## IRRATIONAL NUMBERS

All real numbers which are not rational are called irrational numbers. , , are some examples of irrational numbers.

There are decimals which are non-terminating and non-recurring decimal.

Example: 0.303003000300003…

Hence, we can conclude that

An irrational number is a non-terminating and non-recurring decimal and cannot be put in the form where p and q are both co-prime integers and q ¹ 0.

## Solved Examples Based on Irrational Numbers

### Question:

Prove that is not a rational number.

### Solution:

Let is a rational number

\ [p and q are co-prime and q ¹ 0]

Squaring both sides

Þ

Þ is even or p is even.

Let p = 2r

Þ

Þ

Þ is even so q is even.

Hence, p and q have 2 as a common factor or p and q are not co-prime.

So, our supposition is wrong.

\ is not a rational number.

### Question:

Prove that is an irrational number.

### Solution:

Suppose is a rational number and can be taken as , b ¹ 0 and a, b are co-prime.

Þ [Rational]

Squaring both sides

Þ

Þ

LHS is which is irrational while RHS is rational.

So, our supposition is wrong.

Hence, is not a rational number.

### Question:

Show that there is no positive integer n for which is rational and can be expressed in the form , b ¹ 0 and a & b are co-prime.

### Solution:

Let there be a positive integer n for which is rational.

Which means … (i)

or

Rationalizing LHS, we get

Þ

Þ

Þ

or …(ii)

Adding and subtracting (i) and (ii) we get

and

Þ and

Þ RHS of both are rational.

\ and are also rational.

Þ and are perfect squares of positive integers.

This is impossible as any two perfect squares differ at least by 3.

Hence, there is no positive integer n for which () is rational.