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.

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.

1 They … an attempt to overcome difficulties of this complicated procedure.

2 The science adviser … the whole situation under control.

3 They … not all the examination material yet.

4 I … this article on Maths very helpful.

5 In the given example he … the validity of these principles.

6 The post-graduate … the previous chapter of his report.

7 He … a first-year student of the Mechanics and Mathematics faculty this year.

8 My friend … already the diagram which has certain advantages.

9 The teacher … an example of an algorithm.

10 … you … able to name these rational numbers by decimal numerals?

Ex. 10. Choose the correct tense form.

1. (Have you ever heard/ Did you ever hear) about continued fractions?

2. We (have started/ started) learning fractions when we were at school.

3. By the end of the lecture the students (had learnt/ has learnt) that in Higher Mathematics a fraction is viewed as an element of a field-of fractions.

4. I (have looked up/ looked up) in a dictionary just now the meaning of the word ‘a solidus’(a slanting line).

5. She just (has written/ wrote) this mixed number in another way; i.e. as an improper fraction.

6. She (will have studied/ will study) carefully the chapter about complex fractions by the end of the week.

7. When you come back I (will have divided/ have divided) these fractions using the rule “invert and multiply”.

8. I need to subtract two fractions but I (haven’t got/ didn’t get) the least common denominator yet.

9. Before he solved the problem he (had had to/ have had to) convert the repeating decimals into fractions.

10. You (have remembered/ remembered) since school that a fraction is in its lowest terms when its numerator and denominator have no common factor other than one.

Ex. 11. Ask special questions.

1. By 5 o’clock the experiment will have been over. (by what time) 2. By the age of 41 Sophia Kovalevskaya had won recognition of mathematicians all over the world. (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)