Let us suppose that √2 is a rational number.

So, √2 = a/b, where a and b are both integer and b ≠ 0 and have no common factor other than 1

=> √2 = a/b, where a and b are coprime numbers.

Now squaring on both side, we get

=> (√2)² = (a/b)²

=> 2 = a² /b²

=> 2b² = a²

=> b² = a² /2

=> 2 divides a² 

=> 2 divides a

So, we can write, a = 2c, where c is a constant.

Now squaring on both side, we get

=> a² = (2c)² 

=> a² = 4c²

=> 2b² = 4c² 

=> b² = 2c²

=> c² = b² /2 

=> 2 divides b² 

=> 2 divides b

So, a and b have at least 2 as a common factor but it is stated that a and b has no common factor other than 1.

This contradicts our assumption that √2 is rational.

So, √2 is an irrational number.