Question

prove that it is irrational
$\sqrt{2}$

Answer 1

We will prove whether √2 is irrational by contradiction method.
Let √2 be rational 
It can be expressed as √2 = a/b ( where a, b are integers and co-primes. 
√2 = a/b
2= a²/b² 
2b² = a²
2 divides a²
By the Fundamental theorem of Arithmetic
so, 2 divides a .

a = 2k (for some integer) 

a² = 4k² 
2b² = 4k² 
b² = 2k² 

2 divides b²
2 divides b. 

Now 2 divides both a & b this contradicts the fact that they are co primes. 
this happened due to faulty assumption that √2 is rational. Hence, √2 is irrational.