Пособие по английскому языку "English Course for Mathematician". Семестр 1









Учебно-методическое пособие по английскому языку "English Course for Mathematician". Семестр 1 для студентов 1 курса Механико-математического факультета БГУ.
Текстовая версия:


Grammar: verb “to be”, there is/are. Possessive and Personal Pronouns.

























her noun











Ex. 1. Analyze the following sentences and translate them into Russian.

1. They live together with their parents. 2. Oxford is famous for its University. 3. I didn’t have an umbrella, so Ann gave me hers. 4. Are those people friends of yours? 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. I can’t find my keys. Where are they? 10. We want the photographs. Can you give them to us?

Ex. 2. Put in my/our/your/his/her/their/its/mine/yours/ours.

Pronoun “it

Ex. 3. a) We use a personal pronoun “it” as a subject or an object

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 can’t find it. 5. Your new car is nice. Is it expensive? 6. I want that book. Please give it to me.

b) We use “it” as a subject with expressions that refer to time, weather, day temperature or distance.

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?

c) Preparatory “it”.




will be



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.

the verb “to be”













shall be

will be


















will be

Analyze the interrogative and negative forms.

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 yesterday. 3. She was ready for the seminar last week. 4. Their plans for the holiday were clear. 5. The 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) This book will be interesting. 2. I shall be late for the lesson 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. It will be a nice block of flats here. 9. The students will be free in some minutes. 10. I shall be ready for the next seminar.

Ex. 5. a) Say the following sentences in the past.

1. The facts from the newspaper article are old. 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. Their plans are simple and short. 6. The weather is good at this time of the year. 7. Bob and James are foreign students. 8. A city is a place with big industry and busy streets. 9. The article in the newspaper is interesting. 10. Sport is an essential part of Mike’s life.

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 hot in winter. 5. The 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 tiring. 9. The workers are at the meeting at this time of the day. 10. At the end of the lecture she is very tired.

The Construction there is … , there are …

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 по телевизору был вчера интересный.

3. A lot of people will be present at the party on Saturday. Много людей придет на вечеринку в субботу.

There is a teacher’s desk in the room.

В комнате стоит стол учителя.

There was an interesting film on TV last night. Вчера по телевизору шел интересный фильм.

There will be a lot of people at the party on Saturday. В субботу на вечеринке будет много людей.

Ex. 6. Analyze 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. I 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. I 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.

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]

ощущение, чувство

support (v, n) [sq'pLt]

содержать (семью); поддержка

seem (v) [sJm]


common (adj) [`kOmqn]

общий; обыкновенный

an only child

единственный ребенок

fairly large

довольно большой

impressive (adj) [Im`presIv]

производящий глубокое впечатление

research (adj) [rI`sWC]


take after (mother, father)

быть похожим на (мать, отца)

determined (adj) [dI`tWmInd]


strong-willed (adj.) [`strONwIld]

решительный, волевой

trust (v) [trAst]


be interested in


rely on (v) [rI'laI]

полагаться на

diligent (adj) [`dIlIGqnt]

прилежный, старательный

lively (adj.) ['laIvlI]

живой, веселый

cheerful (adj.) ['tSIqfVl]

веселый, жизнерадостный

restless (adj.) [`restlIs]

непоседливый, неугомонный

enter a university (college)

поступить в университет (колледж)

sociable (adj) ['sqVSqbl]


honest (adj) ['OnIst]


helpful (adj) ['helpfVl]

услужливый, готовый помочь

stubborn (adj) ['stAbqn]


deal with (v) [dJl]

иметь дело с

occupation (n) [Okju`peISqn]

род занятий, профессия

play the piano (the guitar, the violin)

играть на пианино (на гитаре, скрипке)

be fond of smth


remote (adj) [rI'mqut]

отдаленный, далекий

aunt (n) [Rnt]


uncle (n) ['Ankl]


cousin (n) ['kAzn]

двоюродный брат (сестра)

delicious (adj) [dI`lISqs]

очень вкусный, приятный

get along

ладить, относиться друг к другу хорошо

united (adj) [ju:'naItId]


Memorise the following word combinations

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.

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, 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 mother's 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) ['bjHtIful]

красивый (о женщинах)

blond/fair [feq]/ginger [GInGq]/ dark hair

светлые/русые/рыжие/черные волосы

blue/grey/hazel eyes

голубые/серые/карие глаза



curly ['kWlI] /straight [streIt] hair

кудрявые/прямые волосы

handsome (adj) ['hxnsqm]

красивый (о мужчинах)

height (n) [haIt]


look like (smb.)

быть похожим на (кого-либо)

of medium/short/tall height


cреднего/низкого/высокого роста

plain (adj) [pleIn]


plump (adj) [plAmp]


pretty (adj) ['prItI]


slender (adj) ['slendq]


slim (adj)


straight [streIt] /snubbed nose

прямой /курносый нос

Features of Character

be in good/bad mood [mu:d]

быть в хорошем (плохом) настроении

brave (adj) [breIv]


devoted (adj) [dI'vqVtId]


faithful (adj) ['feITful]

дружеский, дружественный

gentle (adj) ['Gentl]


gloomy (adj) ['glHmI]


hard-working (adj) ['hRd'wWkIN]


kind-hearted (adj) ['kaInd'hRtId]

добрый, добросердечный

lazy (adj) ['leIzI]


open-minded (adj) ['qVpn'maIndId]

открытый, искренний

polite (adj) [pq'laIt]


reserved (adj) [rI'zWvd]


rude (adj) [rHd]


shy (adj) [SaI]

робкий, застенчивый, тихий

Interests and Ambitions

ambition (n) [xm'bISqn]


be keen [kJn] on smth

увлекаться чем-либо

desire (n), (v) [di'zaiq]

желание, желать

dislike (v) [dIs'laIk] smth

(to do smth)

не нравится что-либо (делать что-либо)

do sports

заниматься спортом

dream (of) (v), (n) [drJm]

мечтать о, мечта

hate (v) [heIt]


intend (v) [In'tend] to do smth

намереваться делать что-либо

intention (n) [In'tenSqn]


make a career [kq'rIq]

делать карьеру

play football (chess)

играть в футбол (шахматы)

wish (v)


Family Members and Relations in the Family

average ['xvqrIG] /small/large family



consist of (v) [kqn'sIst]

состоять из

nephew (n) ['nevju:]


niece (n) [ni:s]


relatives (n) (relations) ['relqtIvz]


stepdaughter (n) ['stepdLtq]


stepfather (n) ['stepfRDq]


stepmother (n)['stepmADq]


stepson (n) ['stepsAn]


twins (n) [twInz]


admire (v) [qd'maIq] smb

любить, обожать кого-либо

be attached [q'txCt] to

быть привязанным к

be devoted [dI'vqutId] to

быть преданным

blame (v) [bleIm] smb

винить, обвинять

caring (adj) [`kFqriN]


close (adj) [klqus]


cordial (adj) ['kLdjql]

сердечный, радужный

difficulty (n) [`dIfIkqltI]


distant (adj) [`dIstqnt]


elderly (adj) [`eldqlI]


frank (adj) [frxNk]

откровенный, искренний

friendly (adj) [`frendlI]


hostile (adj) [`hOstaIl]


impartial (adj) [im'pRSql]


intolerant (adj) [In'tOlqrqnt]


loving (adj)[`lAvIN]


provide (v) for the family [prq'vaId]

обеспечивать семью

reliable (adj) [rq'laIqbl]


respectful (adj) [rIs'pektful]


share the domestic chores [CLz]

делить, разделять домашние обязанности

take care [kFq] of smb

заботиться о ком-либо

warm (adj) [wLm]


wonderful (adj) ['wAndeful]


worry (v) about smb ['wArI]

беспокоиться о ком-либо

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.

