Q) Prove that √2 is an irrational number.

Ans: Let us assume that √2 is a rational number

Let √2 =  ; where q ≠ 0 and let p, q are co-primes.

  2q² = p²………………. (i)

It means p² is divisible by 2

p is divisible by 2

Hence, we can write that p = 2a, where a is an integer……. (ii)

Substituting this value in equation (i), we get:

2q² = (2a)²

2q² = 4a²

q² = 2a²

It means that q² is divisible by 2

q is divisible by 2

Hence, we can write that q = 2b, where b is an integer…… (iii)

From equation (ii) and (iii), we get that p and q are not co-primes, which contradicts to our initial assumption.

Therefore, √2 is an irrational number………… Hence Proved !