite sides parallel, and four right angles. It is also
a parallelogram, since it has two pairs of parallel sides.
3.	A ... has two pairs of parallel sides, four right angles, and all four sides are equal.
It is also a rectangle and a parallelogram.
4.	A ... is defined as a parallelogram with four equal sides. It does not have to have
4 right angles.
5.	A ... only has one pair of parallel sides. It’s a type of quadrilateral that is not
a parallelogram.
6.	A ... has two pairs of adjacent sides that are equal.
Ex. 11. Find out whether the statements are true or false. Use the introductory
phrases.
I think it is right.	I am afraid it is wrong.
Quite so. Absolutely correct.	I don't quite agree to it.
I quite agree to it.	On the contrary. Far from it.
1.	Plane figures bounded by four sides are called triangles.
2.	The area of a square is the product of the length of its two sides.
3.	A rectangle is a parallelogram that has four obtuse angles.
4.	Every interior angle in a convex polygon has a measure greater than 180°.
150

5.	Rhombus is a parallelogram with four congruent angles.
6.	Every square is a rectangle, but not every rectangle is a square.
7.	To find the perimeter of a regular polygon, divide the length of the sides by the
number of sides.
8.	The area of a geometric figure is a quantity expressed by negative numbers.
9.	A quadrilateral is a square if and only if it is both a rhombus and a rectangle.
10.	The base and the altitude of a rectangle are called its dimensions.
Ex. 12. Ask special questions using the question words in brackets.
Venn Diagram
1.	Let us use a Venn diagram to group the types of quadrilaterals (why). 2. A Venn
diagram uses overlapping (частично совпадающие) circles. It shows relationships
between groups of objects (what). 3. All quadrilaterals can be separated into three
sub-groups: general quadrilaterals, parallelograms and trapezoids (how many).
4.	Since all four sides of a rectangle don’t have to be equal a rectangle isn’t always a
rhombus (why). 5. However, the sets of rectangles and rhombuses intersect (which).
6. Their intersection is the set of squares (whose). All squares are both a rectangle
and a rhombus. 7. We can put squares in the intersection of the two circles (where).
8.	From this diagram, you can see that a square is a quadrilateral, a parallelogram, a
rectangle, and a rhombus (from what). 9. A trapezoid isn’t a parallelogram because
it has only one pair of parallel sides (how many). 10. That is why we must show
the set of trapezoids in a separate circle on the Venn diagram (in what way). Let’s
consider kites. 11. Kites are quadrilaterals that can be parallelograms, (what type
of). 12. If its two pairs of sides are equal, it becomes a rhombus (in what case).
151

Ex. 13. Choose the best alternative to the English sentence.
1.	They’re used to coming back late.
a)	Они часто возвращались домой поздно.
b)	Они привыкли возвращаться поздно.
c)	Они воспользовались своим поздним возвращением.
2.	I can’t help doing it myself.
a)	Ничем не могу помочь, так как делаю это сама.
b)	Не может быть, что я сделала это сама.
c)	Не могу не сделать это сама.
3.	Не regrets leaving school at fourteen.
a)	Он сожалеет о том, что ушел из школы в 14 лет.
b)	Он сожалеет о том, что уходит из школы в 14 лет.
c)	Покидая школу, он сожалел, что ему только 14 лет.
4.	I object to being treated like a child.
a)	Я боюсь лечения как ребенок.
b)	Я против того, чтобы со мною обращались как с ребенком.
c)	Я возражал, чтобы меня лечили как ребенка.
5.	I feel like going for a walk.
a)	Я чувствую, что мне нравится ходить на прогулки.
b)	Вероятно, я пойду на прогулку.
c)	Я хочу (испытываю желание) пойти на прогулку.
Ех. 14. Translate into English.
1.	Простая замкнутая фигура, образованная отрезками, называется многоу¬
гольником. 2. Само слово «многоугольник» состоит из двух греческих слов:
“poly” - много, “gon” - угол. 3. Правильный многоугольник - это многоу¬
гольник, у которого все углы и стороны конгруэнтны. 4. Простейшим мно¬
гоугольником является треугольник, имеющий три стороны, три угла и три
вершины. 5. Пятиугольник имеет пять сторон, шестиугольник - шесть, вось¬
миугольник - восемь, десятиугольник - десять. 6. Многоугольники могут
быть сгруппированы в соответствии с величиной их углов и сторон. 7. Четы¬
рехугольник - это также плоская фигура. 8. Если стороны четырехугольника
равны и все углы прямые, то такой четырехугольник называется квадратом.
9.	Линия вокруг плоской фигуры называется периметром. 10. Периметр пло¬
ской фигуры равен сумме длин его сторон.
152

Ex. 15. Let us revise modal verbs and their equivalents.
Complete the sentences with have to, has to, had to and the verbs from the box.
Some of them are interrogative or negative:
do, get, help, do , take, wear, learn, study, correct, look.
1.	“I don’t want to take exams.” - “You ... them. You have no choice.”
2.	Why... I	this exercise? It’s really boring.
3.	Yesterday he ... me with my homework. It was difficult.
4.	She is going to bed now. She ... up early tomorrow morning.
5.	What uniform ... they	in Oxford university?
6.	The task was very important, and he understood that he ... it in a hurry.
7.	She ... everything by heart.
8.	He doesn’t speak English very well. He ... more.
9.	How often ... your teacher	your homework?
10.	Some students ... up a lot of words in their dictionaries.
Ex. 16. Choose the correct modal verb to fill in each gap.
(Modals of deduction/criticism.)
1.	You ... have had a terrible fright. You’re still trembling.
a) might	b) should	c) must
2.	I... be able to come. I’ll have to ask my parents though.
a) must	b) might	c) could
3.	That... be Mary. She is supposed to be at university.
a) shouldn’t	b) can’t	c) ought not to
4.	You ... take a coat. It’s going to get cold later.
a) can	b) shall	c) should
5.	He ... have forgotten again. I reminded him about fifty times.
a) can’t	b) mustn’t	c) shan’t
6.	That... be Michael. He always gets home from work at about this time.
a) may	b) must	c) could
7.	You ... have told me. I would have got you a present.
a) must	b) might	c) should
8.	She ... know. I certainly haven’t told her.
a) might not	b) can’t	c) mustn’t
153

9.	You ... have spent it. I only gave it to you yesterday.
a) might not	b) can’t	c) shouldn’t
10.	They... have seen the film. It’s been on for a couple of weeks.
a) might	b) can	c) ought to
Ex. 17. Choose the correct modal verb.
1.	You must/have to get a good night’s sleep before the exam in trigonometry.
2.	She needn't/rnustn't find the sum of the interior angles of this polygon. She has
already found it.
3.	I didn't need to/couldrit solve the problem after all - it was too difficult for me.
4.	How wonderful! I don't have to/mustn't revise the properties of special
quadrilaterals. I know them very well.
5.	What you should/may have done is to learn formulas for perimeter and areas
for basic figures.
6.	Her student has done so little work, he shouldn't/needn't have bothered to
come to class today.
7.	You could/will be able to focus on understanding the basic geometric formulas
and their applications at the next lecture.
8.	One may/can remember an old and wise saying “Practice makes a man perfect”.
9.	Geometry сап/ought to be made quite fascinating if you follow some rules
while studying it.
10.	Riemann was able to!had to collect and systematize his work at the end of the
19th century.
11.	You are allowed/should bear in mind that every plane section of a sphere is a
circle.
Ex. 18. Find and correct the mistakes.
1.	I checked the timetable so I mustn’t be wrong about the time of my lessons.
2.	You needn’t to worry about carrying out this experiment.
3.	Do I ought to use a ruler for drawing a line? I may do without it.
4.	You must having read about the Arabs who gave us a present form for writing
fractions.
5.	You need spend as much time as you can on writing your report.
6.	May you tell me where I may get Euclid’s chief work Elements?
7.	You mustn’t write anything down unless you want to.
8.	Don’t be silly, you not ought to use this inaccurate and restricted method for
obtaining a trigonometric function.
9.	Finding the area of the circle was a problem for her. You must have given her
the formula.
10.	It was Sunday and he stayed at home. He needn’t have gone to university.
154

Ex. 19. Read the text and find the answers to the following questions.
1.	What are the main properties of the areas of geometric figures? 2. What numbers
is the area of a geometric figure expressed by? 3. How is measuring areas done?
4. What is called the dimensions of a rectangle?
Areas of Polygons
We all have some idea about the quantity called area, from everyday life. We
will establish here more precisely the concept of area of geometric figures, and
develop methods for its computation. Assume that the area of a geometric figure is
a quantity, expressed by positive numbers, and is well-defined for every polygon.
Further assume that the areas of figures possess the following properties:
1.	Congruent figures have equal areas. Figures of equal area are sometimes
called equivalent. Thus, according to this property of areas, congruent figures are
equivalent. The converse can be false: equivalent figures are not always congruent.
2.	If a given figure is partitioned into several parts, then the number expressing
the area of the whole figure is equal to the sum of the numbers expressing the areas
of the parts. This property of areas is called additivity. It implies, that the area of
any polygon is greater than the area of any other polygon enclosed by it.
3.	The square, whose side is a unit of length, is taken for the unit of area,
i.e. the number expressing the area of such a square is set to 1. When the unit of
length is taken to be, say, 1 meter (centimeter, foot, inch, etc.), the unit square
of the corresponding size is said to have the area of 1 square meter (respectively
square centimeter, square foot, square inch, etc.), which is abbreviated as 1 m2
(respectively crn2,/t2, m2, etc.). Measuring areas is done not by direct counting of
unit squares or their parts fitting into the measured figure, but indirectly, by means
of measuring certain linear sizes of the figure. Let us agree to call one of the sides of
a triangle or parallelogram the base of those figures, and a perpendicular dropped
to this side from the vertex of the triangle, or from any point of the opposite side of
the parallelogram, the altitude. In a rectangle, the side perpendicular to the base
can be taken for the altitude. In a trapezoid, both parallel sides are called bases, and
a common perpendicular between them, an altitude. The base and the altitude of
a rectangle are called its dimensions.
Theorem. The area of a rectangle is the product of its dimensions.
Tliis brief formulation should be understood in the following way: the number
expressing the area of a rectangle in certain square units is equal to the product of
the numbers expressing the length of the base and the altitude of the rectangle in
the corresponding linear units. It should be pointed out that the lengths of the base
and the altitude (measured by the same unit) are expressed by whole numbers.
155

UNIT 12
Инфинитив (The Infinitive)
Формы инфинитива
Active
Passive
Indefinite
to write
to be written
Выражает действие, одновременное
с действием глагола-сказуемого
Continuous
to be writing
-
Выражает одновременное длитель¬
ное действие
Perfect
to have
written
to have been
written
Выражает действие, предшествующее
действию глагола-сказуемого (пере¬
водиться прошедшим временем)
Perfect
Continuous
to have been
writing
-
Выражает предшествующее длитель¬
ное действие
Функции инфинитива
Подлежащее
(subject)
То learn two foreign languages
simultaneously is difficult.
Its useless to discuss the
question.
Изучать два иностранных
языка одновременно трудно.
Бесполезно обсуждать этот
вопрос.
Часть сказуемого
(part of Predicate)
Her task was to pass the
exams.
He began to read the book two
weeks ago.
Ее задача состояла в том,
чтобы сдать экзамены.
Он начал читать книгу две
недели назад.
Дополнение
(object)
He doesn’t like to be asked
personal questions.
Ему не нравится, когда
задают вопросы личного
характера.
Определение
(attribute)
(инфинитив пере¬
водится с оттенком
модальности или
будущности)
The problem to be discussed
is interesting.
He was the first to do the
exercise.
There are some important
things to be considered at the
lesson.
Проблема, которая будет
обсуждена (нужно, пред¬
стоит обсудить), интересна.
Он первым сделал это
упражнение.
Существует несколько
важных моментов, которые
нужно рассмотреть на уроке.
156