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

Ex. 12. Translate into English making use of the words from the box.






hazel eyes

pale skin


plump cheeks



curly hair

broad shoulders



snubbed nose

take after father


1. У ее брата очень широкие плечи. 2. Она симпатичная, а вот сестра ее некрасивая.3. Сегодня она чувствует себя нехорошо и кожа у нее бледная.4. Невысокого роста и толстые дети часто бывают стеснительными. 5. Моей подруге нравятся молодые люди с бородой и карими глазами.6. У близнецов моей сестры кудрявые волосы. 7. Какие девушки нравятся твоему двоюродному брату? – Рыженькие. 8. Все люди хотят быть стройными. 9. У моего племянника курносый нос и полноватые щечки, он похож на отца. 10. Мой друг добрый и честный, но очень уж ленивый.

Ex. 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 can't go skiing. … isn't any snow. 6. … is hot in this room. Open a window. 7. I 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. … wan’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 answerletter. 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 aboutjobs 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 with the necessary form of the verb to be.

1. I … 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 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?

Ex. 18. Writing. Using the words given in the list for the text do the following assignments.


Grammar: The verb “to have” Present, Past,

Future Simple; Types of questions

The verb “to have”









have got











shall have/

will have





has got






will have

In questions and negative sentences the following forms are used.


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)


Did they have a car last year?

They didn’t have a car last year.


Will the students have a seminar tomorrow?

The students won’t have a seminar tomorrow.

The verb to HAVE is also used for many actions and experiences.


breakfast / dinner / a cup of coffee / a cigarette/a drink / a meal.

a bath / a shower / a swim / a rest / a party / a holiday / a nice time / a good journey / a good flight / a good trip

an accident / an experience / a dream/ a sleep / a lie-down / a look (at something) / a chat (with somebody) / a talk / a fight

a baby (= give birth to a baby)

difficulty / trouble / 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.

8. The boy had his father’s blue eyes.

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. 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 friend’s father has got a fine collection of pictures.

Present, Past, Future Simple

Present Simple















do not






















does not






The Present Simple tense denotes:

1. Repeated actions indicated by adverbials of frequency such as 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

2. Universal truths (laws of nature) and permanent characteristics, situations or states.

The sun sets in the west.

She teaches English at school.

Do you like rainy weather?

His parents live in London.

3. Present actions and states, going on at the moment of speech with the so-called stative verbs which include

a) verbs of sense perception: see, hear, notice, taste, smell, etc.

b) verbs of mental activity: understand, think, believe, remember, know, forget, mean, suppose, recognize, etc.

c) verbs of feelings and emotions: like, dislike, hate, love, wish, want, prefer, care, etc.

d) verbs of possession: have, belong, own, possess, etc.

It smells like a hospital in here.

The meat tastes spicy.

I don't see anyone in the room.

Do you recognize me ?

What does he mean ?

Who do you think will win the game?

Do you know what he is speaking about?

I prefer dogs to cats.

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?

They have a big new house.

4. Scheduled facts and events such as flights, train arrivals, departures, itineraries

The flight leaves at 2 p.m. (according to the timetable)

You arrive in Basel at 6.30 a.m. local time. (according to the itinerary)

Ex. 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.

Ex. 5. Write sentences using these words. Put the verb in the right form (arrive or arrives etc.).

Past Simple

























did not
























The Past Simple tense describes:

1. A single action or a state, or a succession of single past actions with time adverbials such as 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 for five years before the war.

I entered the office, looked around and came up to the secretary.

2. A contrast between the past and the present, or something that was true but is not true any more

used to + Infinitive

бывало, раньше

He 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 happy lively girl (but no longer now).

The shops didn't use to open on Sundays in those days.

Ex. 6. Make the following sentences interrogative and negative.

Ex. 7. Write sentences about the past (yesterday / last week etc.).

Future Simple

Positive / Negative





(shall(‘ll)/shall not/shan’t) will(‘ll)/will not/won’t













The Future Simple tense denotes:

1. A predicted future action, a happening which is inevitable and out of anybody’s control with the adverbials of time such as tomorrow, the day after tomorrow, in a week (month, year), next year, in 2008, etc.

Next year I’ll be 18.

Spring will come soon.

In 100 years’ time there will be a lot more people than there are now.

Spring has come, so the snow will start melting, the birds will come back home.

2. An action which the speaker regards as possible, probable or likely to happen in future.

I’m 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.

3. An action which is spontaneous, not part of a plan.

Don't lift the suitcase. I'll help you.

It looks like rain. I'll take my umbrella then.

What would you like to drink? – I'll have a coke, please.

4. A future action in complex sentences in the main part. But after when, while, before, after, as soon as, until / till we use Present Simple, Present Perfect

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 wont recognize her.

Come on! Mum will be worried if we are late again.

I won’t send the parcel until I hear from you.

As soon as Bob and Ashton have got married, they'll move to California.

I shan't phone you until I have done my homework.

Facts to be remembered

1. We use shall I … ? / shall we … ? to ask somebody’s opinion (especially in offers or suggestions)

Shall I open the window?

I’ve got no money. What shall I do?

Where shall we go this evening?

2. You can use won’t to say that somebody / something refuses to do something

The car won’t start. I wonder what’s wrong with it.

Ex. 8. Make the following sentences interrogative and negative.

Types of Questions







Auxiliary verb


Predicate or part of it


Adverbial modifier





in the evening?




free time

in summer?




her hair

in the morning?

Special (except the subject)







Interrogative Pronoun

Auxiliary Verb


Predicate Part of Predicate


Adverbial Modifier





to school

so early?





the money from?






Special (to the subject)





Interrogative Pronoun (+ a noun)



Adverbial Modifier




the information?

Which bus


to the city centre?

Whose friends

will visit


in hospital?

Pre-Reading Activity

Guess the meaning of the following words.

Mathematician [,mxTImq`tIS(q)n], founder [`faundq], progress [`prqugres]

information [Infq`meIS(q)n], privilege [`prIvIlIG], union [`jHnIqn], especially [I`speS(q)lI], comfortable [`kAmf(q)tqbl].

Read and learn the basic vocabulary terms.

delay (n) [dI`leI]


recent (adj) [`rJs(q)nt]

последний, недавний

manage (v) [`mxnIG]

суметь (сделать)

belong (v) [bI`lON]

принадлежать, быть частью группы

outstanding (adj) [aut`stxndIN]

выдающийся, знаменитый

room-mate (n) [`rHm,meIt]

товарищ по комнате

describe (v) [dI`skraIb]


routine (n) [rH`tJn]

определенный режим,

заведенный порядок

flash by (v) [`flxS]

пронестись, промчаться

drag (v) [`drxg]

тянуться, затягиваться

instructive (adj) [In`strAktIv]

содержательный, поучительный

compete (v) [kqm`pJt]


facilities (n) [fq`sIlItIz]

благоприятные условия, возможности

area (n) [`e(q)rIq]

район; область; сфера

various (adj) [`ve(q)rIqs]

различный, всевозможный

society (n) [sq`saIqtI]

общество, объединение

depend on (v) [dI`pend]

зависеть от

imagine (v) [I`mxGIn]

воображать, представлять себе

nowadays (adj) [`nauqdeIz]

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


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

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

места отдыха и развлечений

and entertainments

my best regards to your parents

мой сердечный привет 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 A-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 everyday. 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.

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.

The 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 following questions.

1. Do you take a cold or a hot shower in the morning? 2. What time do you leave home to get to 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.

Ex. 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. I 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.

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.

… it for you.

2. Enjoy your holiday.

Thank you. … you a postcard.

3. I don't want this banana.

Well, I'm hungry. … it.

4. Do you want a chair?

No, it's OK. … on the floor.

5. Did you phone Jenny?

Oh no, I forgot. … it now.

6. Are you coming with me?

No, I don't think so. … here.

