Question

Show that the number $3-\sqrt{5}$ is irrational.

Answer 1

Let $3-\sqrt{5}$ be a rational equal to $\frac{a}{b}$ Then,

$\Rightarrow 3-\sqrt{5}=\frac{a}{b}$

$\Rightarrow \sqrt{5}=3-\frac{a}{b}$

$\Rightarrow \sqrt{5}=\frac{3b-a}{b}$

$\Rightarrow \sqrt{5}$ is rational.

This is a contradiction.

Hence, $3-\sqrt{5}$ is irrational.