Q) Prove that √3 is an irrational number.

Ans: Let us assume that √3 is a rational number

Let √3 =  ; where q ≠ 0 and let p, q are co-primes.

  3q² = p²………………. (i)

It means p² is divisible by 3

p is divisible by 3

Hence, we can write that p = 3a, where a is an integer……. (ii)

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

3q² = (3a)²

3q² = 9a²

q² = 3a²

It means that q² is divisible by 3

q is divisible by 3

Hence, we can write that q = 3b, 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, √3 is an irrational number………… Hence Proved !