7. How do you use this camera?

Give it to me and … you.

Ex. 13. Write sentences beginning I think... or I don't think... .

10 (she won’t be up to her neck in work)

Ex. 14. Ask special questions.

1. 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) 11. I 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. I 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. I used to go fishing when at school, but now I haven’t got any time to do that.

Ex. 16. Translate into English.

1. Летом я принимаю душ по утрам, а зимой я часто принимаю ванну. 2. В Минске есть много мест отдыха и развлечений. 3. Твоя квартира выходит окнами на юг или на север? 4. Я занимаюсь 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 днём. 9. Здесь у меня есть возможность играть в теннис два раза в неделю. 10. Как правило, рабочая неделя пролетает очень быстро. 11. Катя живёт в студенческом общежитии №7, и с ней в комнате живут ещё две девушки.

Ex. 17. Writing.


Grammar: Continuous tenses. Pronouns: some, any, no

Continuous tenses

Present Continuous

Past Continuous

Future Continuous





am working

is working

are working


was working





will be working



were working

Present Continuous Tense

The Present Continuous Tense is used:


For an action happening at the moment of speaking usually with time expressions: at this moment at the time, now, at present, just now

The younger children are sleeping now, but the elder children are not sleeping. They are watching TV. What are their parents doing?

For a temporary action

Jack is studying at Oxford University. Is his friend Bill studying there too? No, he is not studying at Oxford. He is working for a company in London.

For a pre-arranged, planned and intended action

My father is leaving for Paris tonight. My mother is not going with him. Is she staying at home?

Ex. 1. Compare and analyze the usage of Present Simple Tense and Present Continuous Tense.

1. I 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. I usually don’t rest after university, but today I’m very tired and I am having a rest. 7. I 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. The foreign scientists are flying back to Europe tomorrow. Some passenger planes fly faster than sound.

Past Continuous Tense

The Past Continuous Tense is used:


For an action in progress at a definite moment in the past usually with time expressions

at five o’clock yesterday ,all day yesterday, the whole evening, all day long, when he came, while

1. What were you doing at one o’clock yesterday? – At one o’clock yesterday I was having lunch. 2. Were you waiting for a bus when I noticed you at the bus stop? – No, I wasn’t. I was just passing by it.

For a temporary action in the past

1. When Alex met Mary in Minsk last year, she was not having preparatory courses for University. 2. Was she already studying at BSU? Yes, she was studying at the faculty of Mechanics and Mathematics.

Ex. 2. Compare and analyze 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 was 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.

Future Continuous Tense

The future Continuous Tense is used:


For an action which will be going on at a definite moment in the future: at 6 o’clock tomorrow, the whole day tomorrow, at this time tomorrow

I will be working in the library at 10 o’clock tomorrow. And you? What will you be doing at this time? I will not (won’t) be working in the library, I’ll be preparing for my exams.

For an action which will be going on during a certain period of time in the future: when he comes

The children will not (won’t) be sleeping when I come home from work.

Will they be having a party?

No, they won’t. They will be having supper.

Ex. 3. Compare and analyze 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 I’ll 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. This time tomorrow evening my friends will be flying over France. They’ll probably go to the UK too. 5. I think I’ll take an umbrella because it is still raining. They say it will be raining the whole week-end.

Ex. 4. Make the following sentences interrogative and negative.

This 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. 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.

Pronouns some, any, no

Affirmative sentences


Some and its compounds (somebody, something, somewhere, someone) is used with the meaning “small amount of something” and “какой-то, несколько

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.

Я хочу задать тебе несколько вопросов.

Any is used with the meaning “любой

You can come any day you like.

Вы можете прийти в любой день.

Some is used before numerals with the meaningприблизительно, несколько

There were some ninety people at the concert.

На концерте было около девяноста человек.

Any is used in general reported questions

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 is used after hardly, without, seldom, never, rarely etc.

I hardly know anybody in the neighborhood.

Я едва знаю кого-либо из соседей.

Interrogative sentences


Any and its compounds( anybody, anyone, anything, anywhere) are normally used in interrogative sentences

Have you got any free time tonight?

У тебя есть свободное время сегодня вечером?

Some and its compounds are used in interrogative sentences when we make an offer or request

May I have some more coffee?

Могу ли я взять еще кофе?

Would you like some tea?

Не хотели бы вы чаю?

Some is used with the meaning “a part of

Can I take some of these oranges?

Могу ли я взять апельсины?

Any can modify comparatives

Can you go any faster?

Не мог бы ты идти немного быстрее?

Some is used in special questions

Why haven’t you given me something to cover with?

Почему ты не дал мне, чем укрыться?

Negative sentences


No is equivalent to not any

I have no free time left. (I don’t have any free time left.)

У меня не осталось свободного времени.

No can modify comparatives

I’m afraid the weather is no better than it was yesterday.

Я боюсь, что погода нисколько не лучше, чем она была вчера.

Not any is not normally used with subjects, no and none of are used instead

No tourists ever came to our village.

Никакие туристы никогда не приезжали в нашу деревню.

None of my friends lives near me.

Никто из моих друзей не живет рядом со мной.

Ex. 5. Analyze the following sentences and translate them into Russian.

1. Are there any English books in the library? 2. I can’t 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. It’s very tasty. 8. Would you like some time to finish your work? 9. I know some funny jokes. 10. No students are happy to have extra seminars. 11. I 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. Chose the right word.

Ex. 7. Answer the following questions using the pattern below.

- Have you got any sisters?

- Yes, I have some.

- No, I have no sisters.

- 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. ['sIstqm], symbol n. [sImbl], positive adj. ['pOzItIv], diagram n. ['daIqgrxm], complex adj. ['kOmpleks], rational adj. ['rxSqnl], fundamental adj. [ֽfAndq'mentl], fact n. [fxkt], express v. [Iks'pres], negative adj. ['negqtiv], start n. [sta:t], position n. [pq'zISn], direction n. [dI'rekSn], occupy v. [`OkjupaI], zero n. [`ziqrou], different adj. ['dIfrent], basic adj. [`beIsIk].

Read and learn the basic vocabulary terms.

number (n) [`nAmbq]- число, количество, номер

date back to (v) [`deIt] – датироваться, относиться к определенному времени

antiquity (n) [xn`tIkwItI]- древность, античность

integer (n) [`IntIGq]- целое число

aid (n) [`eId] - помощь

complete (v) [kqm`plJt] - завершать, делать полным

fraction (n) [`frxkSn] - дробь

imaginary (adj) [I`mxGInqrI] - мнимый

count (v) [`kaunt] - считать

real (adj.) [rIql] - действительный

unity (n) [`jHnItI] - единица, единство

establish (v) [Is`txblIS] - устанавливать

ratio (n) [`reISIOu] - отношение, пропорция

negative (adj) [`negqtIv] - отрицательный

division (n) [dI`vIZn] - деление

either (conj.) [`aIDq] - любой, каждый

allow (v) [q`lau] - позволять, допускать

divisor (n) [dI`vaIzq] - делитель

quotient (n) [`kwOuSnt] - частное, отношение

include (v) [In`klHd] - заключать, содержать в себе

special (adj.) [`speSql] - особый, специальный

compose (v) [kqm`pouz] - составлять

Memorize the following word combinations

Reading Activity


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.

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 plus 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.

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. The numbers in this uncountable set are known as the real numbers. The integers correspond to a small subset of the reals.

The 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; -√3 and π are examples of such numbers.

The 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, -√-1, 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+ √-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.

a. основная система; b. система чисел; c. действительные и мнимые части; d. в любом направлении; e. заменяется; f. обозначать действительную часть; g. состоит из; h. мнимые числа; i. частное двух чисел; j. через, посредством; k. соответствовать подмножеству

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, действительная и мнимая части.

