If possible, let √3 be a rational number.
(i) ∴ a/b = √3 , where a and b are integers and co-primes
Squaring both sides, we have
From equation (i) and (ii), we have 3 is a factor of a and b which is contradicting the fact that a and b are co-primes.
Thus, our assumption that √3 is rational number is wrong.
Hence, √3 is an irrational number.
(ii) Let us assume to contrary that 7 + 2√3 is a rational number.
p – 7q and 2q both are integers, hence √3 is a rational number.
But this contradicts the fact that √3 is irrational number.
Hence 7 + 2√3 is an irrational number.