Обстоятельство
(adverbial modifier)
I'm studying English in order
(so as) to get a better job.
To understand the importance
of this event you should know
all the facts.
Я изучаю английский язык
для того, чтобы устроиться
на лучшую работу.
Чтобы понять важность
этого события, вы должны
знать все факты.
Часть составного
модального
сказуемого
(part of compound
Modal predicate)
He must be working now.
She must have translated the
text.
They must have been reading
the text-book for an hour.
He may (might) be at his
studies now.
She may (might) have done
the exercise.
Он, вероятно, сейчас рабо¬
тает.
Должно быть, она уже пере¬
вела текст.
Они, вероятно, читают
этот учебник уже в течение
часа.
Возможно, он на занятиях
сейчас.
Она, возможно, выполнила
упражнение.
Глаголы,
после которых
употребляется
инфинитив:
agree, refuse,
promise, threaten,
offer, attempt,
manage, fail, decide,
plan, arrange,
hope, appear, seem,
pretend, afford,
forget, learn (how),
dare, tend
We promised not to be late for
our classes.
He hoped to solve the problem
at once.
Мы пообещали не
опаздывать на занятия.
Он надеялся решить
проблему сразу.
Ex. 1. Read these sentences and state the form and the function of the Infinitive.
Translate into Russian.
1. To solve the equation was not difficult for her. 2. The speaker at the conference
didn’t like to be interrupted. 3. The article is difficult to translate. 4. They must
have attended his lecture before. 5. He is always the first to come to the University.
6.	The method to be applied is rather complicated. 7. He worked hard in order
not to be behind the other students. 8. Hie topic may have been considered at the
previous lesson. 9. Our aim is to extend the definition. 10. It isn’t easy to speak any
foreign language. 11. He must be improving his knowledge of mathematics. 12. Hie
scientist might have been working on this problem for many years.
157

Ex. 2. Open the parentheses and give the correct form of the Infinitive.
1.	I am glad (read) this book now.
2.	I hope (award) a scholarship for the coming semester.
3.	He is happy (work) at this company for more than five years.
4.	He does not like (interrupt) by anybody.
5.	Ann was surprised (pass) the exams.
6.	Hie question is too unexpected (answer) at once.
7.	I want (solve) these equations.
8.	This theorem was the first (prove).
9.	She might (forget) to translate the text yesterday.
10.	The question must (settle) an hour ago.
11.	Hie article is (write) in time.
12.	(Understand) the situation one must (know) the details.
Ex. 3. Complete the sentences by using infinitives. Supply a preposition after
the Infinitive if necessary. Hie first is done for you.
1.	I’m planning to fly to the USA next year.
2.	Hie student promised not... late for the lecture.
3.	I need ... my homework tonight.
4.	I want... computer games after my classes.
5.	He intends ... a programmer when he graduates from the university.
6.	I hope ... all of my courses this term. So far my grades have been pretty good.
7.	I try ... class on time every day.
8.	I learned (how)... when I entered the university.
9.	I like ... a lot of e-mails from my friends.
10.	I hate ... in front of a large group.
11.	My roommate offered ... me with my English.
Ex. 4. Write the correct form (gerund or infinitive) of the verbs given in
parentheses. Sometimes more than one answer is possible.
1.	He regrets (not study) harder when he was at school.
2.	The teacher was very strict and nobody dared (talk) during his lessons.
3.	She suggested (go) to the University by taxi.
4.	(learn) English involves (speak) as much as you can.
5.	(Solve) this equation multiply each term in it by the quantity preceding it.
6.	On (obtain) the data the scientists went on working.
7.	The procedure (follow) depends entirely on the student.
8.	This equation must (solve) at the previous lesson.
9.	Euclid was the first (bring) all the known facts about geometry into one whole
system.
158

10.	We don’t mind (give) further assistance.
11.	Hie method (apply) is rather complicated.
12.	(prove) this theorem means (find) a solution for the whole problem.
13.	Students are (study) the laws of mathematics and mechanics.
Pre-Reading Activity
Guess the meaning of the following words.
Segment ['issrrisr.z], mathematician [,rri3e9rris':z ki], formula ['fc zrjzzls'],
hypotenuse [ha:zpctr/~z], cosine ['kc^sazs], special [zspeV], Pythagorean
[pi:,S2=;s'.-:; ; s'], sine ['sa:r.].
Read and learn the basic vocabulary terms:
triangle (n) ['zrazsszzcl]
shape (n) ['’szp]
edge(n) [s ]
denote (v) [dzzrzc'2z]
determine (v) [dz'zs
relative (adj) ['relszzv]
equilateral (adj) [zi kwflstsrsl]
isosceles (adj) [az'scsskz]
namely (adv) ['r.szzrkz]
whereas (cj) [wssz'sz]
define (v) [dz'fazzz]
scalene (adj) ['skezkzz]
internal (adj) [zr.'rs r_I]
external (adj) [zk'srs r_l]
formerly (adv) [zfc sssiz]
leg(n) ['.ec]
obey (v) [s'bez]
square (n, v) [skwss]
property (n) [Zp.-c-s:z]
therefore (adv) [z?ssfc ]
degenerate (adj) [dd zher.Grzr]
compasses (n) [Zk.\22p=-5zz]
straightedge (n) [Zstzsz:sz<]
base (n) [d ezs ]
area (n) ['sszzs]
arbitrary (adj) [Zz bzrzszz]
треугольник
форма, вид, очертание
грань, край
обозначать, значить
определять, устанавливать
относительный, соответственный
равносторонний
равнобедренный
а именно, то есть
тогда как
определять
нера вносторон н и й, разносторон н и й
внутренний
наружный, внешний
прежде
катет
подчиняться
квадрат (величины), возводить
в квадрат
свойство
поэтому, следовательно
вырожденный
циркуль
угольник
основание
площадь
произвольный, случайный
159

bound (v) [ba'Jf.d]
derive (from) (v) [d:'ra:v]
depend (on, upon) (v) [zfpszz]
trace (to) (v) [:re:s]
cathetus (n) ['kssGsrsiJ (pl. catheti)
ограничивать
происходить(от)
зависеть (от)
восходить (к), находить, прослежи¬
ваться)
катет (катеты)
Memorise the following word combinations:
a right triangle - прямоугольный треугольник
an acute triangle - остроугольный треугольник
an obtuse triangle - тупоугольный треугольник
an oblique triangle - косоугольный треугольник
a regular polygon - правильный многоугольник
Reading Activity
TRIANGLES
A triangle is one of the basic shapes of geometry: a polygon with three corners
or vertices and three sides or edges which are line segments. A triangle with vertices
A, B, and C is denoted A ABC.
There exist different types of triangles. They can be classified:
1)	according to the relative lengths of their sides:
•	In an equilateral triangle all sides have the same length. An equilateral
triangle is also a regular polygon with all angles measuring 60°. It is possible
to construct an equilateral triangle of a given side length using just compasses
and a straightedge.
•	In an isosceles triangle two sides are equal in length. An isosceles triangle
also has two angles of the same measure; namely, the angles opposite the
two sides of the same length; this fact is the content of the Isosceles triangle
theorem.
• In a scalene triangle all sides are unequal. The three angles are also all
different in measure. Some (but not all) scalene triangles are also right
triangles.
Equilateral Isosceles	Scalene
160

2)	according to their internal angles, measured here in degrees:
•	A right triangle (or right-angled triangle, formerly called a rectangled
triangle) has one of its interior angles measuring 90° (a right angle). The side
opposite the right angle is the hypotenuse; it is the longest side of the right
triangle. The other two sides are called the legs or catheti of the triangle.
•	Triangles that do not have an angle that measures 90° are called oblique
triangles.
•	A triangle that has all interior angles measuring less than 90° is an acute
triangle or acute-angled triangle.
•	A triangle that has one angle that measures more than 90° is an obtuse
triangle or obtuse-angled triangle.
•	A triangle with an interior angle of 180° (and collinear vertices) is degenerate.
Right	Obtuse	Acute
ч	v	/
Oblique
Triangles have some distinctive properties which are common to all types. Hie
two most common properties are: 1) the interior angles of a triangle always add up
to 180°; 2) the exterior angles of a triangle always add up to 360°.
The area of a triangle can be calculated by using the following geometric
formula: area of a triangle = !6 base x heigh.
Hie perimeter of a triangle is equal to the sum of its three sides.
For all triangles, angles and sides are related by the law of cosines and law of
sines (also called the cosine rule and sine rule).
Hie law of cosines is a statement about a general triangle that relates the lengths of
its sides to the cosine of one of its angles. Hiis law states that c2 = a2 + b2 - 2ab cosy,
where у denotes the angle contained between sides of lengths a and b and opposite
the side of length c.
Hie law of cosines generalizes the Pythagorean theorem, which holds only for
right triangles: if the angle у is a right angle, then cosy = 0, and thus the law of
cosines reduces to с2 = a2 + Ь2.
161

The law of cosines is useful for computing the third side of a triangle when two
sides and their enclosed angle are known, and in computing the angles of a triangle
if all three sides are known.
The law of sines is an equation relating the lengths of the sides of an arbitrary
triangle to the sines of its angles. According to the law, —-— =	= —-— , where
sin A sin В sinC
а,	by and c are the lengths of the sides of a triangle, and A, B, and C are the opposite
angles. Tie law of sines can be used to compute the remaining sides of a triangle
when two angles and a side are known.
Post-Reading Activity
Ex. 5. Answer the following questions.
1.	What is a triangle? (Give the definition of a triangle?) 2. How many types of
triangles are there? 3. According to what can they be classified? 4. What triangle is
called equilateral? 5. How many sides are equal in length in an isosceles triangle?
б.	In what triangle are all three angles different in measure? 7. Does a right triangle
contain three right angles? 8. What triangles are referred to as oblique? 9. Can
any triangle be called acute-angled? 10. How do we call a triangle with an interior
angle of 180°? 11. Do triangles have any properties which are common to all types?
12.	How can the area and the perimeter of a triangle be calculated?
Ex. 6. Fill in the blanks with the words from the box. Mind there are two extra
words:
a) equilateral
e) right
i) hypotenuse
b) isosceles
f) legs
j) Pythagorean theorem
c) interior
g) oblique
k) degenerate
d) scalene
h) triangle
1) acute
1.	A ... is one of the basic shapes of geometry.
2.	In a ... triangle all sides are unequal.
3.	In an ... triangle two sides are equal in length.
4.	A ... triangle has one of its interior angles measuring 90°.
5.	Tie other two sides are called the ... of the triangle.
6.	Right triangles obey the ....
7.	In an ... triangle all sides have the same length.
8.	A triangle with an interior angle of 180 is ....
9.	Triangles that do not have an angle that measures 90° are called ... triangles.
10.	The side opposite to the right angle is the ....
162

Ex. 7. Find out whether the statements are true or false according to the
information in the text. Use the introductory phrases:
I think, it’s right.	Гт afraid, it is wrong.
Quite so. Absolutely correct.	I don’t quite agree to it.
I quite agree to it.	On the contrary. Far front it.
1.	A triangle is a polygon with three vertices and three edges which are not line
segments.
2.	Triangles can be classified according to the relative lengths of their sides and
according to their external angles.
3.	An equilateral triangle is a regular polygon with all angles measuring 60°.
4.	An isosceles triangle has three angles of the same measure.
5.	In a scalene triangle its three angles are all different in measure.
6.	Triangles having an angle that measures 90° are called oblique triangles.
7.	The longest side of the right triangle is called the hypotenuse.
8.	A triangle having one angle that measures more than 90° is an acute triangle.
9.	A triangle with an exterior angle of 180° is degenerate.
10.	Triangles have only one distinctive property which is common to certain types.
11.	The interior angles of a triangle always add up to 360°.
12.	The perimeter of a triangle is equal to the sum of its three sides.
Ex. 8. Match the English words and word combinations with their Russian
equivalents:
1)	regular polygon
2)	to construct an equilateral triangle
3)	to define an isosceles triangle
4)	different in measure
5)	to find a scalene triangle
6)	to measure in degrees
7)	the right angle
8)	an interior angle
9)	the legs of the right triangle
10)	to construct an oblique triangle
11)	distinctive properties
12)	height and base of a triangle
a)	отличительные свойства
b)	прямой угол
c)	высота и основание треугольника
d)	внутренний угол
e)	измерять в градусах
f)	найти неравносторонний
треугольник
g)	строить равносторонний
треугольник
h)	построить косоугольный
треугольник
i)	катеты прямоугольного
треугольника
j)	различные по величине
k)	правильный многоугольник
l)	определять равнобедренный
треугольник
163

Ex. 9. Complete the grid with the nouns from the text.
1)	the side opposite to the right angle
2)	area or a figure having four equal sides and right angles
3)	corner points of any figure or angle
4)	form or figure
5)	a measurement of distance or dimension;
6)	the unit of angle
7)	a plane figure that is bounded by a closed path or circuit;
8)	the measurement of vertical distance
9)	the other name for “catheti”
Down
10)	figures that have three sides and three angles
Ex. 10. Match the left and the right parts of the sentences.
1.	Hie interior angles of a triangle
2.	A triangle is a polygon with
3.	In a scalene triangle
4.	A right-angled triangle has
5.	Hie three angles of a scalene triangle
6.	A triangle with an interior angle of 180'
7.	Hie exterior angles of a triangle
8.	In an equilateral triangle
9.	In an isosceles triangle
10.	Angles and sides in triangles are
related by
a)	are all different in measure.
b)	is degenerate.
c)	two sides are equal in length.
d)	always add up to 360°.
e)	the cosine rule and sine rule.
f)	all sides have the same length.
g)	always add up to 180°.
h)	three vertices and three edges
i)	all sides are unequal.
j)	one of its interior angles
measuring 90°.
164