1. … is occupied by the number zero. 2. The irrationals can not be expressed as …. 3. Positive fractions or numbers can be expressed as…of two of these integers. 4. We are not allowed to use the number zero as… 5. The beginning of our number system dates back to… 6. The real number system is formed of… 7. Numbers √-2, -√-1 are called… 8. The expression 2+√-3 is composed of… 9. To denote the real part of a complex number we use…

Ex. 11. Open the brackets putting the verbs in the correct form.

1. Why you (look at) me like that? 2. We (read) a book while he (cook) lunch at that time yesterday. 3. This time next month I (to cross) the Pacific Ocean. 4. I (wait) for you when you come out. 5. He (sit) in a café when I saw him. 6. I (go) to the cinema tonight. 7. Look! She (wear) the same dress as me. 8. Yesterday at 8 o’clock we (watch) the football match. 9. He (drive) his car himself today. 10. Great news! Jack (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. There is … time left at all. 12. We can rarely meet as brave as he is.

Ex. 13. Ask special questions.

1. The students are applying irrationals in the given mathematical problems. (Who?) 2. The teacher is explaining how to mark off the line with numbers. (What?) 3. There are some positive numbers in the expression. (What?) 4. He was trying to find the common ratio of the three fractions from 3 until 5 o’clock. (How much time?) 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?) 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. The 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. The delegation of scientists is coming to Minsk in a week. 9. This time next term the students of MMF will be celebrating the birthday of the faculty. 10. I don’t see any other ways to arrange this matter. 11. I 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.


Grammar: Modal verbs and their equivalents



Can, cannot (can’t);

am/are/is (not) able to


Can; shall/will (not) be able to


Could, could not (couldn’t);

Was/were (not) able to

Mental, physical or circumstancial ability to do something

знать как сделать, уметь, мочь, иметь право сделать что-либо; to be able to – быть способным к чему-либо, быть в состоянии, иметь силу, власть, ум, возможность что-либо сделать

He can swim.

I am able to help you now. (particular situation)

I can’t hear anything.

I can do it tomorrow.

I shall be able to help you.

He could speak three languages before he was twelve.

(repeated action, ability in the past)

He was able to win the game.

(managed to do; past single action)


Can you tell me the time? (informal)

Could you give me a lift? (polite)

Permission/Asking for permission (informal)

You can take the book now.

Can I go to the party too, Mother?

Prohibition нельзя (in informal speech or writing)

You can’t cross the street here.


Can I help you? (informal)

Strong doubt не может быть, чтобы; вряд ли, неужели

You can’t be cold. It’s very hot here.

She can’t be Sam’s sister. She is in Kiev now.


This road can get very busy. (general possibility) The roads could get very busy tomorrow because there is a demonstration. (specific situation)

Ex. 1. Analyze 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. I can speak three languages but I can’t 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 can’t 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 the library. 11. Anybody can make mistakes. 12. Can (could) you show me these data? – I’m afraid I can’t. 13. You can speak to Jane now. 14. Could you spell your name for me?



May, may not



Shall/will (not) be allowed to


Might, might not;

Was/were (not) allowed

Permission/Asking for permission можно, можете

You may answer the questions later. (formal)

May I come in? (polite)

You may call me tomorrow. (formal)

They will be allowed to go to the concert.

He said that I might borrow his pen.

He was allowed to enter the country.

Prohibition нельзя (in formal speech and writing)

You may not talk during the test. (formal)

You will not be allowed to take the exam.

We were not allowed to tell her everything.

Doubt/Uncertainty может быть, возможно

The rain may (might) stop later in the day.

He may come tomorrow.


In the museum you may (can) see some interesting things.

Ex. 2. Analyze 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. The answer may give the key to the whole problem. 5. Will he be allowed to take part in the conference? 6. The 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, must not (mustn’t)


Shall/will (not )have to


Had to, did not have to;

Did you have to…?

Necessity, duty, obligation должен, нужно, надо

You must obey these rules.

I’m afraid I must go now.

She will have to come later.

We had to get more exercise last week.

Near certainty, logical assumption вероятно, должно быть

She must be about twenty.

BUT: Probably, he doesn’t know English.

BUT: The weather is likely to change.

BUT: Evidently, she didn’t know my address.


You mustn’t walk on the grass.

Order, strong advice

You must revise for your test.

Ex. 3. Analyze 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 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?



Have/has to, do/does not have to; Do you have to…?


Shall/will (not) have to;

Will you have to…?


Had to, did not have to

Did you have to…?

Necessity, duty, obligation должен, нужно, надо, приходится, вынужден

She has to find a new job.

They’ve got to sell their car.

I shall have to speak to them about this plan.

He had to return home.

Absence of necessity не нужно, не надо

We don’t have to attend classes on Sunday.

Sam won’t have to come back till April.

She didn’t have to take a taxi.

Ex. 4. Analyze 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.



Am/is/are (not) to do…


Am/is/are (not) to do…


Was/were (not) to do…

Necessity, an expected, planned action, a predestined event

должен, предстоит, суждено

What am I to do?

The train is to come at 7.

You were to stay here.

He was never to see her again.

Order, formal instructions не делай, не смей делать

You are not to say a word to anyone.

No-one is to live the room.

Ex. 5. Analyze 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.



Need, needn’t


Need, needn’t


Necessity нужно, надо

Need I call the doctor tomorrow?

(Also: Do I need to call the doctor?)

(Also: The plants needed to be watered.)

Absence of necessity не нужно, не надо

I needn’t speak to her about it.

(Also: She didn’t need to work as hard as me.)

(Also: He didn’t need to stay in a hotel.)

Ex. 6. Analyze 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 (not), ought (not) to


Should (not), ought (not) to


Should (general advice) следует, должен

You should try again. Why should I go there?

Ought to (moral obligation, laws, rules) следует, должен

You ought to tell your parents the truth.

You ought to keep to the speed limit.


It’s 10 o’clock. He should/ought to be at work.

It shouldn’t be difficult to get there.

Ex. 7. Analyze 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 [`desIm(q)l] adj, combination [,kOmbI`neISqn] n, positive [`pozItIv] adj, negative [`negqtIv] adj, symbol [`sImbql] n, plus [plAs] n, minus [`maInqs] n, indicate [`IndIkeIt] v, sum [sAm] n, product [`prodqct] n, numeration [,nju:mq`reISqn] n, base [beIs] n, arithmetic [q`rITmqtIk] n, system [`sIstIm] n, represent [,reprI`zent] v

Read and learn the basic vocabulary terms.

number (n) [`nAmbq] число, количество; v перечислять

numeral (n) [`nju:mqrql] цифра, символ, число

digit (n) [`dIGIt] цифра

value (n) [`vxlju:] величина, значение; (v) ценить

derive (v) [dI`raIv] происходить, получать

introduce (v) [,Intrq`djHs] вводить, представлять, знакомить

denote (v) [dI`nqut] обозначать, отмечать, означать

determine (v) [dI`tE:mIn] определять, устанавливать

invent (v) [In`vent] изобретать, создавать, придумывать, открыть

enable (v) [I`neIb(q)l] давать возможность, делать возможным

appear (v) [q`pIq] появляться

sign (n) [saIn] n знак, v подписывать

addition (n) [q`dIS(q)n] сложение

add (v) [xd] складывать

addend (n) [x`dend] слагаемое

subtraction (n) [sqb`trxkS(q)n] вычитание

subtract (v) [sqb`trxkt] вычитать

minuend (n) [`mInjuqnd] уменьшаемое

subtrahend (n) [`sAbtrqhend]вычитаемое

difference (n) [`dIfqrqns] разность

inverse (adj) [In`vE:s] обратный, противоположный

multiplication (n) [,mAltIplI`keIS(q)n] умножение

multiply (v) [`maltIplaI] умножать

multiplicand (n) [,mAltIplI`kxnd] множимое

multiplier (n) [`mAltIplaIq] множитель

factor (n) [`fxktq] сомножитель; v разложить на множители

division (n) [dI`vIZqn]деление

divide (v) [dI`vaId] делить

dividend (n) [`dIvIdent] делимое

divisor (n) [dI`vaIzq] делитель

quotient (n) [`kwquSqnt] частное

meaningless (adj) [`mJnINlIs] бессмысленный

Memorize the following word combinations

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. The 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.

