What Are Rational Numbers?

The word ‘rational’ arises from the term ‘ratio’. You know that a ratio like 3:2 can also be written as 32. Here, 3 and 2 are natural numbers.

Similarly, the ratio of two integers p and q (q ≠ 0), i.e., p:q can be written in the form pq. This is the form in which rational numbers are expressed.

Thus, 45 is a rational number. Here, p = 4 and q = 5.

Is −34 also a rational number? YesNo

Yes , because p = – 3 and q = 4 are integers.

You have seen many fractions like 38,48,123 etc. All fractions are rational numbers. Can you say why?

How about the decimal numbers like 0.5, 2.3, etc.? Each of such numbers can be written as an ordinary fraction and, hence, are rational numbers. For example, 0.5 = 510, 0.333 = etc.

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Numerator and Denominator

In pq, the integer p is the numerator, and the integer q ( q ≠ 0) is the denominator.

Thus, in −37, the numerator is –3 and the denominator is 7.

Are integers also rational numbers?

Any integer can be thought of as a rational number. For example, the integer –5 is a rational number, because you can write it as −51. The e integer 0 can also be written as −51. The integer 0 can also be written as 0 = 02 or 07 etc. Hence, it isis not also a rational number.

Thus, rational numbers include integers and fractions.

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Equivalent rational numbers

A rational number can be written with different numerators and denominators. For example,

consider the rational number −23.

By multiplying the numerator and denominator of a rational number by the same non zero integer, we obtain another rational number equivalent to the given rational number.

This is exactly like obtaining equivalent fractions.

Just as multiplication, the division of the numerator and denominator by the same non zero integer, also gives equivalent rational numbers. For example,

10−15 = 10÷−5−15÷−5 = −23,1224 = −12÷−1224÷12 = −

We write −23 as −23, −1015 as −1015 etc.