Ex. 11. Let’s revise Perfect Tenses. Complete the sentences using the words from
the box:
already, before, ever, for, just, by, since, so, still, yet, never.
1.	Have you ... dreamt of going to London?
2.	I haven’t worked out how to set the timer on the video ....
3.	My dad’s lived in the same house ... he was born.
4.	The film’s only been on ... a couple of minutes.
5.	Kate has passed three exams out of five ... far.
6.	He will have translated the text... 3 o’clock tomorrow.
7.	He’s only ... got home.
8.	It’s eleven o’clock and he ... hasn’t come home. Where could he be?
9.	I’ve ... met Ann .... What’s she like?
10.	He has ... finished doing his homework.
Ex. 12. Transform the sentences from Perfect Active into Perfect Passive.
I.	She has just typed her report for the conference. 2. The teacher told us that she
had checked all the tests. 3. The student will have written his degree work by May.
4.	They have learnt a lot of new English words. 5. He hasn’t found the answer yet.
6. I’ve just received my exam results. 7. By the end of the conference, the participants
had discussed a number of important questions concerning the problem. 8. They
will have read two books on topology by the end of the month. 9. We had planned
the meeting months in advance, but we still had problems. 10.1 had discussed the
plan of my work with my science adviser before the end of the class.
Ex. 13. Find mistakes in the following sentences. Mind the use of Perfect Tenses
in the Active and Passive Voice.
1.	They finished their experiment by 5 o’clock yesterday.
2.	The production of such computers has reduced by the end of the previous year.
3.	I can’t do the exercise. I had forgotten my text-book at home.
4.	The article just translates by all the students.
5.	By the time Kate returned from her studies, her brother goes to his friends.
6.	His graduation paper will present by 3 o’clock tomorrow.
7.	He is doing this work by tomorrow;
8.	The solution for the problem is found by the end of the meeting yesterday.
9.	The students already pass their credits.
10.	She is written her course-paper by next month.
II.	Hie advantages of this program already spoke of by the scientists at the
conference.
12.	The algorithm carefully hadn’t wrorked out at the recent seminar yet.
165

Ex. 14. Ask special questions using question words given in parentheses.
The Development of Geometry
1.	The earliest recorded beginnings of geometry can be traced to early predecessors,
(to whom) 2. They discovered obtuse triangles in the ancient Indus Valley and
ancient Babylonia from around 3000 BC. (where, when) 3. Early geometry was
a collection of empirically discovered principles concerning lengths, angles, areas,
and volumes, (what collection) 4. In geometry a spatial point is a primitive notion
upon which other concepts may be defined, (where) 5. Points have neither volume,
area, length, nor any other higher dimensional analogue, (what) 6. In branches of
mathematics dealing with a set theory, an element is often referred to as a point,
(where, how) 7. A point could also be defined as a sphere which has a diameter of
zero, (how)
Ex. 15. Choose the correct variant of translation.
1.	It is difficult to study a foreign language.
a)	Это трудный иностранный язык для изучения.
b)	Трудно изучать иностранный язык.
c)	Изучать иностранный язык было трудно.
2.	Не hopes to pass his examination in mathematical analysis.
a)	Он надеется сдать экзамен по математическому анализу.
b)	Он надеялся на сдачу экзамена по математическому анализу.
c)	Он будет надеяться на сдачу экзамена по математическому анализу.
3.	She was writing the dictation very carefully in order not to make mistakes.
a)	Она написала диктант очень внимательно, не сделав ошибок.
b)	Она писала диктант очень тщательно и в правильном порядке, не делая
ошибок.
c)	Она писала диктант очень внимательно, чтобы не сделать ошибок.
4.	Гт sorry not to have seen this film in English at the lesson.
a)	Мне жаль, что на уроке я не посмотрела этот фильм на английском языке.
b)	Я сожалею о том, что не посмотрю этот английский фильм на уроке.
c)	Я не сожалею о том, что не посмотрела этот фильм на уроке английского
языка.
5.	Не read the rule several times to understand it better.
a)	Он читает правило несколько раз, чтобы понять его лучше.
b)	Он прочитал правило несколько раз, чтобы лучше понять его.
c)	Он читал правило несколько раз и понимал его лучше.
166

6.	Tliis is just the person to speak to on this problem.
a)	Вот человек, о котором говорится в этой проблеме.
b)	Это как раз тот человек, с которым можно поговорить на эту тему.
c)	Только с этим человеком говорят об этой проблеме.
Ех. 16. Translate these sentences from Russian into English.
1.	Треугольник - это плоская фигура, ограниченная тремя линиями и
содержащая три угла.
2.	Треугольники бывают равносторонние, разносторонние, равнобедрен¬
ные.
3.	Треугольник с вершинами N, О, Р обозначается Л NOP.
4.	У равнобедренного треугольника два угла имеют одинаковую величину.
5.	Название треугольника происходит от латинского слова «триангулум» =
= треугольный.
6.	Существуют семь видов треугольников в зависимости от формы и
градусной меры углов.
7.	Равносторонний треугольник - это треугольник, у которого три стороны
равны.
8.	Другие математики определяют равнобедренный треугольник как
треугольник, по крайней мере, с двумя равными сторонами.
9.	Площадь треугольника может быть вычислена при помощи формулы.
10.	Треугольник, у которого все внутренние углы меньше 90°, является
остроугольным.
Ех. 17. Read the text below and find the answers to the following questions.
1. Was Pythagoras the first to know about the theorem that bears his name? In
what was he the first? 2. What is the earliest indicator showing knowledge of
the relationship between right triangles? 3. Who had a mechanical device for
demonstrating the converse of the Pythagorean Theorem? 4. Who apart from
the Egyptians knew of specific instances of the Pythagorean Theorem? 5. Is the
Pythagorean Property true for all right triangles? 6. What must one do to prove
that c2 = a2 + b2 for the triangle under consideration?
The Pythagorean Theorem
Pythagoras was not the first in antiquity to know about the remarkable theorem
that bears his name, but he was the first to formally prove it using deductive
geometry and the first to actively ‘market’ it (using todays terms) throughout the
ancient world. One of the earliest indicators showing knowledge of the relationship
167

between right triangles and side lengths is a hieroglyphic-style picture, of a knotted
rope (связанная узлом верёвка) having twelve equally-spaced knots (узел).
The rope was shown in a context suggesting its use as a workmans tool (рабочий
инструмент) for creating right angles, done via (через, посредством) the
fashioning (придание вида, формы) of a 3-4-5 right triangle. Thus, the Egyptians
had a mechanical device for demonstrating the converse of the Pythagorean
Theorem for the 3-4-5 special case.
Not only did the Egyptians know of specific instances (примеры, отдельные
случаи) of the Pythagorean Theorem, but also the Babylonians and Chinese some
1000 years before Pythagoras definitively institutionalized (устанавливать на
практике) the general result circa (приблизительно) 500 ВС.
The Pythagorean Theorem is a central theorem which states: in any right
triangle the sum of the squares of the lengths of the two legs is equal to the square
of the length of the hypotenuse: a2 + b2 = c2, where a and b are the lengths of the
legs and c is the length of the hypotenuse. The Pythagorean Property is true for all
right triangles. There exist several proofs of the Pythagorean Theorem. Lets discuss
one of them.
A rectangle encloses the basic right triangle as shown below. The three triangles
comprising the rectangle are similar, allowing the unknown dimensions x,y, to be
solved via similarity principles in terms of a, b, and c. Once we have x, y, and z in
hand, the proof proceeds as a normal dissection (разбиение, рассечение).
A Rectangular Dissection Proof:
1)	xlb = ale -> x- able, ylb = b/c => у - b2!c & z!a - ale ->z- a2!c,
2)	A = c{ab/c} = ab;
3)	A = \/2{ablc x b2/c + ab + able x a2/c} = ab/2,
{ЬЧс2 + 1 + t^/c2};
4)	ab - abl2{b2/ c2+ 1 + al/c2} => 2ab = ab{b2/c2 + 1 + a2! c2} ->
=> 2 = {УЧс2 + 1 + a^c2} -> 1 - УЧс2* a2!c2-> b2/c2 + a2!^ - 1 => a2 + b2 = c2.
168

UNIT 13
Инфинитивные обороты
(The Infinitive Constructions)
Объектный падеж с инфинитивом
(Complex Object)
Глагол	Существительное (общ. п.)
(действительный залог) +	+ Infinitive
Местоихмение (объект, п.)
Глагол-сказуемое
(в действительном залоге)
Пример
Перевод
1. Глаголы, выражающие вос¬
приятие посредством органов
чувств (употребляются с инфи¬
нитивом без частицы to):
to see - видеть
to hear - слышать
to watch )	>
}. наблюдать
to observe J
to notice - замечать
to feel - чувствовать
The professor saw the
students (them) carry
out the laboratory work
We heard them discuss
a new lecture in
geometry.
Профессор видел,
что студенты
(они) выполняли
лабораторную работу.
Мы слышали, что
они обсуждали новую
лекцию по геометрии.
2. Глаголы, выражающие
желание:
to want ч
to wish > хотеть, желать
to desire J
to like - нравиться, любить
to hate - не любить, ненавидеть
to require - требовать от
should (would) like - хотел бы
to enable - давать право,
возможность
They required us to take
an examination in time.
Hed like his colleagues
to complete the
experiment soon.
Они потребовали,
чтобы мы сдали
экзамен вовремя.
Он хотел бы,
чтобы его коллеги
скорее закончили
эксперимент.
3. Глаголы, в
положение:
to think
to suppose
to consider
to believe
ыражающие пред¬
полагать,
считать
The teachers believe
first-year students to
study English with great
pleasure.
Преподаватели по¬
лагают, что перво¬
курсники изучают
английский язык с
большим удоволь¬
ствием.
169

Ex. 2. In the sentences to follow look for the Complex Object and then translate
them into Russian.
1.	We know all points on the circle to be equidistant from a given fixed point.
2.	They can assume a line to be defined as an infinitely large set of points.
3.	The teacher heard them discuss a new theorem.
4.	We consider both theories to be necessary, though they are contradictory.
5.	I expect this equation to have a different solution.
6.	Some prominent scientists believe many problems of maths to be solved in the
21sl century.
7.	We want the students to learn and revise the rules regularly not only before the
exams.
8.	The professor desired his postgraduate student to apply a new method of
investigation.
9.	We watched the professor draw a new axis in order to prove the theorem
considered.
10.	They expected a solution to be found as soon as possible.
11.	Mathematicians have found the ratio of the circumference to a diameter to be
the same for all circles.
12.	They didn’t expect trigonometric functions to be so complicated.
Именительный падеж с инфинитивом
(Complex Subject)
Существительное (общ. падеж)
Местоимение (именит, падеж)
+ Глагол:	+ Infinitive
a)	в страд, залоге
b)	в действ, залоге
c)	to be + прилагательное
Глагол-сказуемое
Пример
Перевод
1. В страдательном залоге:
believe - полагать, считать
expect - ожидать, рассчиты¬
вать
know - знать
say - говорить, утверждать
report - сообщать
suppose	"I
consider	1	считать,
think	|	полагать
understand	J
The problem is considered
to be complicated.
The Internet is reported to
be in great demand.
Mathematics is known to
be the language of science.
The explanation was found
to be convincing.
Считают, что задача
трудна.
Сообщают, что
интернет пользуется
большим спросом.
Известно, что мате¬
матика - это язык
науки.
Объяснение оказалось
убедительным.
171

Глагол-сказуемое
Пример
Перевод
find - оказываться, обнару¬
живать
The students are supposed
to know the rules well.
They were understood to
agree with our viewpoint.
Полагают, что сту¬
денты хорошо знают
правила.
Считали, что они
согласятся с нашей
точкой зрения.
2. В действительном залоге:
seem 1 казаться,
appear J по-видимому
prove
turn out ? оказываться
come out J
happen | оказываться,
chance J случаться
Hie data proved to be
wrong.
The book does not appear
to be difficult.
My group-mate happened
to have prepared for the
exam better.
There seems to be some
confusion of tenses in his
test.
He turned out to be a good
friend.
Оказалось, что данные
некорректные.
По-видимому, эта
книга нетрудная.
Оказалось, что
мой одногруппник
приготовился к
экзамену лучше.
Кажется, в его
контрольной есть
некоторая путаница
во временах.
Оказалось, что он
хороший друг.
3. Глагол-связка
be + прилагательное:
be likely - вероятно, может
be not likely | маловероятно,
be unlikely [ вряд ли
be sure 1 наверняка,
be certain j несомненно
They are likely to come in
time.
This scientist is sure to
get a Noble Prize for his
outstanding discovery.
This problem is certain to
arise.
Вероятно, они придут
вовремя.
Несомненно, этот
ученый получит
Нобелевскую премию
за его выдающееся
открытие.
Несомненно, эта
проблема возникнет.
Ex. 3. In the sentences to follow find the Complex Subject and translate them
into Russian.
1.	Every point at a distance R from point 0 is said to be on the circle.
2.	This rule does not appear to hold for all operations of arithmetic.
3.	When two angles have the same vertex and the line between them is a side of
both, the angles are said to be adjacent.
4.	When one of the angles of a triangle is obtuse, the triangle is considered to be
an obtuse one.
5.	Like terms are expected to be arranged in a similar way.
6.	A proper solution of this equation is likely to be obtained.
172