The 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.

There are special signs to indicate operations with numbers. They are plus (+), minus (-), multiplication (x) and division (:) signs.

The 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. The 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. The 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. The number resulting from the multiplication is known as the product. You must remember 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. The 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. present notation; 4. numeration system; 5. decimal system; 6. to be based on tens; 7. inverse operation; 8. capital letters; 9. arithmetic sentence; 10. to measure the distance

a. основываться на десятках; b. обратная операция; c. заглавные буквы; d. измерять расстояние; e. десятичная система; f. представить числа; g. использовать одни и те же числа; h.система счисления; i. арифметическое выражение; j. современная система записи

Ex. 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. The Roman system of numbers is based upon the letters A, B, 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 larger units of ten, hundred, thousand and so on. 8. We work both with positive and negative numbers in arithmetic. 9. The 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. The result of division is known as …. 10. The difference is the number that when added to the … 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. The 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. They 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. I must discuss the details of my dissertation with the science adviser. (Who(m)) 8. You need to follow your teacher’s 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)

Ex. 14. Choose the phrase closest in meaning to the given statement.

1. Dan can’t be a teacher.

a) I’m 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/won’t 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 ill, he (could/was able to/may) finish all the paperwork. 8. You (can/must/ought to) go and see that movie. It’s very interesting. 9. Don’t worry, you (don’t have to/mustn’t/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.

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 поделить 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 обозначили (указали) операцию сложения знаком плюс. 9. Ни один студент не мог решить задачу, заданную профессором. 10. Им придется использовать двоичную систему счисления. 11. Данный элемент может быть обозначен тем же символом.


Grammar: Present, Past, Future Perfect (Active Voice)

The Present Perfect Tense

The Past Perfect Tense

The Future Perfect Tense

have/ haven’t + V3

has/ hasn’t + V3

had/ hadn’t + V3

shall/ will have + V3

shan’t/ won’t have + V3

just, yet, already, ever, never, lately, recently, today, this week, this month, etc.

They have not done it yet. Они еще это не сделали.

(to be, to have, to know) for, since, how long

How long have you known him? Как давно ты его знаешь?

by 6 o’clock yesterday, by that time, by the end of last year, before he came

They had done it by the time we came.

Они сделали это к нашему приходу.

They hadn’t done it by the time he came.

Had they done it by the time you came?

by 6 o’clock tomorrow, by this time tomorrow, by the end of next year, before he comes

They will have done it by 6 o’clock tomorrow.

Они сделают это завтра к шести часам.

They won’t have done it by 6 o’clock tomorrow.

Will they have done it by 6 o’clock tomorrow?

Ex. 1. Compare Russian and English Tense forms.

Grammar Rules Patterns.

Я решил 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.)

Я решу эти задачи …

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 following 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.

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.

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) When we meet again, I am sure, she will have become a student of the faculty of Cybernetics.

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 21st 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 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. This century scientists have made a lot of discoveries about the universe.

Pre-Reading Activity

Guess the meaning of the following words.

Fraction ['frxkS(q)n] n, fractional ['frxkSqnl] a, equivalent [i'kwivqlqnt] a, rational ['rxSqnl] a, process ['prquses] n, concept ['kOnsept] n, arithmetic [q'riTmqtik] n, operation [Opq'reiSn] n, horizontal [,hOri'zontl] a, separate ['sepqreit] v ['seprit] a, decimal ['desiml] a, type ['taip] n.

Read and learn the basic vocabulary terms.

part [pa:t] n


proper ['prOpq] a


improper [im'prOpq] a


change [tSeindZ] n

замена, заменять

proportional [prq'pLSnl] a


equal ['i:kw(q)l] a


reduce [ri'dju:s] v


relation [rI'leISn] n


ratio ['reISIqu] n

пропорция, отношение

numerator ['nju:mqreitq] n

числитель (дроби)

denominator [di'nOmineitq] n

знаменатель (дроби)

slanting ['slRntIN] a


define [dI'faIn] v


quantity ['kwOntiti] n

величина, количество

mixed [mikst] a


imply [Im'plaI] v


point [pOint] n


repeat [ri'pi:t] v


repeating (fraction) [ri'pi:tiN] a

периодическая (дробь)

terminating [,tWmi'neitiN] a

конечная (дробь)

agreement [q'gri:mqnt] n


produce [prq'djHs] v


consider [kqn'sidq] v

рассматривать, полагать, считать

the former (of) ['fLmq] adj

первый (из)

compare [kqm'pFq] v


irreducible [iri'dju:sqbl] a

несократимая (дробь)

nought [nO:t] n


whole ['hqul] a

весь, целый

power ['pauq] n


Memorize the following word combinations.

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.

A fraction is a quotient of numbers, the quantity obtained when the numerator is divided by the denominator. Thus, ¾ represents three divided by four.

The 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 a slanting line (e.g. ½), or may be written above and below a horizontal line.

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 <Object: word/embeddings/oleObject1.bin> where a and n 0.

Usually there are several ways of reading fractions. One may say ‘three quarters’ for <Object: word/embeddings/oleObject2.bin> and ‘one sixth’ for <Object: word/embeddings/oleObject3.bin>. In strictly mathematical contexts these fractions might 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’ (e.g. <Object: word/embeddings/oleObject4.bin>).

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 the denominator (e.g. <Object: word/embeddings/oleObject5.bin>, <Object: word/embeddings/oleObject6.bin>). A mixed number is the sum of a whole number and a proper fraction. This sum is implied without the use of ‘’ sign (e.g. 1<Object: word/embeddings/oleObject7.bin>).

Fractions which represent the same fractional number like <Object: word/embeddings/oleObject8.bin>, <Object: word/embeddings/oleObject9.bin>, <Object: word/embeddings/oleObject10.bin>, and so on, are called equivalent fractions.

You have already known that any fractional number multiplied by one has the same value as the original number (e.g. 1 <Object: word/embeddings/oleObject11.bin> <Object: word/embeddings/oleObject12.bin> = <Object: word/embeddings/oleObject13.bin>).

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 <Object: word/embeddings/oleObject14.bin>. When both the numerator and the denominator are both multiplied by two, the result is <Object: word/embeddings/oleObject15.bin> which has the same value as <Object: word/embeddings/oleObject16.bin>.

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 lowest or simplest terms. Consider the following fractions: <Object: word/embeddings/oleObject17.bin> and <Object: word/embeddings/oleObject18.bin>. The former is not in the lowest terms because both 3 and 9 can be divided by 3. In contrast <Object: word/embeddings/oleObject19.bin> 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. It’s 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 know 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 we multiply it by one? 9. In what way can we reduce a fraction? 10. What is a decimal fraction?

Ex.6. Find the English equivalents for the Russian words and word 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. часть целого; 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; j. an improper fraction; k. the least common denominator (LCD); l. 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 can’t 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. It’s 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.

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

Ex. 10. Choose the correct tense form.

