Proof by contradiction : show that √ 2 is irrational.

Step-by-step explanation:

Let's assume that is rational,

So, it should be expressed in the form of  

where a and b are two prime integers.

So, we can write,

As represents a even number, must also be even.  

Since is even number, so a is also an even number.

Let assume again that a=2c

As is even, so is also an even number, and so is b.  

And since and b both are even number here, two even numbers can not be relatively prime, so can not be expressed in the form of . So, is considered as an irrational number.