7.	The students appeared to be unable to carry out these complex calculations.
8.	The line drawn perpendicular to a radius through its endpoint is known to be
a tangent to a circle.
9.	Two tangents can always be drawn to a circle from any point outside the circle,
and these tangents are said to be equal in length.
10.	A circle is known to be a set of points in a plane each of which is equidistant
from some given point called the center.
11.	Every point at a distance less than R from 0 is said to be inside the circle.
12.	The sets are supposed to be designated by capital letters А, В, C.
13.	Two circles which have two common points are said to intersect each other.
Ex. 4. Change the sentences according to the model using the Complex Subject.
It is believed that he is a hard-working student.
He is believed to be a hard-working student.
1.	It is expected that all second-year students will pass the exams successfully.
2.	It is known that the experiments have been finished in time.
3.	It is certain that this problem will be solved very soon.
4.	It is likely that he has given them wrong instructions.
5.	It is reported that the postgraduates of our Faculty have finished their
investigations.
6.	It seems that she is a very talented researcher.
7.	It happened so that the error was easily detected.
8.	It is expected that the delegation of prominent foreign mathematicians will
arrive at our University next week.
Pre-Reading Activity
Guess the meaning of the following words.
Coordinate [kca'c da::], diameter [da:'sra::s]. arc [a k], perpendicular
Read and learn the basic vocabulary terms:
circle (n) [ss kl]
curve [ks v] (curved line)
half-line (n) ['ha r" laza]
measure (v) ['aas3 s ]
reference-line (n) ['zefzsas Ta:a]
equidistant (adj) [':kw:'a:st=at]
круг, окружность
кривая линия
полупрямая, луч
мерить, измерять
базисная линия
равноудаленный
173

chord (n) [kc 2]
secant (n) [zsi ksr.:], (secant line)
come from (v) ['кл~' Tress ]
inscribe (v) [:s'skra:b]
bisect (v) [kafsskt]
arc (n) [2 kJ
annulus (n) ['ssrjrf.ssJ
ratio (n) [rs: :e'j]
tangent (n) ['tssehss:], (tangent line)
radius (n) [':s:£:ss]> pl. radii ['reizza:]
хорда
секущая
происходить, иметь,
происхождение
вписывать
делить пополам
дуга
(плоское) круговое кольцо
соотношение, коэффициент,
отношение
касательная (линия)
радиус, радиусы
Memorise the following word combinations:
a major arc - большая из двух дуг (окружности)
a minor arc - меньшая из двух дуг
internally tangent circles - круги, касающиеся внутренним образом
externally tangent circles - круги, касающиеся внешним образом
a circumscribed polygon - описанный многоугольник
an inscribed polygon - вписанный многоугольник
a regular octagon - правильный восьмиугольник
to arrive at a more precise definition - (для того) чтобы прийти к более точному
определению
are subtended - стянуты
by a similar process - подобным образом
in other words - иными словами
as closely as desired - так близко, как хотелось бы
no matter how short an arc is - какой бы короткой не была дуга
Reading Activity
THE CIRCLE
Lets turn now to the simplest of all curved lines, the circle. We shall study its
properties and its relation to straight lines and to figures made up of straight lines,
especially polygons.
In a plane all the points at a given distance from a given fixed point are said to
form a circle. A circle is a set of points in a plane, all of which are the same distance
from a given point.
The fixed point О is called the center of the circle, from which all other points
are equidistant. The distance R is called the radius. A radius is a line segment from
the center of a circle to a point on the circle.
174

Every point at a distance R from О is said
to be on the circle. Every point at a distance less
than R from О is located inside the circle, and
every point at a distance greater than R from О
lies outside the circle. If the center of the circle
is taken as the origin of a rectangular network, it
follows from the Pythagorean Theorem that the
coordinates (x, y) of every point P of the circle
will satisfy the equation x2 +y* = r2. This equation
is the equation of the circle.
On any half-line with end-point О there is a
Fig- 1
point at the distance r from O. We may select one such half-line, for example, OX
in Fig. 1 as a reference line from which to measure the angles to all other such half¬
lines. If we measure these angles in degrees, then on every half-line which makes
an angle between 0 and 360° degrees with OX there is a point of the circle.
All the points of the circle which lie on half-lines from p to q (Fig. 2) are said
to form an arc PQ of the circle. Hie word “arc” comes from a Latin word meaning
«I	»
bow.
In Fig. 2 arc PQ corresponds to angle POQ. Angle POQ is called a central
angle because its vertex is at the center of the circle.
A chord of a circle is a line segment whose two endpoints lie on the circle. Hie
diameter, passing through the circle centre, a circle. A tangent to a circle
is a straight line that touches the circle at a single point, thus guaranteeing that
all tangents are perpendicular to the radius and diameter. A secant is an extended
chord: a straight line cutting the circle at two points.
A Fig. 2 notice that the half-lines OP and OQ form two angles whose sum
is 360°. Ordinarily when we speak of angle POQ we refer to the lesser of these
two angles; only rarely do we mean the greater angle. Similarly, when we speak
of the arc PQ, we ordinarily mean the arc that corresponds to the lesser central
angle POQ; but occasionally we mean the arc that
I P	corresponds to the greater central angle. Except for
q /	the end-points P and Q, all the points of the first arc
PQ are sometimes called the minor arc which are
/	/	\	distinct from the points of the second arc PQ which
0 are sometimes called the major arc. If the two central
1	;	angles POQ are equal, each of the two corresponding
\	/	area PQ is called a semicircle.
Instead of speaking of the perimeter of a circle,
'' —we usually use the term circumference to mean
pXg 2	the distance around a circle. We cannot find the
175

circumference of a circle by adding the
measure of the segments, because a circle
does not contain any segments. No matter
how short an arc is, it is curved at least
slightly. Mathematicians have discovered,
that the ratio of the circumference (c) to a
diameter (d) is the same for all circles and is
c	c c
expressed ~. The number ~ or — (since
d = 2r - the length of a diameter is equal to
twice the length of a radius), which is the
same for all circles, is designated by л.
c	c
— = л or — = Л .
d	2r
By using the multiplication property of equation, we obtain the following:
c = nd or c - 2nr.
Post-Reading Activity
Ex. 5. Answer the following questions.
1.	How can we define a circle? 2. What is the center of a circle? 3. What is a radius?
4.	When do we say that a point is on the circle, inside the circle and outside the
circle? 5. How can you formulate the equation of the circle using the Pythagorean
Theorem? 6. What is an arc? 7. When is an angle called a central angle? 8. What
kind of line segment is a chord? 9. What is the longest chord in a circle? Give the
definition of this figure. 10. What is a tangent to a circle? 11. What kind of chord
is a secant? Give the definition of a secant. 12. What is a circumference? Give the
formula of a circumference.
Ex. 6. Match the English words
equivalents:
1)	a reference line
2)	equidistant
3)	an inscribed polygon
4)	come from
5)	a chord
6)	externally tangent circles
and words combinations with their Russian
a)	(плоское) круговое кольцо
b)	иметь происхождение
c)	хорда
d)	секущая
e)	круги, касающиеся внешним образом
f)	вписанный многоугольник
176

7)	by a similar process
8)	a regular octagon
9)	a circumscribed circle
10)	a concentric circle
11)	an annulus
12)	a secant
g)	подобным образом
h)	равноудаленный
i)	описанный круг
j)	правильный восьмиугольник
k)	базисная линия
l)	концентрический круг
Ех. 7. Mark the following as true or false. Use the introductory phrases.
The statement is true.
It's correct to say.
I share this viewpoint.
Quite the contrary (the reverse).
I can't agree with the statement.
You are wrong there, I am afraid.
1.	In a plane all the points at a given distance from a given fixed point are said to
form a chord.
2.	The diameter passing through the circle centre is the shortest chord in a circle.
3.	Angle POQ is called a central angle because its vertex is at the centre of the
circle.
4.	A tangent to a circle is a curved line that touches the circle at two points.
5.	A secant is an extended chord a straight line cutting the circle at a single point.
6.	Mathematicians have discovered that the ratio of the circumference to a
diameter is the same for all circles.
7.	Hie circumference of a circle may be defined as the limit of the perimeter of an
circumscribed regular polygon.
8.	Hie equation x1 2 3 4 5 + y2 = r2 is the equation of the semicircle.
Ex. 8. Fill in the blanks with necessary words and word combinations. Mind
there are two extra words:
a)	a line segment
b)	only
c)	circumscribed
d)	vertex
e)	circumference
f)	through
g)	in a plane
h)	are said
i)	bisect
j)	arc
k)	a chord
l)	given
m)	on the circle
n)	intersect
o)	a set of points
p)	fixed
q)	meaning
r)	the measure
s)	connects
t)	perimeter
1.	A circle is ... in a plane, all of which are the same distance from a ... point.
2.	A radius is ... from the center of a circle to a point....
3.	Tangent circles are two circles that	at one point.
4.	A ... circle is a circle passing ... each ... of a polygon.
5.	... all the points at a given distance from a given ... point... to form a circle.
177

6.	The word ... comes from a Latin word ... “bow”.
7.	A diameter is ... which ... the center to any points on the circle.
8.	We cannot find ... of a circle by adding ... of the segments.
Ex. 9. Match the definitions of the circles with their names.
Based upon their relative positions, two circles in a plane or a circle and a polygon
have special names.
1.	Tangent circles	a) is the region between concentric circles.
b) are circles that have different centers.
2. Concentric circles
3. A circumscribed circle
c) are both circles which are on the opposite
sides of the tangent line.
4. Externally tangent circles
d) are two or more circles in a plane with the
same center, but the lengths of their radii
vary.
5. An inscribed circle
e) are two circles that intersect only at one point.
f) is a polygon that is inside a circle so that each
of its vertices lies on the circle.
6. Eccentric circles
7. Inscribed polygon
g) is a circle to which all the sides of a polygon
are tangents.
8. A circumscribed polygon
h) is a polygon that is outside the circle in such
a way that all of its sides are tangent to the
circle.
9. Internally tangent circles
i) is a circle passing through each vertex of a
polygon.
10. An annulus
j) are both circles lying on the same side of the
tangent line.
Ex. 10. Let us revise the Degrees of Comparison. Give the best English
equivalents for the words in parentheses.
1.	A circle is (самая простая) of all curved lines.
2.	Every point at a distance (больше) than radius is said to be outside the circle.
3.	A secant segment is a line segment with an endpoint in the exterior of a circle,
and the other endpoint on the circle, (самой далекой) from the external point.
4.	Tom comes top in all the exams - he must be (самый умный) student in the
group.
5.	(Чем дольше) he refuses to recognize the impossibility of the solution, (тем
хуже) for him.
6.	It is (легче) to say what mathematics is not, than what it is.
7.	Is this proof (более правильно)?
178

8.
9.
10.
Peter speaks English (лучше) of all the students in this group.
Hie symbol of non-equality implies either (больше) than or (меньше), than.
(Чем скорее) the problem is solved, (тем лучше).
Hi is contribution of the ancient Greeks is (намного больше, чем) the formulas
of the Egyptians.
2.
3.
Ex. 11. Translate into Russian, paying attention to the degrees of comparison.
The diameter is the longest possible chord of any circle.
Tliis lecture is not so complicated as the one the professor delivered last month.
The science advisor was very glad that his postgraduates continued further
discussion.
The mathematical lessons are getting more and more difficult in the third year
of studies.
5.	For the 8.15 class, the latest time the students must leave the hostel is 7.40.
The diameter is twice as long as the radius.
A major arc is an arc greater than a semicircle but less than 360°.
His argument is less convincing than his proof.
The ruler is the simplest instrument for drawing.
The proof is not so valid as he supposed at first.
4.
6.
7.
8.
9.
10.
Ex. 12. Choose the correct variant.
1.	These rules are ... to understand than others.
a) so difficult	b) more difficult
2.	We have never been ... than on that day.
a) more happier	b) the happiest
3.	This is ... translation I have ever done.
a) the most worse	b) the worst
4.	Can you repeat... sentence.
a) the latest	b) the least
5.	Which is ... textbook of the two?
a) better	b) the best
6.	I don’t think this is ... thing for us at the moment.
a) a more important b) much more important
7.	She is getting ... with every day.
a) more weaker	b) the weakest
c) much difficult
c) happier
c) the worser
c) the last
c) more better
c) the most important
c) much weaker
179

