 ## Rational Numbers

Pure mathematics is, in its way, the poetry of logical ideas.
— Albert Einstein, German theoretical physicist

A number is called Rational if it can be expressed in the form p/q where p and q are integers (q > 0). It includes all natural, whole number and integers.

Example: 1/2, 4/3, 5/7,1 etc.

### Properties of Rational Numbers

#### 1. Closure Property

This shows that the operation of any two same types of numbers is also the same type or not.

##### a. Whole Numbers

If p and q are two whole numbers then

 Operation Addition Subtraction Multiplication Division Whole number p + q will also be the whole number. p – q will not always be a whole number. pq will also be the whole number. p ÷ q will not always be a whole number. Example 6 + 0 = 6 8 – 10 = – 2 3 × 5 = 15 3 ÷ 5 = 3/5 Closed or Not Closed Not closed Closed Not closed

##### b. Integers

If p and q are two integers then

 Operation Addition Subtraction Multiplication Division Integers p+q will also be an integer. p-q will also be an integer. pq will also be an integer. p ÷ q will not always be an integer. Example – 3 + 2 = – 1 5 – 7 = – 2 – 5 × 8 = – 40 – 5 ÷ 7  = – 5/7 Closed or not Closed Closed Closed Not  closed

##### c. Rational Numbers

If p and q are two rational numbers then

 Operation Addition Subtraction Multiplication Division Rational Numbers p + q will also be a rational number. p – q will also be a rational number. pq will also be a rational number. p ÷ q will not always be a rational number Example p ÷ 0 = not defined Closed or Not Closed Closed Closed Not closed

#### Representation of Rational Numbers on the Number Line

On the number line, we can represent the Natural numbers, whole numbers and integers as follows

By,

Middle School

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