Reading Activity Numbers

The beginning of our number system dates back to antiquity where symbols, which we call positive integers, were used as an aid in counting, and only in the nineteenth century the system, which we know today, was completed. As an aid in studying this number system, let’s use the diagram.

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.

UNIT 4

Grammar: Modal verbs and their equivalents

CAN

Present Can, cannot (can’t); am/are/is (not) able to	Future Can; shall/will (not) be able to	Past 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)
Request
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.
Offer
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.
Possibility
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

Present May, may not	Future May; Shall/will (not) be allowed to	Past 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.
Possibility
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

Present Must, must not (mustn’t)	Future Shall/will (not )have to	Past 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.
Prohibition
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?