8.	Girls tend to be ... than boys.
a)	more tidier	b) tidier	c) much tidy
9.	... than twenty students passed all the exams.
a)	a fewer	b) a few	c) fewer
Ex. 13. Ask special questions using the words in parenthesis.
1.	An arc is usually named by its endpoints, (how)
2.	A chord is a line segment connecting any two points on the circle, (what)
3.	They attended the lectures on geometry twice a week, (how often)
4.	Most mathematical proofs can be given in many different ways, (how)
5.	In geometry we separate all geometric figures into two groups: plane figures
and space figures or solids, (how many)
6.	Later you ought to do some measurements to check your calculations, (when)
7.	We are already familiar with the basic concepts of geometry through our high
school studies of maths, (what)
8.	Hie points of geometry have no size and no dimensions, (what)
9.	Numbers became abstract when we began to reason about their nature and
enumerate their properties using arithmetical and logical operations, (when)
10.	A straight line extends indefinitely only in one direction, (where)
11.	Every mathematical problem must be settled either in the form of a direct
answer to the question, or by the proof of the impossibility of its solution, (how)
12.	The Greeks were able to carry out many constructions with two tools, (how
many)
13.	Hie theory in question was developed successively by different scientists, (who)
Ex. 14. Choose the correct variant of translation.
1.	We expect them to solve this problem.
a)	Мы ожидаем, что они решат эту задачу.
b)	Мы ждали, что они решат эту задачу.
c)	Мы ждем, пока они решат эту задачу.
2.	Hiey are believed to have done their best.
a)	Они верят, что сделали все возможное.
b)	Полагают, что они сделали все возможное.
c)	Полагали, что они сделали все возможное.
3.	Hiey appear to have known all about the set theory.
a)	Им кажется, что они узнали все о теории множеств.
b)	Они пришли и узнали все о теории множеств.
c)	Кажется, они узнали все о теории множеств.
180

4.	What made the students do the test quickly?
a)	Что сделали студенты, чтобы выполнить тест быстро?
b)	Что заставляет студентов выполнять тест быстро?
c)	Что заставило студентов выполнить тест быстро?
5.	First-year students are thought to have shown very good results at the exams.
a)	Первокурсники, как считают, показывают очень хорошие результаты
на экзамене.
b)	Считают, что первокурсники показали очень хорошие результаты на
экзамене.
c)	Считали, что первокурсники покажут очень хорошие результаты на эк¬
замене.
Ех. 15. Translate into English.
1.	Отрезок прямой, соединяющий две какие-либо точки окружности, на¬
зывается хордой.
2.	Хорда, проходящая через центр окружности, называется диаметром.
3.	Диаметр равен сумме двух радиусов, и поэтому все диаметры одной
окружности равны между собой.
4.	Любая часть окружности называется дугой.
5.	Существует более точное определение окружности.
6.	Математики нашли, что соотношение окружности к диаметру одно и то
же для всех кругов и обозначено знаком л.
7.	Мы сначала впишем квадрат в круг.
8.	Иными словами, окружность круга может быть определена как предел
периметра вписанного правильного п-угольника.
9.	Линейка - простейший инструмент для черчения.
10.	Угол, вершина которого находится в центре круга, называется централь¬
ным.
11.	Окружность - это множество точек равноудаленных от заданной фикси¬
рованной точки.
Ех. 16. Read the text and find the answers to the following questions.
1.	How do we define the circumference of a circle in traditional mathematics?
2.	What concept is it necessary to introduce if we are going to arrive at a more
precise definition of a circle?
3.	How can we inscribe a regular octagon in a circle?
4.	Give the definition of a circumference using the concept of limits.
181

Circumference of a Circle
In traditional approaches to mathematics, the circumference of a circle
has not always been clearly defined. Sometimes the circle itself was called the
circumference, and at other times, the measure of the distance around the circle
was called the circumference. If we define the circumference as the perimeter of
the circle the measure of the circle is symbolized by the formula c - nd or c = 2nr.
There exist more precise definitions of a circumference. To arrive at a more
precise definition it is necessary to introduce the concept of limits. By using the
limit concept, the circumference of a circle may be defined as the limit of the
perimeter of an inscribed regular polygon. To illustrate this we can first inscribe a
square in a circle. Hie sum of the sides of the square will be an approximation of the
circumference of the circle. Then, bisecting the central angles, which are subtended
by the sides of the square we can inscribe a regular octagon. The sum of the sides
of the octagon will be a closer approximation of the circumference. Next bisecting
the central angles subtended by the sides of the octagon, we can inscribe a regular
16-gon. The sum of the sides of the 16-gon will be an even closer approximation
of the circumference. By a similar process we can then inscribe a regular 32-gon
and 64-gon and so on. Clearly the sum of n sides of an inscribed regular n-gon can
be made to approximate the circumference of the circle as closely as desired by
choosing n sufficiently large.
In other words the circumference of a circle may be defined as the limit of the
perimeter of an inscribed regular n-gon as n increases.
UNIT 14
Условные предложения
(Conditional Sentences)
Придаточное предложение условия соединяется с главным предложе¬
нием союзами if (если), unless (если не), provided (that), providing (that), on
condition (that) (при условии, если; при условии, что), supposing (that), sup¬
pose (that) (предположим, что).
Типы условных предложений
Существует три типа условных предложений.
1. В условных предложениях первого типа глагол в придаточном пред¬
ложении употребляется в Present Indefinite, а в главном предложении - в
Future Indefinite.
182

If it stops raining, we
will go to the beach.
Unless Peter comes
over, I shall be very
upset.
Если дождь перестанет,
мы пойдем на пляж.
Если Петя не придет
к нам, я очень
расстроюсь.
Предложение выражает
реальное предположение,
относящееся к будущему
времени, и соответствует в
русском языке условному
предложению с глаголом в
изъявительном наклонении.
2.	В условных предложениях второго типа глагол в придаточном предло¬
жении употребляется в Past Indefinite, а в главном предложении - сочетание
should или would с Indefinite Infinitive (без to).
If Mary got better
next week, she
would join us for
a picnic.
If John sawr June
tomorrow, he would
invite her for a party.
Если бы Мэри стала
лучше себя чувствовать
на следующей неделе,
она бы поехала с нами
на пикник.
Если бы Джон увидел
Джун завтра, он пригла¬
сил бы ее на вечеринку.
Предложение выражает
нереальное или малове¬
роятное предположение,
которое относится к на¬
стоящему или будущему
времени.
Оно соответствует в
русском языке условному
предложению с глаголом
в сослагательном накло¬
нении.
Глагол to be в форме were употребляется со всеми лицами единственного
и множественного числа.
If he wTere here now, he would arrange Если бы он был здесь сейчас, он ула-
everything.	дил бы все.
3. В условных предложениях третьего типа глагол в придаточном пред¬
ложении употребляется в Past Perfect, а в главном предложении - сочетание
should или would с Perfect Infinitive.
If the child had
been more careful,
he would not have
fallen down.
We would not have
got lost if we had
taken a map.
Если бы ребенок был
более внимательным, он
бы не упал.
Мы бы не заблудились,
если бы взяли карту.
Предложение выражает
предположение, относя¬
щееся к прошедшему
времени и являющееся
поэтому невыполнимым.
Оно соответствует в
русском языке условному
предложению с глаголом в
сослагательном наклонении.
183

Ex. 1. Define the type of conditionals in the following sentences and translate
them:
1.	If I had known you were in hospital, I would have gone to see you.
2.	If I had not gone to the party, they would have been offended.
3.	If Kate phones this evening, I will invite her to the cinema.
4.	What would you do if you won one million pounds?
5.	Would you mind if I used your phone?
6.	If I had been hungry yesterday, I would have eaten something.
7.	Hurry up! If we don’t catch a 10 o’clock coach, we will be late for a flight.
8.	If the driver in front hadn’t stopped so suddenly, the accident wouldn’t have
happened.
9.	If it rains this evening, I will not go out.
10.	If I were taller, I would join the basketball team.
Ex. 2. Choose the correct form.
1.1	will be furious if he ever ... about it.
a) finds out	b)	found out	c)	had found	out
2.	Unless we ... a wrong turning, we would not have arrived late.
a) took	b)	had taken	c)	take
3.	If I had known you were	back from your holiday, I...	you.
a) will phone	b)	would phone
4.	If you ... smoking, you will damage your health.
a) don’t give up b) didn’t give up
5.	What... you ... if you missed the last bus?
a) will ... do	b) would ... do
6.	If he ... me earlier, I would have changed my plans.
a) warns	b) warned
7.1	... to lend you some money if I sell my car.
a) would be able b) would have been able
8.	If I ... you, I wouldn’t go out in such weather.
a) were	b) had been you
9.	If I had more time, I ... tennis.
a) will take up	b) would have taken up
10.	If you ... more exercises, you would feel better.
a) did	b) do
c) would have phoned
c) hadn’t given up
c) would ... have done
c) had warned
c) will be able
c) am
c) would take up
c) will do
184

Ex. 3. Open the brackets using the appropriate verb-forms in both parts of the
conditional sentences.
1.	If you (like) Julia Roberts, you (love) this film.
2.	If he (not see) the other car, there (be) a car accident last night.
3.	Its a shame, Paul is late. If he (leave) home earlier, he (not miss) the train and
would be in time.
4.	If Mark (be) younger, he (join) the army. But he is already 27.
5.	I’m not well-off. If I (earn) enough money, I (buy) a car of my dream.
6.	If the weather (be) good tomorrow, we (go) to the beach.
7.	Leon, your test is bad. If you (be) more careful, you (not make) so many
mistakes.
8.	If I (not have) to work such long hours, I (not be) always so tired.
9.	I (let) you know if the meeting (start) earlier. Don’t worry!
10.	Emily (give) definitely you a ring if she (change) her mind.
Pre-Reading Activity
Guess the meaning of the following words.
Orthogonal [c	vector ['vek:s], inverse [ir.'vs s], basis ['beisis],
transformation Ltrssr.srVnei.'sr.],linear [zhn:=],special ['ipeklj,group [c:" p],
normal ['no msl], isometry [ ai'scooistr:], discrete [ozs'kri t], determinant
Read and learn the basic vocabulary terms:
transpose (n) [trsr-s'psoz]
inverse (n) [ir/vs s]
rotation (n) [rso':si<-]
reflection (n) [rif Iskki]
remainder (n) [rl'meinds]
interchange (v)	]
constrain (v) [ksr/st:sio]
entry (n) ['sntri]
row(n) [.-co] (n)
column (n) ['kclsoi]
транспонированная матрица
инверсия, обращение, обратная величина
вращение
отражение
остаток
обмениваться
ограничить
элемент(матрицы)
строка
столбец
Memorise the following word combinations
a square matrix - квадратная матрица
an orthogonal unit vector - ортогональный единичный вектор
185

an identity matrix - единичная матрица
a linear transformation - линейное преобразование
a unitary transformation - унитарное (единичное) преобразование
an orthogonal group - ортогональная группа
to bring to the identity matrix- привести к единичной матрице
finite-dimensional linear isometries - конечномерные линейные изометрии
an inner product - скалярное произведение
a bottom right entry - элемент матрицы, расположенный в нижнем правом
углу таблицы
a matrix inverse - обратная матрица
simultaneous equations - совместные уравнения, система уравнений
Reading Activity
MATRICES
A matrix is a set of quantities arranged in rows and columns to form a
rectangular array. Matrices don’t have a numerical value. Hiey are used to represent
relations between quantities as well as to represent and solve simultaneous
equations. A matrix of tn rows and n columns is called an (tnn) matrix.
There are a few types of matrices: a square matrix, a row matrix, a column
matrix, a unit matrix, a transpose of a matrix and others. Here we’ll regard more
closely a square matrix.
In algebra a square matrix is an orthogonal matrix with real entries whose
columns and rows are orthogonal unit vectors. Equivalently, a matrix Q is
orthogonal if its transpose - the matrix that results from interchanging the rows
and columns - is equal to its inverse: Q1 = Q-1, which derives QlQ = QQ1 = /,
where / is the identity matrix. Th is type of matrix is a square matrix in which all the
elements in the leading diagonal are ones and the other elements are equal to zero.
An orthogonal matrix is the real specialization of a unitary matrix.
The set of n x n orthogonal matrices forms a group O(n), known as the
orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with
determinant +1 is called the special orthogonal group, and each of its elements
is a special orthogonal matrix. Orthogonal matrices arise naturally from inner
products, and from matrices of complex numbers. Orthogonal matrices preserve
inner product, so for vectors w, v in an «-dimensional real inner product space
<u,v> = <Qu,Qv>.
To see the inner product connection, let’s consider a vector v in an
«-dimensional real inner product space. Written with respect to an orthonormal
basis, the squared length of v is vlv - (Qv)7(Qv) - vlQl v.
186

