Question

Prove that $2+\sqrt{5}$ is an irrational number.

Answer 1

If possible, let us assume $2+\sqrt{5}$ is a rational number.
$2+\sqrt{5}=\frac{p}{q}$ where $p,q\in z,q\ne 0$
$2-\frac{p}{q}=-\sqrt{5}$
$\frac{2q-p}{q}=-\sqrt{5}$
$\Rightarrow -\sqrt{5}$ is a rational number
$\because \frac{2q-p}{q}$ is a rational number
But $-\sqrt{5}$ is not a rational number.
$\therefore$ Our supposition $2+\sqrt{5}$ is a rational number is wrong.
$\Rightarrow 2+\sqrt{5}$ is an irrational number.