Question

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

Answer 1

Let us assume on the contrary that $5-\sqrt{3}$ is
rational. Then, there exist prime positive integers $a$
and $b$ such that

$5-\sqrt{3}=\frac{a}{b}$
$\Rightarrow$ $5-\frac{a}{b}=\sqrt{3}$
$\Rightarrow$ $\sqrt{3}$ is rational $[$$\because a,b$ are integers$\therefore \frac{5b-a}{b}$is a rational number$]$

This contradicts the fact that $\sqrt{3}$ is irrational.
So, our assumption is incorrect. Hence, $5-\sqrt{3}$ is
an irrational number.