The finite-dimensional linear isometries - rotations, reflections, and
their combinations - produce orthogonal matrices. Hie converse is also true:
orthogonal matrices imply orthogonal transformations. However, linear algebra
includes orthogonal transformations between spaces which may be neither finite¬
dimensional nor of the same dimension.
The inverse of every orthogonal matrix is again orthogonal. In fact, the set of
all n x n orthogonal matrices satisfies all the axioms of a group. It is a compact Lie
group of dimension n(n - 1 )/2, called the orthogonal group and denoted by O(n).
Hie orthogonal matrices whose determinant is +1 form the special orthogonal
group SO(n) of rotations. Now lets consider (n + 1) x (n + 1) orthogonal matrices
with bottom right entry equal to 1. Hie remainder of the last column (and last row)
must be zeros, and the product of any two such matrices has the same form. Hie
rest of the matrix is an n x n orthogonal matrix; thus O(n) is a subgroup of O(n + 1)
(and of all higher groups).
0
O(n) :
0
0-01
Since an elementary reflection can reduce any orthogonal matrix to this
constrained form, a series of such reflections can bring any orthogonal matrix to
the identity matrix; thus an orthogonal group is a reflection group.
Orthogonal matrices are important for a number of reasons, both theoretical
and practical.
Post-Reading Activity
Ex. 4. Answer the follow ing questions.
1.	What is a matrix? 2. Do matrices have a numerical value? 3. An orthogonal matrix
is the real specialization of a unitary matrix, isn’t it? 4. What types of matrices
do you know? 5. What is an orthogonal group? 6. Do orthogonal matrices arise
from inner products or from matrices of complex numbers? 7. What is a finite¬
dimensional isometry? 8. What does a linear algebra include? 9. Hie inverse of every
orthogonal matrix is again orthogonal, isn’t it? 10. What produces an orthogonal
matrix? 11. Do the orthogonal matrices whose determinant is +1 form the special
orthogonal group SO(n) of rotations or do they form the orthogonal group 0(h)?
12. What reasons are orthogonal matrices important for?
187

Ex. 5. Match the English words and word combinations with their Russian
equivalents.
1)	both theoretical and practical
2)	preserve inner product
3)	matrices arise from
4)	finite-dimensional linear
isometries
5)	bring any orthogonal matrix to
the identity matrix
6)	orthogonal unit vectors
7)	with respect to the basis
8)	include the identity matrix
9)	the converse is also true
a)	противоположное также верно
b)	привести любую ортогональную
матрицу к единичной матрице
c)	ортогональные единичные векторы
d)	матрицы возникают из
e)	как теоретический, так и
практический
f)	сохранять скалярное произведение
g)	по отношению к базису
h)	конечномерные линейные
изометрии
i)	включать единичные матрицы
Ех. 6. Define the following statements as true or false. Use the introductory
phrases.
I think it’s right.
I ant afraid it is wrong.
Quite so.
I don’t quite agree to it.
Absolutely correct.
On the contrary.
I quite agree to it.
Far front it.
1.	A matrix is a set of quantities arranged in rows and columns to form a rectangular
array.
2.	Simultaneous equations can be solved by means of matrices.
3.	Only two types of matrices are considered in mathematics.
4.	Matrices of complex numbers are derived from orthogonal ones.
5.	The horizontal lines in a matrix are called columns.
6.	Matrices are applied only in mathematics.
Ex. 7. Fill in the blanks with the necessary words and word combinations given
bellow. Mind there are two extra ones:
a) finite-dimensional linear isometries
g) orthogonal unit vectors
b) a linear transformation
h) spaces
c) an orthogonal group
i) orthogonal
d) a unitary matrix
j) an orthogonal matrix
e) the special orthogonal group
k) inner products
f) determinant
1) a transpose
188

1.	In algebra a square matrix is ... with real entries whose columns and rows are
2.	The set of n x n orthogonal matrices forms a group O(n) known as ....
3.	An orthogonal matrix is the real specialization of....
4.	Orthogonal matrices arise naturally from ....
5.	Thus ... - rotations, reflections, and their combinations - produce orthogonal
matrices.
6.	Linear algebra includes orthogonal transformations between ....
7.	The inverse of every orthogonal matrix is again ....
8.	The orthogonal matrices whose ... is +1 form ....
Ex. 8. Match the left and the right parts of the follow ing statements.
1.	A matrix consists of
2.	A matrix of tn rows and n columns
3.	The remainder of the last column
4.	A square matrix is an orthogonal
matrix
5.	A series of such reflections
6.	Matrices are important for
a)	both theoretical and practical reasons.
b)	with real entries.
c)	must be zeros.
d)	row vectors and column vectors.
e)	is called an (tnn) matrix.
f)	can bring any orthogonal matrix to
the identity matrix.
Ex. 9. Let us revise all tenses in the Active and Passive Voice.
1.	You can borrow my dictionary. I (use) it a lot but I (not use) it now.
2.	A lot of new houses (build) nowadays in Belarus provinces.
3.	Helen (travel) to Kiev next month to attend a conference.
4.	A newr underground station (complete) in Minsk by 2020.
5.	I (meet) my group-mate when I (go) to university yesterday.
6.	A new drug for cancer (find) last year.
7.	They (make) two attempts to pass the exam in algebra this week.
8.	A newr university campus (open) in a week.
9.	Mr. Grey (wrork) for a software company for 20 years already.
10.	The workshop on mechanical modelling (be held) tomorrow.
Ex. 10. Translate the sentences according to the models.
Model 1: One can - можно; one cannot (one can’t) - невозможно
One can use this numerical method.
Можно воспользоваться этим числовым методом.
One can’t swim a fewr miles without having a pause.
Невозможно плыть несколько миль не отдыхая.
189

1.	One can speak many languages.
2.	One can have a few jobs at a time.
3.	One can take exams in advance.
4.	One cant be back into the childhood.
Model 2: One has to - необходимо; one doesn’t have to - нельзя
One has to remember the relationship between these facts.
Необходимо помнить о соотношении этих фактов.
One doesn’t have to cross the road at a red light.
Нельзя переходить дорогу на красный свет.
1.	One has to sleep seven hours at night at least.
2.	One has to bear in mind the formulae while solving equations.
3.	This week one has to work longer hours.
4.	One doesn’t have to drive without having a driving licence.
Model 3: One needs - необходимо; one doesn’t need - нет необходимости
One needs to mend the roof.
Необходимо починить крышу.
One doesn’t need to come here every day.
Нет необходимости приходить сюда каждый день.
1.	One needs to inform them about the cancellation of the conference.
2.	One doesn’t need to drive fast. There is plenty of time.
3.	One doesn’t need to take an umbrella. It is not raining.
Model 4: One may - можно; one may not - нельзя
One may work in the reading hall at any time.
В читальном зале можно работать в любое время.
One may not feed animals in zoo.
Нельзя кормить животных в зоопарке.
1.	One may get a free museum admission on Sundays.
2.	Next Saturday one may have lectures by choice.
3.1 don’t know if one may or may not miss his lectures regularly.
4.	One may not smoke aboard the plane.
Model 5: One must - необходимо; one must not (one mustn’t) - нельзя
One must invite her for a meeting.
Необходимо пригласить ее на встречу.
I’m sure that one mustn’t be rude to anybody.
Я уверен, что нельзя быть грубым ни с кем.
190

1.	One mustn’t be late for an appointment.
2.	Tomorrow one must come to university half an hour earlier than today.
3.	One mustn’t pollute the environment.
Model 6: One should - следует, надо; one should not (one shouldn’t) - нельзя
One should go to the dentist twice a year.
К дантисту следует ходить два раза в год.
One shouldn’t put such questions.
Нельзя (не следует) задавать такие вопросы.
1.	One should be more careful while writing a dictation.
2.	One should think over the plan once again.
3.	One shouldn’t leave little children alone.
Ex. 11. Match the left and the right parts of the conditional sentences.
1.	If I were you,
2.	If you had told me that you were
coming,
3.	If you run into Peter by chance,
4.	If I could have your attention for a
moment,
5.	If the lake freezes,
6.	If he had arrived a minute later,
7.	Would it be all right
8.	Take care! You will hurt your leg
9.	Unless he were with me now,
10.	He would have passed the exam
a)	we will go skating.
b)	I could be in trouble.
c)	if I had a day off?
d)	I would be very grateful.
e)	he would have missed the train.
f)	I wouldn’t waste my time reading this
novel.
g)	I would have met you at the station.
h)	tell him to call me.
i)	if he had done more work.
j)	if you fall off the bike.
Ex. 12. Ask special questions.
1.	A matrix Q is orthogonal if its transpose is equal to its inverse, (when)
2.	An elementary reflection can easily reduce any orthogonal matrix to this
constrained form, (how)
3.	Matrices never have a numerical value, (what)
4.	A compact Lie group of dimension n(n - 1 )/2 is denoted by O(n). (what... by)
5.	Matrices are represented in rows and columns, (how)
6.	If there are m rows and n columns, the matrix is an m x n matrix, (when)
7.	Matrices are used in many fields of mathematics, (where)
8.	In geometry every theorem must be proved, (what)
9.	The identity property is being considered by the students, (by whom)
10.	All the digits have been given in a line, (what)
191

Ex. 13. Translate from English into Russian.
1.	If the team had trained more last season, it would not have lost the match.
a)	Если бы команда тренировалась больше в прошлом сезоне, она не про¬
играла бы матч.
b)	Если команда будет тренироваться больше, она не проиграет матч.
c)	Если бы команда тренировалась больше в прошлом сезоне, она не про¬
игрывала бы матч сейчас.
2.	One must remember about the family members’ birthdays.
a)	Нужно помнить о днях рождения членов своей семьи.
b)	Вы можете помнить о днях рождения членов своей семьи.
c)	Вы, возможно, помните о днях рождения членов своей семьи.
3.	If the experiment comes to nothing, we will be very upset.
a)	Мы очень огорчены, если эксперимент оканчивается неудачно.
b)	Мы были очень огорчены, когда эксперимент окончился неудачно.
c)	Если эксперимент окончится неудачно, мы будем очень огорчены.
4.	We have been asked to leave the hall as soon as possible.
a)	Нас попросили покинуть зал как можно скорее.
b)	Мы попросили покинуть зал как можно скорее.
c)	Нас просят покинуть зал как можно скорее.
5.	When we came to the city again, the palace had already been built.
a)	Когда мы приехали в город опять, дворец еще строился.
b)	Когда мы опять приехали в город, дворец уже был построен.
c)	Когда мы приедем в город опять, дворец все еще будет строиться.
Ех. 14. Translate into English the following sentences.
1.	Мы можем представить, что нулевая матрица определена линейной
комбинацией А-А.
2.	Матрица порядка |х|, состоящая из одной строки и одного столбца,
называется единичной матрицей.
3.	Теория матриц и детерминантов возникла из необходимости решения
линейных уравнений.
4.	Матрицы подчиняются некоторым законам элементарной алгебры.
5.	Две матрицы АиВ равны, когда они являются матрицами одного порядка.
6.	Мы продолжаем разрабатывать алгебру матриц.
7.	Векторы могут быть записаны в строки и столбцы.
8.	Если мы изменим любые из столбцов и строк матрицы, то мы получим
новую матрицу.
192

9.	Необходимо отметить, что умножение матриц возможно, если число
столбцов в В такое же, как и число строк в А.
10.	Студенты применили бы матрицы для решения задачи, если бы нашли
этот способ рациональным.
Ех. 15. Read the text and find the answers to the following questions.
1.	Where do matrices find applications?
2.	What matrices can be added and subtracted?
3.	Is a matrix multiplication commutative?
4.	What properties do matrices have?
Application of Matrices
A matrix is a rectangular table of elements (or entries) which may be numbers
or more generally any abstract quantities that can be added and multiplied.
Matrices find many applications. They are a key tool in linear algebra. One use
of matrices in linear algebra is to represent linear transformations. Matrices can
also keep track of the coefficients in a system of linear equations. Physics makes
use of matrices in various domains, for example, in geometrical optics, mechanics.
Graph theory uses matrices to measure distances. Computer graphics uses them to
project a 3-dimensional space onto a 2-dimensional screen.
To apply a matrix correctly one should bear in mind its properties. Matrices
of the same size can be added and subtracted. Matrices of compatible sizes can be
multiplied. These operations have many properties of ordinary arithmetic, except
that a matrix multiplication is not commutative, that is, AB and BA are not equal
in general.
Matrices have the following properties:
•	to add matrices, add corresponding elements together to obtain another
matrix of the same order;
•	only matrices of the same order may be added;
•	to subtract matrices, subtract corresponding elements to obtain another
matrix of the same order;
•	only matrices of the same order may be subtracted;
•	to multiply a matrix by a number (also called a scalar), multiply each element
of it;
•	to multiply matrices, multiply rows by columns and add.
193

