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
|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|
If p and q are two integers then
|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
|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