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Rational Numbers

Concepts Related to Rational Numbers

You have studied fractional numbers in your earlier classes. Some examples of fractional numbers are.

These numbers are also known as rational numbers.

What comes first to your mind when you hear the word rational?

Yes,you are right. It is something related to the ratios.

The ratio 4:5 can be written as, which is a rational number. In ratios, the numerator and denominator both are positive numbers while in rational numbers, they can be negative also.

Thus, rational numbers can be defined as follows.

Any number which can be expressed in the form , where p and q are integers and, is called a rational number.”

For example, is a rational number in which the numerator is 15 and the denominator is 19.

Now, is −34 a rational number?

Yes, it is a rational number. −34 can be written as. It is in the form of and q ≠ 0.

Thus, we can say that every integer is a rational number.

Now, consider the following decimal numbers.

1.6, 3.49, and 2.5

These decimal numbers are also rational numbers as these can be written as

If in a rational number, either the numerator or the denominator is a negative integer, then the rational number is negative.

For example, are negative rational numbers.

If the numerator and the denominator both are either positive integers or negative integers, then the rational number is positive.

For example, are positive rational numbers.


Conventions used for writing a rational number:   

We know that in a rational number, the numerator and denominator both can be positive or negative.

Conventionally, rational numbers are written with positive denominators.

For example, –9 can be represented in the form of a rational number as , but generally we do not write the denominator negative and thus, is eliminated. So, according to the convention, –9 can be represented in the form of a rational number as .


Equality relation for rational numbers:

For any four non-zero integers p, q, r and s, we have

Order relation for rational numbers:

If are two rational numbers suc…

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