Question

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

Answer 1

Let $2+\sqrt{5}$ is a rational number such that

$2+\sqrt{5}=\frac{p}{q}$

Where $p$ and $q$ are co-prime.

Therefore,

$2+\sqrt{5}=\frac{p}{q}$

$\sqrt{5}=\frac{p}{q}-2$

$\Rightarrow \sqrt{5}=\frac{p-2q}{q}$

$\sqrt{5}$ is an irrational number and $\frac{p-2q}{q}$ is a rational number.

$\because$ Rational $\ne$ Irrational.

Thus $2+\sqrt{5}$ is not a rational number.

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

Hence proved.