Properties of rational numbers

Let’s solve the problem. John has read twice as many books as Bill. John has read 7 books. How many books has Bill read?

This problem is easily translated into the equation 2n=7, where n represents the number of books that Bill has read. If we are allowed to use only integers, the equation 2n=7 has no solution. This is an indication that the set of integers does not meet all our needs.

If we try to solve the equation 2n=7 we will find that n=7/2 as to find a multiplier you should divide the product 7 by the multiplier 2. It is the name for a rational number. As we know a rational number is a quotient of two integers p/q where q is not equal to 0. The denominator denotes the number of equal parts into which the whole is divided. The numerator denotes how many of these parts are taken.

Fractions representing values less than 1, like 2/3 for example are called proper fractions. Fractions which name a number equal or greater than 1, like 2/3 or 3/2, are called improper fractions. There are numbers like 1 ½ (one and one second) which name a whole number and a fractional number. Such numerals are called mixed fractions. Fractions which represent the same fractional numbers like ½, 2/4, 4/8 and so on are called equivalent fractions.

We can change such fractions to their higher or lower terms. If the numerator and denominator of the fraction are relatively prime we call it the simplest fraction. The process of bringing a fractional number to lower terms is called reducing a fraction. Multiplying both integers named in the numerator and denominator of the fraction by the same whole number simply produces another name for the fractional number.

To reduce a fraction to lower terms you must determine the greatest common factor. The greatest common factor is the largest possible integer by which the numerator as well as denominator is divisible.

You can also perform all arithmetic operations with rationals. Before performing addition or subtraction with fractional numerals you must bring them to a common denominator. When multiplying with fractions you should find the product of the numerators and the product of denominators. Dividing simple fractions one by another you must multiply the numerator of the dividend by the denominator of the divisor and also multiply the denominator of the dividend by the numerator of the divisor. For example 3/8 : 5/9 = 3/8 ∙ 9/5

From the above we can draw the conclusion that mathematical concepts and principles are just as valid in the case of rational numbers as in the case of integers.

Task 6

Answer the questions on the text:

1. Why does the equation 2n=7 have no solution?

2. When do we say that a problem has no solution?

3. What should we do to find a multiplier?

4. What is a rational number?

5. What does the denominator denote?

6. What does the numerator denote?

7. What is a proper fraction?

8. What fraction do we call improper?

9. What is a mixed fraction?

10. What’s the difference between proper and improper fraction?

11. What’s an equivalent fraction?

12. How can we transform equivalent fractions?

13. What is the simplest fraction?

14. How to reduce a fraction?

15. Explain what is the greatest common factor?

16. When do we say that a number is relatively prime?

17. How do we add and subtract two simple fractions?

18. How do we multiply fractions?

19. How can we divide a fraction by another one?

20. Do mathematical concepts work in the case of rational numbers or integers?

Task 7

Fill in the gaps: