Introduction

You began your study of numbers by counting objects around you. The numbers used for this purpose were called counting numbers or natural numbers. They are 1, 2, 3, 4, ... and so on. By including 0 to natural numbers, we got the whole numbers, i.e., 0, 1, 2, 3, ... The negatives of natural numbers were then put together with whole numbers to make up integers.

Integers are ..., –3, –2, –1, 0, 1, 2, 3, .... We, thus, extended the number system, from natural numbers to whole numbers and from whole numbers to integers rational numbers.

You were also introduced to fractions.

These are numbers of the form

numeratordenominator

where the numerator is either 0 or a positive integer and the denominator, a positive integer. You compared two fractions, found their equivalent forms and studied all the four basic operations of addition, subtraction, multiplication and division on them.

In this Chapter, we shall extend the number system further. We shall introduce the concept of rational numbers along with their addition, subtraction, multiplication and division operations.

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Need for Rational Numbers

Earlier, we have seen how integers could be used to denote opposite situations involving numbers. For example, if the distance of 3 km to the right of a place was denoted by 3, then the distance of 5 km to the left of the same place could be denoted by .

If a profit of ₹ 150 was represented by 150 then a loss of ₹ 100 could be written as . There are many situations similar to the above situations that involve fractional numbers.

You can represent a distance of 750 m above sea level as 34 km. Can we represent 750m below sea level in km? Can we denote the distance of 34 km below sea level by −34? We can see −34 is neither an integer, nor a fractional number. We need to extend our number system to include such numbers.