I quite agree with you. On the contrary. Far from it.

1. A group consists a number of entities which can be combined together according to one axiom. 2. In the operation of multiplication two numbers are multiplied to give their quotient. 3. In the abstract operation of multiplication the elements are written close together. 4. In a closure axiom the product of any two elements of the group is a unique element which does not belong to the group.5. When multiplying the elements of a group by an identity element, the former ones are changed. 6. A finite group under multiplication consists of seven elements. 7. There is no need to verify individually the laws of closure and existence of inverses in subgroups.

Ex. 11. Read and translate the following sentences paying attention to the word One used as:

a) an Indefinite Pronoun to talk about people in general, including the speaker and hearer or in very general statements, when we are speaking about “anyone, at any time”.

One can omit this condition.	Можно опустить это условие
One should knock before going into somebody’s room.	Следует постучать в дверь, прежде чем войти в чью-либо комнату.
One believes that…	Cчитаю, что…
One knows that…	Известно, что…

1. One must emphasize here a basic difference between linear and non-linear systems.

2. Analytic geometry is a branch of mathematics in which one studies geometry by means of algebra.

3. To interpret this phenomenon, one must know the structure of the atom.

4. One can simply determine the location of the point in space.

5. One must have the same denominators when adding two fractions.

6. In fact, one can in theory prove any theorem directly from the axioms.

7. One believes that the procedure described above will simplify the experiment.

8. When making experiments of this kind, one is faced with still another difficulty.

b) a Substituteword instead of repeating a singular countable noun. One has a plural ones.

This property is not so essential as that one.	Это свойство не является таким важным, как то свойство.
Green apples often taste better than red ones.	Зеленые яблоки на вкус часто лучше, чем красные.

1. The result, like the one just described, is in no way surprising.

2. The procedure is straight-forward and is the one followed throughout the experiment.

3. If a mathematical problem is a strict expression of a physical one, it has a unique solution.

4. This equation essentially differs from the one which we solved at the last lesson.

5. The numerical set is one whose members have numerical values.

6. These are easy questions to answer and those are difficult ones.

7. We shall replace the old equations by new ones.

Ex. 12. In which of the following English sentences the italicized group of words will be translated as:

1. Достигнув успеха …

a. Having been achieved the success did not prevent the scientist from working hard and developing the problem.

b. Achieving success and recognition some scientists stop working hard.

c. Having achieved success and recognition, the scientist went on working hard over his problem.

2. Используя алгебру …

a. Having used algebra, we can reduce complex problems to simple formulas.

b. Using algebra, we can reduce complex problems to simple formulas.

c. Being used, algebra helpedus to reduce complex problems to simple formulas.

3. Поняв идею …

a. Having understood the idea, we can simplify our notation.

b. Understanding the idea, we can simplify our notation.

c. Having been understood, the idea turned out to be a simple one.

Ex. 13. Read and translate the following sentences paying attention to the inversion of the verb. Certain adverbs and adverbial phrases, mostly with a negative sense, can for emphasis be placed first in a sentence and are then followed by the inverted (i.e. interrogative) form of the verb. The most important of these are given below: never никогда, seldom редко, neither, nor a также не, hardly, scarcely … when едва (только) … как, no sooner … than как только, не успел … как, not only … but не только … но, not until (till) и только когда.

Nor should we forget the importance of this argument.	А также мы не должны забывать о весомости этого аргумента.
Never before had I been asked to accept a bribe.	Никогда раньше мне не предлагали взятку

1. Not till he got home did he realize that he had lost that important document.

2. He had no money, nor did he know anyone he could borrow from.

3. Hardly had I arrived when trouble started.

4. No sooner had she agreed to marry him than she started to have doubts.

5. Never in my life have I seen such a proof.

6. The ancients had no knowledge of stellar distances, neither was there then any means by which they could determine them.

7. Since the Moon has no atmosphere, there can be no wind, neither can there be any noise, for sound is carried by the air.

