Prove that

Let us assume, to the contrary, that is rational. So, we can find integers p and q(q ≠ 0), such that , where p and q are coprime.

Squaring both sides, we get

..... (i)

⇒

⇒ 5 divides q²⇒ 5 divides q

So, p and q have at least 5 as a common factor. But this contradicts the fact that p and q have no common factor. So, our assumption is wrong.

5 is irrational.

5 is irrational, 3 is a rational number.

So, we conclude that 3 + 5 is a rational number.