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. (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. The students haven’t attended the course in the history of mathematics this month. (why) 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. Man’s 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. Pr. 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 что-то разделили на несколько равных частей. 2. В дроби число, стоящее над чертой называется 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 его. Таким образом, неправильная дробь равна или больше единицы. 5. Числа, состоящие из целого числа и дроби, называются смешанными. 6. Если вам нужно сократить дробь, то вы должны разделить числитель и знаменатель этой дроби на одно и то же число. Это число называется общим делителем. 7. Преподаватель только что объяснил нам, что дроби, знаменателями которых является числа, выраженные единицей с последующими нулями (одним или несколькими) называются десятичными числами. 8. Десятичные числа, в которых одна или несколько цифр повторяются многократно, называются периодическими, а те, в которых повторяются нули непериодическими десятичными числами (или конечными десятичными числами). 9. Ты уже перенёс 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 ещё не сложил эти дроби, так как не знает, как привести их к общему знаменателю.


Grammar: Degrees of Comparison; Perfect Continuous Tenses

Degrees of Comparison

short words:

-er, -est

big – bigger – the biggest

thin – thinner – the thinnest

short – shorter – the shortest

words ending in – y:

-ier, -iest

easy – easier – the easiest

happy – happier – the happiest

heavy – heavier – the heaviest

long words:

more, the most

important – more important – the most important

general – more general – the most general

both ways for some adjectives (stupid, gentle, friendly, cruel, common, pleasant, quiet, shallow)

simpler – the simplest


more simple- the most simple

narrower – the narrowest


more narrow – the most narrow

cleverer – the cleverest


more clever – the most clever




real - more real - the most real

right - more right - the most right

wrong - more wrong - the most wrong

a bit/ a littlе + comparative (немного)

a bit



more difficult


much/ a lot/ far + comparative (намного, гораздо)

Its much cheaper. – Это намного дешевле.

The film is far better than the book. Фильм гораздо лучше, чем книга.

By far + superlative (гораздо)

He is by far the best student in the class. – Он

гораздо лучше, чем все остальные студенты.

Ex. 1. Analyze 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 student’s 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 this task is wrong. My friend’s 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 ½ . 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.

Irregular Comparatives and Superlatives:

good / well – хороший/хорошо

bad / badly – плохой/плохо

much / many - много

little - мало

far – далекий/далеко

late – поздний/поздно

old – старый

near – близкий/ близко

better – лучший/лучше

worse – худший/хуже

more – больше/более

less – меньше/менее

farther– дальше

further– дальше, дополнительный, добавочный

later – более поздний/позже

latter – последний (из упомянутых)

older – более старше

elder – старше в семье



best – самый лучший

worst – самый худший

most – наибольшее количество

least – наименьшее количество

farthest – самый


furthestсамый дальний, дальше всего

the latest (there may be more to come) – самый поздний, но не последний

the last (final, before this) – последний, окончательный

the oldest – самый старший (о возрасте)

the eldest - старший в семье

the nearest (о расстоянии)

the next (порядок)

Ex. 2. Analyze these sentences and compare the adjectives given there. Translate them into Russian.

1. This 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 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 café is in a five - minute walk from here. The next news bulletin comes in 10 minutes.

Types of Comparisons

There are a number of different sentence patterns with comparative and superlative forms:

Than чем

This book is more interesting than that one. Today is warmer than it was yesterday. The chair is less comfortable than this armchair.

You are two years older than me/than I am.

the most (the least) наибольший, больше всего (наименьший, меньше всего)

This is the most exciting place of all I have ever been to. Carol is the least experienced person in our team.

Asas такой же … как,

так же … как

He is as tall as his father.

Could you come as soon as possible?

Not as…as не такойкак,

(not so …as) не так … как

The weather is not as (so) good as it was yesterday.

He is not so (as) tall as his father.

Twice as…as Three times as…as

в два/три раза больше

The sameas такой же … как

Oil is twice as expensive as it was several years ago.

I'll have the same ice-cream as last time.

The more... , the better чем, тем

The warmer the weather, the better I feel.

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. This 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. The 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. The 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. There 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. The (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

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 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 статьясамая трудная из всех, которые мы когда-либо переводили. 2. Сегодня намного холоднее, чем вчера. 3. Его книга гораздо интереснее вашей. 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 студент 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 английский? 12. В пятой группе больше студентов, чем в во второй.13. У меня меньше книг, чем у Кати. 14. Мария гораздо красивее своей сестры. 15. Я получил дальнейшую (добавочную) информацию по этому делу. 16. У мамы меньше времени, чем у отца.17. Этот текст намного больше, чем предыдущий. 18. Он не так молод, как мой брат.

Perfect Continuous









been working








had been working
















1) for a long time

2) for 5 years

3) since 2 o’clock

4) all morning/day/week

1) for 2 hours when he came.

2) since 2 o’clock when you came

1) for two hours by the time he comes

2) next year for five years already

3) for forty minutes when you ring us up

Ex. 7. Analize 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.

- I have been studying it since September.

- They had been discussing the definition for some time, when the teacher came.

- He will have been speaking at the conference for half an hour, when his scientific adviser comes.

b) Negative

1. He is not doing any experiments right now.

2. She wasn’t watching TV yesterday.

3. Next year they will not be living here as they are moving to another house.

- He has not been doing any experiments since last year.

- She had not been watching TV for some time when her mother came.

- Next year they will not have been living here for 5 years but for 6 years already.

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?

- Have they been working for a long time?

- Had he been translating this text for an hour when his mother came?

- Will she have been doing her research for five years by the end of this year?

Facts to be remembered

For actions started and finished in the past and lasted for some time. The result of the actions is visible in the present.

He looks so tired. He has been studying for his exam.

To express anger, annoyance or irritation.

Who has been using my cup?

We can use the Present Perfect Continuous to talk about longer repeated actions that have finished.

“You look tired.” – “I have been running.”

For certain duration with visible results in the past.

They were wet because they had been walking in the rain.

As past equivalent to the Present Perfect Continuous.

(She is going to the doctor. Her leg has been aching for two days). She went to the doctor. Her leg had been aching for two days.

Note the difference in translation between the Present Perfect Continuous and the Present Perfect Tenses.

Я бежала всю дорогу.

Я учил неправильные глаголы весь день.

Извините за беспорядок – я крашу дом.

Он пробежал всю дистанцию до финиша довольно хорошо.

Я выучил неправильные глаголы.

Я покрасил две комнаты с обеда.

Ex. 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. The 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. Peter’s 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 a. as if he had been running for several hours without a rest.

2. “You look tired. b. she will have been working at the department for 35 years.

3. “Aren’t you hungry?” c. I decided to have a cup of tea.

4. By the 1st of August, 2009 d. because she has been painting the ceiling.

5. He was out of breath e. “I have been writing my course paper for more than a month.

6. “Why are my books all over the floor?” f. because she had been cleaning the flat the whole day.

7. After I had been walking for an hour g. “No, I’ve been eating all day”.

8. Her hair is white h. “He has been walking in the rain”.

9. “Why is his coat wet?” i. when his brother came

10. She looked very dirty 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 университет. 5. Ты выглядишь усталой. – Я стирала белье весь день. 6. Я вымыл свою машину. – Разве она не выглядит чудесно? 7. Сейчас она учит испанский язык, но она еще не очень говорит. 8. Он часами играет эту музыку на фортепьяно. Пусть он перестанет играть. 9. Студенты пишут этот тест уже 20 минут. Только один студент уже написал его. 10. На этой неделе я написала 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 [GenqrqlaI`zeIS(q)n] n, arithmetic [q`rITmetik] n, procedure [prq`sJGq] n, symbol [`sImbql] n, formula [fLmjulq] n, characteristic [,kxrIktq`ristik] n, coefficient [,kouI`fISqnt] n, zero [`zIqrou] n

Read and learn the basic vocabulary terms.

compute [kqm`pjHt]


deal (dealt) with [dJl]

иметь дело с; рассматривать

apply [q`plaI]

использовать, применять

instead of [In`sted]


particular [pq`tIkjulq]


concerning [kqn`sWnIN]

относительно, касательно

replace [rI`pleIs]

заменять, замещать

hold (held) for [hould]

годится для

to be true [trH]

быть верным, справедливым; удовлетворять

likewise [laIkwaIz]

подобно, так же, таким же образом

raising to a power

[reIzIN tq q `pauq]

возведение в степень

term [tWm]


in terms of

на языке... , в переводе на... , с точки зрения

multinomial [mAltI`noumjql]

многочлен, полином

binomial [baI`noumjql]

двучлен, бином

trinomial [traI`noumjql]




Memorize the following word combinations.

