**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.

3q^{2} = p^{2}………………. (i)

It means p^{2} 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:

3q^{2} = (3a)^{2}

3q^{2} = 9a^{2}

q^{2} = 3a^{2}

It means that q^{2} 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 !**