SUPPLEMENTARY READING
THE HISTORY OF GEOMETRY
Geometry is the Greek name for the science which the early Egyptians began
and developed about 5000 years ago. Hie word “geometry” is derived from two
Greek words: “geo” meaning earth and “metron” meaning measure.
For erecting pyramids the early Egyptians needed professional geometers who
were able to locate a line running north and south.
The geometry known to the Egyptians consisted principally of rules and
formulas for finding areas and volumes. Hie Egyptians were principally interested
in the practical application of their rules.
After a time Greek philosophers and teachers developed and perfected the
proofs of the Egyptians. The most important of the early Greek teachers was
Pythagoras who was born about 569 before our era. He founded a school in Italy.
Hie students were divided into two classes - beginners and Pythagorians.
Plato, who lived more than a hundred years later than Pythagoras, was
primarily a philosopher. His interest in geometry was not because of its practical
use, but because of the logic contained in the proofs.
By the 3rd century BC, geometry was put into an axiomatic form by Euclid,
whose treatment - Euclidean geometry - set a standard for many centuries
to follow. Euclid was a teacher of geometry in Alexandria. He used to say that
geometry trained the habits of expressing thoughts accurately. One of his most
important textbooks is called “Hie Elements”. “The Elements” of Euclid has been
used as a basis for all textbooks on geometry since his time.
Archimedes developed ingenious techniques for calculating areas and volumes,
in many ways anticipating modern integral calculus. Hie field of astronomy,
especially as it relates to mapping the positions of stars and planets on the celestial
sphere and describing the relationship between movements of celestial bodies,
served as an important source of geometric problems during the next one and a half
millennia. In the classical world, both geometry and astronomy were considered
to be part of the Quadrivium, a subset of the seven liberal arts considered essential
for a free citizen to master.
Hie introduction of coordinates by Rene Descartes and the concurrent
developments of algebra marked a new stage for geometry, since geometric figures
such as plane curves could now be represented analytically in the form of functions
and equations. Hiis played a key role in the emergence of infinitesimal calculus in
the 17th century. Furthermore, the theory of perspective showed that there is more
to geometry than just the metric properties of figures: perspective is the origin of
194

projective geometry. The subject of geometry was further enriched by the study of
the intrinsic structure of geometric objects that originated with Euler and Gauss
and led to the creation of topology and differential geometry.
In Euclids time, there was no clear distinction between physical and geometrical
space. Since the 19th-century discovery of non-Euclidean geometry, the concept of
space has undergone a radical transformation and raised the question of which
geometrical space best fits physical space. With the rise of formal mathematics
in the 20th century, “space” (whether “point”, “line”, or “plane”) lost its intuitive
contents, so today one has to distinguish between physical space, geometrical
spaces (in which “space”, “point” etc. still have their intuitive meanings) and abstract
spaces. Contemporary geometry considers manifolds, spaces that are considerably
more abstract than the familiar Euclidean space, which they only approximately
resemble at small scales. These spaces may be endowed with additional structures
which allow one to speak about length. Modern geometry has many ties to physics
as is exemplified by the links between pseudo-Riemannian geometry and general
relativity.
HYPERBOLIC GEOMETRY
Hyperbolic geometry is the first and most important form of non-Euclidean
geometry. Th is is a form of geometry in which Euclids first four postulates are true,
but the fifth postulate is replaced by the statement that “there is more than one line
through a point parallel to a line.” The acknowledged discoverers of hyperbolic
geometry are Janos Bolyai (1802-1860), a Hungarian, and Nikolai Lobachevsky
(1793-1856), a Russian, who independently published in books in 1832 and
1829, respectively, demonstrations that this form of geometry has a collection
of theorems that appear to be just as consistent as those of Euclidean geometry.
Both were of the opinion that hyperbolic geometry could provide just as adequate
a description of the space in which we live as does Euclidean geometry. Hie
significance of this discovery is immense. For over two thousand years people had
felt that Euclids geometry was necessarily the geometry of space. Mathematics and
physics were wedded through this belief. Hyperbolic geometry showed that there
are other conceivable descriptions of space. Physics became the study of physical
space, while mathematics became a more abstract science. From this time on, it
was clear that in mathematics one could begin with any set of postulates and study
the abstract consequences of those postulates. The significance of this discovery
is frequently compared with Copernicus’s theory, which said that the earth is not
the center of the universe, or Einsteins theory that time is not a concept that is the
same for all observers. The analogy of course is that each theory freed people from
long-accepted modes of thought.
195

Another reason that the discovery of hyperbolic geometry is considered to be
so important is that it led to yet more forms of geometry. The subject of differential
geometry was an outgrowth of this discovery, and it has been an essential tool for
physicists, most notably Einstein, in their study of the physical universe. This is
a prime example of topics in abstract mathematics that have turned out to have
unexpected applications in the real world.
SPHERICAL GEOMETRY
Consistency
Spherical geometry is not considered to be as important in pure mathematics
as hyperbolic geometry because not all the theorems of absolute geometry are valid
in it, and because it is quite difficult to treat spherical geometry as a formal axiom
system. Nevertheless, because of its intuitive model, it gives a very nice illustration
of how some very strange-sounding axioms can actually yield a consistent body of
mathematics.
There is a fine point here. We should say that spherical geometry is consisient
if Euclidean geometry is consistent. This is true because our model of spherical
geometry has been constructed within 3-dimensional Euclidean geometry. But how
do we know that Euclidean geometry is consistent? No one has ever seen an infinite
Euclidean plane. Hie answer is that Euclidean geometry can easily be reduced to
real analysis, which means just the standard properties of the real numbers, using
coordinate geometry. All the theorems of Euclidean geometry can be reduced to
statements about relationships between numbers, by considering the equations
of the geometric figures. So everything has been reduced to the consistency of
real analysis. Moreover, the consistency of real analysis can be reduced to that of
arithmetic, which means the standard properties of the positive integers. Whether
or not arithmetic has been proved to be consistent depends on what techniques we
allow ourselves to use in our proof. Indeed, the consistency of arithmetic cannot
be proved using just arithmetic. The famous twentieth-century mathematician
Hermann Weyl remarked that God exists because arithmetic is consistent, but the
devil exists because we cannot prove that it is consistent. Almost all mathematicians
are happy to accept the consistency of arithmetic, and hence that of Euclidean and
spherical geometry. Another very significant factor in the importance of spherical
geometry is the possibility that the geometry of the universe is the 3-dimensional
analogue of spherical geometry. Riemann considered this possibility in 1854. The
idea that Euclidean geometry is not necessarily the geometry of the universe, but
only one of several candidates, and that the determination of the geometry of the
universe belongs to physics and not to mathematics, is one of the most important
contributions of non-Eudidean geometry to human thought. Perhaps an even more
196

important consequence of non-Euclidean geometry was the removal of belief in
absolute truth. The realization that Euclidean geometry was not the only possible
type of geometry had ramifications in such diverse areas as religion and physics.
SOLID FIGURES
A solid figure, or a solid, is any part of space uniquely determined by one or
more surfaces. They may be plane or curved.
A polyhedron is a solid bounded by plane surfaces.
Hie planes bounding a polyhedron are called its faces, the lines in which
adjacent faces intersect are called its edges, and the points in which the edges meet
are called its vertices.
A polyhedron with four faces is called a tetrahedron, with six - a hexahedron,
with eight - an octahedron, with twelve - a dodecahedron, with twenty - an
icosahedrons.
A prism is a solid figure bounded by plane faces, two of which are congruent
polygons.
Hie polygons are called the bases and the other faces are called the lateral
faces of the prism. Lateral faces intersect in lateral edges. These lateral faces are
parallelograms because a pair of corresponding sides of the bases are equal and
parallel.
Prisms take their names from the nature of their bases, e.g. a prism with
pentagonal bases is called a pentagonal prism.
If the lateral edges of a prism are perpendicular to the bases, the prism is
called a right prism, and a right section of a prism is a section made by a plane
perpendicular to all the lateral faces of the prism.
A parallelepiped is a prism whose bases are congruent parallelograms. It can
also be defined as a polyhedron bounded by three pairs of parallel lines. Special
cases are the rectangular parallelepiped or cuboid, whose faces are rectangles, and
the cube whose faces are squares.
A pyramid is a polyhedron, one of whose faces is a polygon, the other faces are
triangles with a common vertex.
Hie polygonal face is called the base, the triangular faces are called the lateral
faces and they meet each other in the lateral edges.
Hie vertex of the pyramid is the common vertex of the lateral faces.
Pyramids take their names from the nature of their bases, for example,
a pentagonal pyramid is a pyramid whose base is a pentagon.
A regular pyramid is a pyramid whose base is a regular polygon and whose
faces are isosceles triangles. Hie term right pyramid is also used.
197

A frustum of a pyramid is that part of a pyramid lying between the base and
a plane section parallel to the base.
A right circular cylinder is the solid generated by the revolution of a rectangle
about one of its sides. The side about which the rectangle rotates is called the axis
and the opposite side is called a generator of the cylinder. The other two sides trace
out the bases of the cylinder.
The bases of the cylinder are circles and two generators of the cylinder are
parallel.
The right circular cylinder is the type of cylinder which is most common.
A right cylinder is formed when the generator is perpendicular to the bases.
Such cylinders are characterized by the nature of the bases. A right elliptic cylinder
is a right cylinder with ellipses as bases, and a right circular cylinder is a cylinder
whose bases are circles.
An oblique circular cylinder is a cylinder whose bases are circles to which the
generating lines are oblique.
A truncated cylinder is the portion of a cylinder included between a base and
the section by any plane which does not pass through either base.
A section of a cylinder perpendicular to the generator is called a normal
section.
The word sphere is commonly used in two senses in one it is considered to be
a surface; in the other - as the portion of space inclosed by that surface, but it is
useful to think of the sphere as a surface.
The surface of a sphere is generated by the rotation of a semicircle about its
diameter.
Hie centre of the defining semicircle is called the centre of the sphere, it
follows that every point on the sphere is at a constant distance from the centre,
this distance is called the radius of the sphere. A diameter of the sphere is any line
passing through the centre terminated at both ends of the surface.
Every plane section of a sphere is a circle.
If two spheres intersect, their common section is a circle.
A right circular cone is the solid generated by the revolution of a right-angled
triangle about one of the sides containing the right angle. The side about which
the triangle rotates is called the axis, the surface traced out by the hypotenuse of
the triangle is called the curved or lateral surface and the surface traced out by the
third side is known as the base. A right circular cone is a right cone with a circle as
the base and the axis passes through the centre of the circular base.
An oblique circular cone is a cone whose base is a circle such that the line
jointing its centre to the vertex of the cone is oblique to the plane of the circle.
A frustum of a cone is the portion of a cone included between the base and the
section of the cone by a plane parallel to the base.
198

