# Rational Numbers

Rational numbers are a segment of the real numbers, which can be written in p/q form where p and q are an integer and q (the denominator) is not equal to zero. Rational numbers originated from the concept of ratio.

## Examples of Rational Numbers

Given below are some examples of rational numbers:

• 1/2 or 0.5
• -6/7
• -0.25 or -1/4
• -13/15 or  -0.8666666666666667

## Symbol

The rational numbers are universally represented by the symbol ‘Q’.

## Properties

### Closure Property

Rational numbers are closed under addition, subtraction, multiplication, and division operations.

In simple words, addition, subtraction, multiplication, and division of 2 rational numbers ‘a’ and ‘b’ give a rational number. In rational numbers (p/q form), q ≠ 0. If q = 0, the result is undefined.

For example:

• 6/7 + 2/9 = 68/63
• 3/5 – 1/7 = 16/35
• 2/3  × 3/9 = 2/9
• 7/15 ÷ 10/3 = 7/50

### Commutative Property

Rational numbers are commutative under the operations – addition and multiplication. However, this property does not hold for subtraction or division of 2 rational numbers.

⇒ a + b = b + a, here a and b are 2 rational numbers

And, a × b = b × a

⇒ a – b ≠ b – a,

And, a ÷ b ≠ b ÷ a

For example:

• 1/3 + 2/3 = 2/3 + 1/3

= 3/3 = 1

• 7/8 × 8/9 = 8/9  × 7/8

= 7/9

### Associative Property

Rational numbers have the associative property for only addition and multiplication.

For example:

⇒ a + (b + c) = (a + b) + c, here a and b are 2 rational numbers

And, a × (b × c) = (a × b) × c

• 1/3 + (1/5 + 2/4) = (1/3 + 1/5) + 2/4

= 31/30

• 3/7 × (2/5 × 3/4) = (3/7 × 2/5) × 3/4

= 9/70

### Distributive Property

According to this property, the multiplication of a whole number is distributed over the sum of the whole numbers.

⇒ a × (b + c) = (a × b) + (a × c), here a and b are 2 rational numbers

For example:

• 1/3 x (1/6 + 1/7) = (1/3 x 1/6) + (1/3 x 1/7)

=  13/126

### Identity Property

According to Identity property, 0 is an additive identity and 1 is a multiplicative identity for rational numbers.

⇒ a/b + 0 = a/b (Additive Identity)

a/b x 1 = a/b (Multiplicative Identity)

For example:

• 3/4 + 0 = 3/4 (Additive Identity)
• 3/4 x 1 = 3/4 (Multiplicative Identity)

### Inverse Property

According to Identity property, for a rational number a/b, its additive inverse is -a/y, and b/a is its multiplicative inverse.

⇒ a/b + (-a/b)  = 0 , here the additive inverse of a/b = (-a/b)

And, a/b x b/a = 1, here the multiplicative inverse a/b = b/a

For example:

• 2/9 + (-2/9) = 0, here the additive inverse of 2/9 is -2/9
• 3/8 x 8/3 = 1, here the multiplicative inverse of 3/8 is 8/3.

## How to find rational numbers?

We need to check the following conditions to identify a rational number.

• It is written as p/q, where q ≠ 0.
• The ratio p/q can be simplified further and written in decimal form

Rational numbers:

• Consist of positive, negative numbers, and zero
• It can be written as a fraction

For example:

Is 0.3505350535053505… a rational number?

The number above has a set of decimals 3505, which is repeated continuously here.

## Types

We must not consider any fraction with integers as rational numbers. Below are the different types of rational numbers:

• integers like -1, 0, 3 etc.
• fractions with numerators and denominators are integers such as 2/5, -6/7

(example: 12/45 is not a rational number as it can be further simplified into the standard form as 4/15. So the rational number here is 4/15)

In simple words, rational numbers are of 2 types:

• Standard – a fraction that cannot be simplified further but can be written in decimal form
• Positive and negative rational numbers

So are decimals rational numbers?

Rational numbers can be decimals with

• terminating decimals such as 1/8 (0.125), 1/16 (0.0625), or
• non-terminating decimals with repeating patterns (after the decimal point) such as 1/15 (0.0666666666666667), 2/9 (0.2222222222222222).

So are all integers rational numbers?

Yes. Any integer is a rational number. It can be expressed as a fraction or terminating decimal using the properties of rational numbers.

## Operations with Rational Numbers

### Adding and Subtracting Rational Numbers

We add and subtract rational numbers in the same way we do with fractions. While adding or subtracting any 2 rational numbers, we make their denominators equal and add the numerators.

For example:

2/3 – (-3/5)= 2/3 + 3/5 = 2/3 × 5/5 + 3/5 × 3/3 = 10/15 + 9/15 = 19/15

### Multiplying and Dividing Rational Numbers

We multiply and divide rational numbers in the same way we do with fractions. While multiplying or dividing any 2 rational numbers, we multiply the numerators and the denominators separately. Then we simplify the result.

For example:

1/2 × -2/5 = (1 × -2)/(2 × 5)= -2/10

While dividing any 2 fractions, we convert any one of the fractions into their own reciprocal and then multiply with the other.

For example:

5/7 ÷ 9/28 = 5/7 × 28/9 = 20/9 = ${2\dfrac{2}{9}}$

## List of Rational Numbers

From the discussion above, it is evident that the range of rational numbers is infinite. So we do not have a list of rational numbers. Hence, we cannot find the smallest rational number.

So is 0 a rational number?

Yes. As we know, a number written in p/q form where q ≠ 0, 0 can also be written in p/q form such as 0/1, 0/2, 0/3, 0/-4, 0/5/0.8.

## Difference Between Rational and Irrational Numbers

The numbers which are not rational are considered irrational. We will learn about the differences in the next article.

[link Difference Between Rational and Irrational Numbers article here]

## Solved Examples

Find the rational numbers in the following set – √3/2, 3/7, – 5/8, π, 1.512362309…..

Solution:

As we know,
Simplifying a rational number gives terminating or a non-terminating decimal with repeating pattern
∴ the rational numbers in the following set is 3/7 and – 5/8

Find a rational number among the following-  1/3 and 2/5.

Solution:

As we know,
The average of any 2 numbers is attained from the 2 given numbers.
(1/3 + 2/5) ÷ 2
= 11/15 × 1/2
= 11/30

## FAQs

Q1. Are negative numbers rational?

Ans. Rational numbers consist of all positive and negative numbers and zero. The numbers can be written as p/q form with the signs. These include Whole numbers, integers, fractions, terminating, and repeating decimals.