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 • 20 • 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 • b+b2.

This 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.

The 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.

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 • 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).

Algebraic expressions consisting of more than one term are called multinomials. 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 plus 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 (<Object: word/embeddings/oleObject20.bin>) has the same meaning as in arithmetic. It is frequently replaced by the fraction line; thus <Object: word/embeddings/oleObject21.bin> means the same as 6 <Object: word/embeddings/oleObject22.bin> 3 and in both cases the result or quotient is 2. The two dots above and below the line in the division sign (<Object: word/embeddings/oleObject23.bin>) 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 which means "greater than" or "less than". The 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".

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 are used in algebra and what do they indicate? 10. How is the sign (<Object: word/embeddings/oleObject24.bin>) read? 11. What is the meaning of the multiplication sign, the equality sign and the division sign? 12. What does the expression (a + b) mean?

Ex. 12. Find the English equivalents for the following Russian words and word combinations.

1. утверждение; 2. иметь дело; 3. рассматривать; на языке; 4. вычислять; 5. подобно, таким же образом; 6. алгебраические выражения; 7. полином; 8. для удобства; 9. член; 10. трехчлен; 11. возведение в степень; 12. тогда утверждение справедливо; 13. представлять

a. then the statement is true; b. trinomial; c. raising to a power; d. deal with; e. for convenience; f. term; g. compute; h. likewise; i. in terms of; j. algebraic expressions; k. statement; l. represent; m. polynomial

Ex. 13Give the proper English equivalents for the Russian expressions.

computed, instead of, simpler, ordinary letters, replace, hold, 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 + 2c•d2. 9. Algebraic expressions may be given a более простая form by combining similar terms. 10. Algebraic expressions consisting of more than one term are called полиномами.

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 6x6 + 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” (<Object: word/embeddings/oleObject25.bin>) 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 plus (+) and minus (-) to indicate addition and subtraction. (why, 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 <Object: word/embeddings/oleObject26.bin>, or a power x3, but must never be either a sum or a difference. 4. An algebraic expression which consists of several monomials connected by the plus and minus signs, is known as a polynomial. 5. Such is, for instance, the expression x + yz + c-3 + <Object: word/embeddings/oleObject27.bin> 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 -a, term +b3 and so on. 8. When there is no sign before the first term it is xy or + xy.

Ex. 17. Translate into English.

1. Алгебра – это система правил, касающихся действий с числами. 2. В алгебре числа обозначаются буквами, а не цифрами. 3. Поскольку буквы обозначают числа, все законы арифметики годны для действий с буквами. 4. Знаки, 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 выражению можно придать более простую форму путём приведения подобных членов. 8. Алгебраическая сумма нескольких одночленов называется многочленом. 9. Двучлен – это алгебраическое выражение, состоящее из двух членов, трёхчлен – алгебраическое выражение, состоящее из трёх членов.


Grammar: Present, Past, Future Indefinite, Passive Voice

Present Indefinite Passive

Past Indefinite Passive

Future Indefinite


I am

He/She/It is asked

We/You/They are

I/He/She/It was

We/You/They were asked

I/We shall/will be

He/She/It will be asked

You/They will be

Universal truths, repeated actions indicated by adverbials of frequency usually, always, often, sometimes, seldom, rarely, never, as a rule, every (day, month, etc.), once (a week, a month, etc.).

A single action or a state with time adverbials such as yesterday, ago, last (time, week, month, year), the other day, in 2008.

A predicted future action, an action which the speaker regards as possible to happen in future with the adverbials of time tomorrow, the day after tomorrow, in (a week, month, year), next (week, month, etc.), in 2015.

1. She is often asked at the lessons. – Ее часто спрашивают на уроках.

2. Computers are used everywhere. – Компьютеры используются везде.

1. He was asked at the lesson yesterday. – Его спросили на уроке вчера.

2. A computer was used to solve a difficult problem last week. – Компьютер был использован для решения трудной задачи на прошлой неделе.

1. You will be asked at the lesson tomorrow. – Вас спросят на уроке завтра.

2. This new computer will be used in our laboratory next week. – Этот новый компьютер будет использоваться в нашей лаборатории на следующей неделе.

Ex. 1. Read these sentences. Compare the predicates in these pairs of sentences


All lectures at the University are attended by me.

The Internet is often used by us.

A new method is applied in his research.

Новый метод применяется в его исследовании.

All fundamental discoveries are known to our scientists.


I developed a new system of notation.

The article was finished by him yesterday.

The rule was studied at the lesson last week. Это правило изучили на уроке на прошлой неделе.

A new system of notation was developed by me.


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.

Книга будет опубликована писателем в следующем году.



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?



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. 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 [Iks`preS(q)n], identical [aI`dentik(q)l], conditional [kqn`dIS(q)nl], accuracy [`xkjurqsI], classify [`klxsifaI], linear [`lInIq], transformation [,trxnsfq`meiS(q)n], original [q`rIGqnl], reduce [rI`djHs].

Read and learn the basic vocabulary terms.

equation (n) [i`kweiSqn] – уравнение

statement (n) [`steItmqnt] – утверждение, формулировка

finite (a) [`faInaIt] – конечный

variable (a) [`vFqriqbl] – переменная

identical (a) [aI`dentIkql] – аналогичный

briefly (adv) [`brJflI] – кратко

root (n) [rHt] – корень

aid (n) [eId] помощь

illustrate (v) [`IlqstreIt] иллюстрировать

restriction (n) [rIs`trIkSqn] ограничение

substitute (v) [`sAbstitjHt] замещать, заменять

satisfy (v) [`sxtIsSaI] – удовлетворять

linear (a) [`lInIq] – линейный

quadratic (a) [kwq`drxtIk] – квадратичный

cubic (a) [`kjHbIk] – кубический

integral (n) [`IntIgrql] – интеграл, целое число

fractional (a) [`frxkSqnql] – дробный

rational (a) [`rxSqnl] – 1. рациональный; 2. целесообразный

irrational (n) [i`rxSqnl] – иррациональное число

original (a) [O`rIGqnl] – первоначальный

extraneous (a) [eks`treInjqs] – посторонний, чуждый

Memorize the following word combinations.

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 - 4х + 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. The roots of an equation remain unchanged if the same expression is added to or subtracted from both sides of the equation.

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.

The 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. The integral equation will have all the roots of the original fractional equation but sometimes will include additional roots. These 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. Then 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

2ax + bx + с = 0 (a <Object: word/embeddings/oleObject28.bin> 0) where a, b and с are known and x is unknown.

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. In what types are equations classified according to the degree? 8. In what types are equations classified according to their form? 9. In what way can one solve an equation containing fractions? 10. What operation must one do when solving an equation by the combination of rules?

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

a. независимо от формы; b. линейное уравнение; c. равный для некоторых определенных величин; d. рассматриваемое выражение; e. путем подстановки; f. наименьший общий знаменатель; g. для каждого значения включенной переменной; h. при нахождении; i. может быть верным для; j. удовлетворять уравнению при подстановке

Ex. 5Give the proper English equivalents for the Russian expressions.

to solve, the unknown quantity, substituted into the expression, the value, roots, have been transposed, the Least Common Denominator, is checked, clearing, to satisfy

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. The 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. The 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. The statement of condition may be such that no number can be found to удовлетворить it.