POLAR COORDINATES
In the last article we used two distances, the abscissa and the ordinate, to
determine the position of a point in the plane, and although this is always possible,
it is sometimes not desirable. In order to take care of those cases where the
coordinates are given as a distance and a direction, we shall define the system of
polar coordinates.
Instead of two lines, the frame of reference consists of a point and a line. Hie
point О (Fig. 1) is called the pole, and the line OX, the polar axis. All angles are
generated by revolving OX either clockwise
or counterclockwise, and all distances are
measured from the pole along the terminal
side of the angle. Thus the point PJ is
located by turning the positive angle a and
measuring r units along the line OP^ Then
the point has the coordinates (rp a), where
Tj is usually referred to as the radius vector
or distance, and a, as the vectorial angle.
It is easily seen that P{ may be located
by generating the positive angle P, or the
negative angle y, and measuring the distance
rr This (Fig. 1) raises the question of positive
and negative directions of the radius vector, and we shall lay down the rule thus
this vector is positive it an angle is turned and the distance measured without
going through the pole. On the other hand, we shall say that the radius vector is
negative if an angle is turned and it is necessary to measure backward through the
pole in order to reach the point. With this understanding, it is obvious that the
coordinates of Pt may be written as (rta), (-r(P), (-rp -y) or (rj-6). In like manner
the coordinates of the point P2 may be written (-r2-a), (r2P), (r2-y), or (r2, -6).
Hie angles used in the above
illustrations are all less than 360°, and
this is usually the case, although it is
possible to use (360° + a, ), 2(360°) +
+ a, ... , for instance, in place of a, and
obtain the same point Pv Hence we are
correct in saying that such a point has an
unlimited number of polar coordinates.
A study of Fig. 2 will show some
connection Fig. 2 between the two
coordinate systems which have been
199

described. In this figure we have chosen the frames of reference so that the origin
of the rectangular system and the pole of the polar system coincide, as do also
the X-axis and the polar axis. The point P has rectangular coordinates (x, y) and
polar coordinates (r, 0). By means of the Pythagorian theorem we see at once that
i2 = x2 + y2 regardless of the quadrant in which the point P is located.
CARTESIAN COORDINATE SYSTEM
Illustration of a Cartesian coordinate plane. Four points are marked and
labeled with their coordinates: (2, 3) in green, (-3, 1) in red, (-1,5; -2,5) in blue,
and the origin (0,0) in purple.
A Cartesian coordinate system is a
coordinate system that specifies each point
uniquely in a plane by a pair of numerical
coordinates, which are the signed distances
to the point from two fixed perpendicular
directed lines, measured in the same unit
of length. Each reference line is called a
coordinate axis or just axis of the system,
and the point where they meet is its
origin, usually at ordered pair (0, 0). The
coordinates can also be defined as the
positions of the perpendicular projections
of the point onto the two axes, expressed as signed distances from the origin.
One can use the same principle to specify the position of any point in three-
dimensional space by three Cartesian coordinates, its signed distances to three
mutually perpendicular planes (or, equivalently, by its perpendicular projection
onto three mutually perpendicular lines). In general, n Cartesian coordinates (an
element of real и-space) specify the point in an n-dimensional Euclidean space for
any dimension n. These coordinates are equal, up to sign, to distances from the
point to n mutually perpendicular hyperplanes.
Cartesian coordinate system with a circle of radius 2 centered at the origin
marked in red. The equation of a circle is (x - a)2 + (y - b)2 = t2 where a and b are
the coordinates of the center (a, b) and r is the radius.
The invention of Cartesian coordinates in the 17th century by Rene Descartes
(Latinized name: Cartesius) revolutionized mathematics by providing the first
systematic link between Euclidean geometry and algebra. Using the Cartesian
coordinate system, geometric shapes (such as curves) can be described by
Cartesian equations: algebraic equations involving the coordinates of the points
lying on the shape. For example, a circle of radius 2, centered at the origin of the
200

plane, may be described as the set of all points whose coordinates x and у satisfy
the equation x2 + y2 = 4.
Cartesian coordinates are the foundation of analytic geometry, and provide
enlightening geometric interpretations for many other branches of mathematics,
such as linear algebra, complex analysis, differential geometry, multivariate
calculus, group theory and more. A familiar example is the concept of the graph of a
function. Cartesian coordinates are also essential tools for most applied disciplines
that deal with geometry, including astronomy, physics, engineering and many
more. They are the most common coordinate system used in computer graphics,
computer-aided geometric design and other geometry-related data processing.

NUMERALS
Cardinal Numerals
(Коли не cm венные ч исли телън ые)
1-12
100 и далее
1 one [was]
100 a (one) hundred ['Las.drs d]
2 two [VS ]
101 a (one) hundred and one
3 three [9:1 ]
125 a (one) hundred and twenty-five
4 four [f2 ]
200 two hundred
5 five [fazv]
500 five hundred
6 six [s'ks]
1,000 a (one) thousand [z9a'SZ=nd]
7 seven [sevs]
1,001 a (one) thousand and one
8 eight [ev]
1,256 a (one) thousand two hundred and fifty-six
9 nine [nazr_]
2,000 two thousand
10 ten [sen]
2,075 two thousand and seventy-five
11 eleven [z'levs]
25,000 twenty-five thousand
12 twelve [:welv]
100,000 a (one) hundred thousand
13-19
1,000,000 a (one) million [ mz:iss]
1,000,000,000 a (one) milliard (в Англии);
13 thirteen ['9?kis]
a (one) billion (в США)
14 fourteen ['fs kin]
15 fifteen ['flfvsj
• Каждые три разряда отделяются запятой.
16 sixteen [ s;ks ris]
17 seventeen ['sevskls]
• В пределах каждых трех разрядов перед
18 eighteen [ eikin]
десятками (а если их нет, то перед
19 nineteen ('s&:s.'tis]
единицами) ставится союз and:
—
1,225,375 one million two hundred and
20-90
twenty-five thousand three hundred and
20 twenty [kweniz]
seventy-five.
21 twenty-one [z:w5S.:i'w.is]
• Числительные hundred, thousand, million
22 twenty-two
не приобретают окончание -s, когда перед
30 thirty ['9s V]
ними стоят two, three и т. д.: two hundred,
40 forty ['fc ::]
four thousand, five million.
50 fifty ['f*::i]
Однако когда они выражают неопределен-
60 sixty ['s:ks::J
ное количество сотен, тысяч, миллионов
70 seventy ['ssvnvzj
и превращаются в существительные, то
80 eighty [ si!l]
во множественном числе добавляется -s и
90 ninety [si’SV]
после них употребляется существительное
с предлогом of:
hundreds of people сотни людей,
thousands of words тысячи слов.
202

Ordinal Numerals
(Порядковые числительные)
1st first [fs ST]
2nd second ['sekr.d]
3rd third [9= d]
4th fourth [fs 9]
5th fifth [f-:9]
6th sixth [szks9]
7th seventh [zsev2.9]
8th eighth [ezz9]
9lh ninth [rzazr.9]
10th tenth [T6s9]
11th eleventh [i'levr.9]
12th twelfth [zwelz’9]
13th thirteenth [ 9s kls9]
14th fourteenth ['fs 'zizi9]
15th fifteenth ('fzf'Tir.9]
16th sixteenth [S’ksTZ r.9]
17th seventeenth [sevr/zi r.9]
18th eighteenth [rZZl z.9]
19th nineteenth ['r.azr/zi 2.9]
20th twentieth ['Twe2.T"9]
21st twenty-first ['TwenTz'fs st]
22nd twenty-second
23rd twenty-third
24th twenty-fourth
25th twenty-fifth
30lh thirtieth [ 9s tzz9]
31s* thirty-first
32nd thirty-second
40th fortieth ['fc T"9]
50lh fiftieth ['fzfTZz9]
60lh sixtieth [ s:ksT"9]
70lh seventieth ['sevrzTZz9]
80lh eightieth [4ztzz9]
90lh ninetieth [ r.azr.TZz9]
100th hundredth [Zh.\rzdz9d9]
101st hundred and first
102 nd hundred and second
125th hundred and twenty-fifth
200th two hundredth
500th five hundredth
1,000th thousandth [z9a'Jzsr.9]
1,001st thousand and first
1,256th thousand two hundred and fifty-sixth
2,000th two thousandth
25,000th twenty-five thousandth
100,000th hundred thousandth
1,000,000th millionth ('~.Z?S2.9]
1,000,000,000th milliardth или billionth
['bzks-9]
•	Существительное, определяемое поряд¬
ковым числительным, употребляется
с определенным артиклем. Артикль
сохраняется перед порядковым
числительным и в том случае, когда
существительное не упомянуто:
March is the third month of the year.
Март - третий месяц года.
His second book is better than the first.
Его вторая книга лучше первой.
•	Перед порядковым числительным
может стоять и неопределенный
артикль. В этом случае числительное
приобретает значение другой, еще один:
We have sent her a second letter.
Мы послали ей второе (еще одно) письмо.
203

Fractional Numerals
(Дробные числительные)
Common Fractions
(Простые дроби)
Decimal Fractions
(Десятичные дроби)
1/2
1/3
1/4
1/5
1/10
1/25
1/100
2/3
3/4
4/7
7/18
9/10
2lA
3 */4
1/2 ton
2/5 ton
2 A
a (one) half
a (one) third
a (one) fourth
a (one) quarter
a (one) fifth
a (one) tenth
a (one) twenty-fifth
a (one) hundredth
two thirds
three fourths
three quarters
four sevenths
seven eighteenths
nine tenths
two and a half
three and a quarter
three and a fourth
half a ton
two fifths of a ton
two and a half tons
two tons and a half
0.2
.2
0.003
0.5
1.22
3.4
3.215
14.105
0.25 ton
1.25 tons
23.76 tons
nought point two
(zero) point two
о ['c‘2j point two
point two
nought point nought nought three
о [’c‘2] point о ['2'2] о [r2*2] three
point zero zero three
(zero) point five
one point two two
three point four
three point two one five
one four point one nought five
fourteen point one nought five
nought point two five of a ton
one point two five tons
two three point seven six tons
twenty-three point seven six tons
•	В десятичных дробях в английском языке целое число отделяется от дроби
точкой (читается point) вместо запятой.
•	При чтении десятичных дробей каждая цифра читается отдельно.
•	Ноль читается как nought [г.2 :], (амер.) zero ['zi? ”2'2] или о ['2'2].
•	Ноль целых в американском варианте можно совсем не читать, произнося
только point.
Хронологические даты
1900 - nineteen hundred
1904 - nineteen four / nineteen о ['2'2] four
1915 - nineteen fifteen
1949 - nineteen forty-nine
2000	- two thousand
2001	- two thousand and one / twenty о [rC*J] one
2067 - twenty sixty-seven
204

Слово year после обозначения года не употребляется, но иногда употребляется перед
ним:
in 1915 - in the year nineteen fifteen
in 1992 - in nineteen ninety-two
Даты обозначаются порядковыми числительными:
15th May, 1948
May 15th, 1948
May 15, 1948
The fifteenth of May, nineteen forty-eight
или
May the fifteenth, nineteen forty-eight
the 70s - the seventies - семидесятые годы
the 60s - the sixties - шестидесятые годы

IRREGULAR VERBS
(НЕСТАНДАРТНЫЕ ГЛАГОЛЫ)
Infinitive
Past Indefinite
Past Participle
Перевод
arise
arose
arisen
возникать
be
was, were
been
быть
bear
bore
borne
носить, выносить
become
became
become
становиться
begin
began
begun
начинать(ся)
bend
bent
bent
гнуть(ся)
bind
bound
bound
связывать
break
broke
broken
ломать
build
built
built
строить
choose
chose
chosen
выбирать
come
came
come
приходить
cost
cost
cost
стоить
cut
cut
cut
пересекать, резать
deal
dealt
dealt
иметь дело (с)
do
did
done
делать
draw
drew
drawn
чертить, тащить
fall
fell
fallen
падать
feel
felt
felt
чувствовать
find
found
found
находить
fight
fought
fought
бороться
Пу
flew
flown
летать
foresee
foresaw
foreseen
предвидеть
forget
forgot
forgotten
забывать
get
got
got
получать, становиться
give
gave
given
давать
grow
grew
grown
расти, выращивать
have
had
had
иметь
hear
heard
heard
слышать
hold
held
held
иметь силу, держать
keep
kept
kept
держать, хранить
206

Infinitive
Past Indefinite
Past Participle
Перевод
know
knew
knowm
знать
lay
laid
laid
класть
lead
led
led
вести
learn
learnt (learned)
learnt (learned)
узнавать, учиться
leave
left
left
оставлять
let
let
let
позволять
lose
lost
lost
терять
make
made
made
делать, заставлять
mean
meant
meant
значить, подразумевать
meet
met
met
встречать
put
put
put
класть
read
read
read
читать
run
ran
run
приводить в движение,
бежать
say
said
said
говорить, сказать
see
saw’
seen
видеть
send
sent
sent
посылать
set
set
set
помещать, ставить
show
show'ed
shown
показывать
sit
sat
sat
сидеть
speak
spoke
spoken
говорить, разговаривать
spend
spent
spent
тратить
spread
spread
spread
распространять(ся)
stand
stood
stood
стоять
strike
struck
struck(stricken)
ударять, бастовать
swing
swrung
swung
качать(ся)
tear
tore
torn
разрывать
tell
told
told
рассказывать, сказать
think
thought
thought
думать
throw’
threw’
thrown
бросать
understand
understood
understood
понимать
w'ear
wore
worn
носить
win
won
won
выигрывать
w’rite
wrote
written
писать

Учебное издание
Бизюк Людмила Константиновна
Зенченко Валентина Алексеевна
Потапова Наталья Леонидовна и др.
ENGLISH FOR FUTURE MATHEMATICIANS
АНГЛИЙСКИЙ ДЛЯ БУДУЩИХ МАТЕМАТИКОВ
Учебно-методическое пособие
На английском и русском языках
Ответственный за выпуск Т. М. Турчиняк
Художник обложки Т. Ю. Таран
Технический редактор Т. К. Раманович
Компьютерная верстка О. В. Гасюк
Корректор Л. С. Мануленко
Подписано в печать 26.05.2017. Формат 60x84/16. Бумага офсетная.
Ризография. Усл. печ. л. 12,09. Уч.-изд. л. 11,06.
Тираж 200 экз. Заказ 319.
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