# Rational Numbers

Rational numbers are represented in p/q form where q is not equal to zero. It is one of the most important maths topics. Any fraction with non zero denominators is a rational number. These numbers can also be represented on a number line. Apart from this definition, will also learn about its properties, the difference between rational and irrational numbers. Solved examples help to understand the concepts in a better way.

## What is a Rational Number?

Rational number can be defined as any number which can be represented in the form of p/q where q is greater than 0. Also, we can say that any fraction fit under the category of rational numbers, where denominator and numerator are integers and the denominator is not equal to zero.

Rational numbers represented in the form of a/b, where b≠0.

The set of rational numbers –

1. Involve positive and negative numbers, and zero.
2. Can be expressed as a fraction.

A number is rational if we can write it as a fraction, where both denominator and numerator are integers.

Below diagram helps us to understand more about the number sets.

• Rational numbers (R) include all the real numbers (Q).
• Real numbers include the integers (Z).
• Integers involves the natural numbers(N).
• Every whole number is a rational number because every whole number can be expressed as a fraction.

## Positive and Negative Rational Numbers

Positive Rational Numbers Negative Rational Numbers
If both the numerator and denominator are of same signs. If numerator and denominator are of opposite signs.
All are greater than 0 All are less than 0
Example: 12/17, 9/11 and 3/5 are positive rational numbers Example: -2/17, 9/-11 and -1/5 are negative rational numbers

## Rational Numbers Properties

Some of the properties of rational numbers are discussed below:

• The result of two rational numbers is always a rational number is we multiply, add or subtract them.
• A rational number remains the same is we divide or multiply both numerator and denominator with the same number.
• Sum of zero and a rational number revert the same number itself.

## Difference Between Rational and Irrational Numbers

Rational Numbers and Irrational Numbers

There is a difference between rational and Irrational Numbers. A fraction with zero denominators is called a rational number. The number ½ is a rational number because it is read as integer 1 divided by the integer 2. All the numbers that are not rational are called irrational.

Rational numbers can be either positive, negative or zero. While specifying a negative rational number, the negative sign either in front or with the numerator and that is the standard mathematical notation. For example, we denote negative of 5/2 as -5/2.

An irrational number cannot be written as a fraction but can be written as a decimal. It has endless non-repeating digits after the decimal point. Some of the irrational numbers are mentioned below.

π = 3.142857…

√2 = 1.414213…

### Rational Numbers Examples

Example 1: Identify each of the following as irrational or rational: ¾ , 90/12007, 12 and √5 rational or irrational numbers

Solution: Since a rational number is the one that can be expressed as a ratio. This indicates that it can be expressed as a fraction wherein both denominator and numerator are whole numbers.

• ¾ is a rational number as it can be expressed as a fraction.
• Fraction 90/12007 is rational.
• 12, also be written as 12/1. Again a rational number.
• Value of  √5 = 2.2360…. , does not end. Cannot be written as a fraction. It is an irrational number.

Example 2:  Identify whether mixed fraction, 1½ is a rational number.

Solution: the Simplest form of 1½ is 3/2

Numerator = 3, which is an integer

Denominator = 2, is an integer and not equal to zero.

So, yes, 3/2 is a rational number.

### How to write Standard form of Rational Number

The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive. For Example, 12/36 is a rational number. But it can be simplified as 1/3; common factors between the divisor and dividend is only one. So we can say that rational number ⅓ is in standard form.

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#### Practise This Question

Between two given rational numbers, we can find