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. The 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 term of 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 алгебраических выражений. 2. Корень уравнения остаётся прежним, если к обеим частям уравнения прибавить или от обеих частей уравнения вычесть одно и то же выражение. 3. Корень уравнения остаётся 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 значения неизвестного, при которых равенство будет верным). 5. После подстановки эти значения неизвестного удовлетворяют уравнению. 6. Значения неизвестного, которые удовлетворяют уравнению, называют корнями или решениями уравнения. 7. Величина, обозна-ченная через х, является неизвестной. 8. Такие задачи обычно решаются алгебраически, и используются определённые 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 уравне-ние. 12. Оба члена умножаются на выражение, содержащее неиз-вестную. 13. Как называется система линейных уравнений?



Present, Past Continuous. Present, Past, Future Perfect. Passive Voice

Present Continuous Passive

Past Continuous Passive

Future Conti-nuous


I am

He/She/It is being asked

We/You/They are

I/He/She/It was

being asked

We/You/They were


now, at this moment, while (в то время как, пока), at present

at 6 o’clock, when she came, from 6 till 7 o’clock, the whole evening

1. He is being asked by a teacher now.

Его сейчас опрашивает учитель.

2. These natural numbers are being multiplied at the moment. –

Эти натуральные числа умножают в данный момент.

1. He was being asked by a teacher at 7. Его опрашивал учитель в 7ч.

2. The problems were being discussed at 3 o’clock yesterday. –

Проблемы обсуждали вчера в 3 часа.

Present Perfect Passive

Past Perfect Passive

Future Perfect Passive

I have


She has

It been asked


You have





It had been asked




I shall/

We will


She have been

It will asked



already, yetуже (в вопро-се), ещё (в отриц. предл.), this month, recently, for along time, just

by (к), before

already, yet, by, before

1. He has already been asked by a teacher.

Учитель его уже опросил.

2. The axiom has just been accepted.Аксиома только что была принята.

1. He had been asked by a teacher by 6.

Учитель его опросил к 6ч.

2. The whole chapter had been studied by the end of the semester. – Вся глава была изучена к концу семестра.

1. He will have been asked by a teacher by 4.

Учитель его опросит к 6ч.

2. A new method will have been introduced by the end of the month. – Новый метод будет представлен к концу месяца.

Ex.1. Analyze these pairs of sentences and compare the predicates given there.

1. They are solving the equation at the moment.

2. He was dividing these numerals at 2 o’clock yesterday.

3. We have already reduced the fraction.

4. When I came back you had already replaced the terms in the equation.

5. 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.

1. Are you changing the improper fraction to the whole number now?

2. Was she writing the decimal fractions at that moment?

3. Has he omitted the plus sign in this sentence?

4. 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?

1. The students are not multiplying these integers right now.

2. We were not subtracting the fractions when the teacher came in.

3. You have not divided the numerator yet.

4. The student had not proved the theorem by the end of the class yesterday.

5. 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. I 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.

Pre-Reading Activity

Guess the meaning of the following words.

Algebraic adj. ["xlGI`breIIk]; polynomial n. ["pOlI`nqumIql]; integral adj. [`IntIgr(q)l]; constant n. [`kOnstqnt]; coefficient n. ["kquI`fISqnt]; exponent n. [eks`pqunqnt]; fundamental adj. ["fAndq`mentl]; process n. [`prquses].

Read and learn the basic vocabulary terms:

term n. [tE:m]

член, термин;

upper adj. [`Apq]

верхний, высший;

degree n. [dI`grJ]


appear v. [q`pIq]

показываться, появляться;

above prep. [q`bAv]

над, выше, сверх;

monomial n. [mO`nqumIql]

одночлен, adj. – одночленный;

binomial n. [baI`nqumIql]

двухчлен, adj. – двухчленный;

trinomial n. [traI`nqumIql]

трёхчлен, adj. – трёхчленный;

power n. [`pauq]

показатель степени;

place v. [pleIs]

размещать, ставить;

obtain v. [qb`teIn]

получать, приобретать;

arrange v. [q`reInG]

размещать, располагать, расставлять;

arrangement n. [q`reInGmqnt]

размещение, расположение;

ascend v. [q`send]

подниматься, восходить;

descend v. [dI`send]

спускаться, снижаться;

state v. [steIt]

сообщать, формулировать;

concern v. [kqn`sE:n]

касаться, иметь отношение.

precede v. [prI`sJd]

предшествовать, стоять перед чем-л.

Memorize the following word combinations.

Reading Activity


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 2×3, –5×yz, and xy/z.

In the expression 2×3 – xyz – xy/z there are three terms. The expression c×(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 +, 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+a/x; all the other expressions are called integral ones.

An algebraic expression such as 5x³–7x² + 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+5x³yz³-9xyz+2x+7 is a polynomial in x, y 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.

Let’s 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.

The 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.

The 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.

The 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.

The 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. образовывать многочлен; 4. знаки, предшествующие им; 5. состоять из одного или нескольких членов; 6. разместить таким образом; 7. сложить коэффициенты; 8. разместить делимое по возрастающим или убывающим показателям степени; 9. касающееся операции деления; 10. сложить произведения

a. to form a polynomial; b. to be compose of one or more terms; с. to consist of several monomials; d. to place in such a way; e. an algebraic expression; f. to add 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

Ex6. 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 членов. 2. In the algebraic expression 3x³+ 2x²+5 the constant multipliers 3, 2, 5 are called коэффициенты. 3. In the polynomial 2x³+ 5x²+9 the upper numbers 3 and 2 are called показателями степени. 4. A polynomial consisting of three terms is called трёхчлен. 5. One of the fundamental operations that had been applied to those polynomials before other operations was вычитанием. 6. The algebraic expression 2y³-3y²+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 a дробным expression. 10. Adding two polynomials we obtain a сумму.

Ex. 7. Make the following sentences negative and interrogative.

1. The result of the 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. The necessary information has just been obtained. 6. The whole material about polynomials has been learned by the student recently. 7. Those algebraic expressions have been carefully arranged in descending powers. 8. The report on four operations of polynomials is being discussed during the meeting. 9. The remainder in this expression will have been found by the end of the lesson. 10. The experiment was being conducted when you came in.

Ex. 8. Mark the following as True or False.

Ex. 9. Ask special questions.

1. A polynomial is an algebraic expression composed of one or more terms. (What) 2. In the expression 2×3 – xyz – xy/z there are three terms. (How many) 3. A polynomial of two terms is called a binomial. (How) 4. The polynomial 3x³+4x²+5 is of the third degree in x. (What) 5. The 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, how) 9. The remainder found in the result of the subtraction is used as the dividend. (How) 10. The subtraction with those polynomials hadn’t been done correctly 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. The terms of a polynomial are taken with the signs preceding them. 4. The 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. The 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. The 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 выражение, которое содержит только действия умножения, деления и возведения в степень, называется одночленом. 3. Алгебраическая сумма нескольких одночленов называется многочленом. 4. Трёхчлен – алгебраическое выражение, состоящее из трёх членов. 5. Числа при неизвестных х, у, называются коэффициентами многочлена. 6. Многочлены можно складывать, вычитать, умножать и делить. 7. Чтобы разделить многочлен на одночлен, нужно делимое и делитель разместить в убывающем или возрастающем порядке общего неизвестного. 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 остаток при делении равен нулю, то деление называют точным или без остатка.

Таблица нестандартных глаголов


Past Indefinite

Past Participle







was, were






носить, выносить








































пересекать, резать




иметь дело (с)








чертить, тащить
































получать, становиться








расти, выращивать












иметь силу, держать




держать, хранить














learnt (learned)

learnt (learned)

узнавать, учиться
















делать, заставлять




значить, подразумевать
















приводить в движение, бежать




говорить, сказать












помещать, ставить












говорить, разговаривать
















ударять, бастовать












рассказывать, сказать