Let us assume, to the contrary, that √5 is a rational number. Then, there exist co-prime positive integers a and b such that
It means 5 divides b2 and so 5 divides b.
So, 5 is a common factor of both a and b which is a contradiction.
So, our assumption that √5 is rational is wrong.
Hence, we conclude that √5 is an irrational number.
