Question

Prove that √5 is an irrational number.

Answer 1

Let us assume, to the contrary, that √5 is a rational number.
Then, there exist co-prime positive integers a and b such that

Squaring both sides, we have

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.