Question

Prove that 3 + $\sqrt{5}$ is an irrational number.

Answer 1

Let us assume that 3 + $\sqrt{5}$ is a rational number.

$\Rightarrow$3 + $\sqrt{5}$ = $\frac{p}{q}$, where p and q are the integers and q $\ne$ 0.

$\Rightarrow$$\sqrt{5}$ = $\frac{p}{q}-3=\frac{p-3q}{q}$

Since, p, q and 3 are integers. So, $\frac{p-3q}{q}$ is a rational number.

$\Rightarrow$$\sqrt{5}$ is also a rational number.

but this contradicts the fact that $\sqrt{5}$ is an irrational number.

This contradiction has arisen due to the wrong assumption that 3 + $\sqrt{5}$ is a rational number.

Hence, 3 + $\sqrt{5}$​ is an irrational number.