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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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