8. Scarcely had the professor started his lecture when the lights in the room went off.

Ex. 14. Translate the following word combinations into English using either Participle I or Participle II.

1. решения, отвечающие нашим требованиям; 2. наука, обеспечивающая высокий уровень жизни общества; 3. приборы, изобретенные нашими инженерами; 4. методы исследований, хорошо известные ученым; 5. мир, созданный наукой; 6. функция, определенная посредством формулы; 7. прямая, соединяющая две точки; 8. угол, делящий плоскость; 9. разделенная диагональ; 10. примененный метод; 11. работа, продолженная на следующий день; 12. предмет, взятый в качестве модели.

Ex. 15. Read and translate the following sentences. Write out (in row) the numbers of sentences in which the Participle is used as:

a) an Attribute b) an Adverbial Modifier

1. Given two sets X and Y, there is a set whose elements are those which belong only to one of the two given sets.

2. Expressed in math terms, this theorem gives a general method of calculating the area.

3. The sense implied in this statement is not clear.

4. Certain properties of the real world can be described using numbers.

5. When finding the product of multinomials, we make use of the distributive law.

6. The group of integers under addition has subgroups comprising all even integers.

7. Lobachevsky wrote a new geometry asserting that there could be several parallels.

8. Having calculated the area, we can say now that the formula is exact.

9. Parallel lines are lines extending in the same direction and being the same distance apart no matter how far extended.

10. Having supposed the inequality, we obtained the necessary results.

11. Considering specific physical phenomena, we may see that one and the same quantity in one phenomenon is a constant while in another it is a variable.

12. Having started from a system of axioms, we then could make certain logical deductions.

13. The statements followed by some illustrations were rather convincing.

Ex. 16. Ask special questions using the words in parentheses.

1. There are really two types of problems involved here. (How many?)

2. Having understood the ideas, we can simplify our notation. (When?)

3. Being interested in set theory, he never missed his special course. (Why?)

4. Rational functions are functions involving an additional operation of division. (What?)

5. A point representing a variable is called a variable point. (How?)

6. The students studying the theory of sets find this statement interesting. (Who?)

7. Equations containing one or more variables to the first power only are linear in one or two variables. (What?)

8. We can find that some elements form a smaller group inside the big one. (What?)

9. Groups can arise in many quite distinct situations. (In what cases?)

10. When speaking of quantities, we shall have in view their numerical values. (What?)

11. The meanings of these words are often confused in speech. (Where?)

Ex. 17. Translate into English the following sentences.

1. Сейчас теория группы разрабатывается абстрактно, так что ее можно применять во многих различных ситуациях.

2. Чтобы суметь исследовать структуру группы более детально, необходимо ввести сложение как еще одну операцию между элементами группы.

3. Феликс Кляйн (Felix Klein) продемонстрировал, что понятие группы может оказаться полезным при классификации многих областей математики.

4. Развитие этих двух ветвей алгебры привело к качественно новым проблемам науки, связанным с возникновением теории Галуа и теории групп.

5. Группа – это математическая система, элементы которой удовлетворяют четырем основным правилам.

6. Таким образом, предмет «алгебра» определился в XVIII в., превратившись в науку об алгебраических уравнениях.

7. Термин «группа» обозначает особый вид математической системы, и он не имеет ничего общего с разговорным значением, приписываемым слову «группа».

Ex. 18. Topics for discussion.

1. The history of developing the group concert.

2. The working mechanism of such a structure as a group.

3. The four rules characterizing a group.

4. Dwell on the notion of a subgroup.

Ex. 19. Read the text and find the answers to the following questions.

1. What was the main problem concerning algebraic equations in the XVI-XVIII centuries? 2. How could equations of degree 3 and 4 be solved? 3. In those years the scientists considered the idea of solving equations of degree 5 and higher by using roots of higher order absolutely natural, didn’t they? 4. Who settled the problem once and for all? 5. What method did he offer? 6. What is the criterion for solvability of equations?

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