Question

$3-\sqrt{5}$ is an irrational number
If true then enter $1$ and if false then enter $0$

Answer 1

Let assume that on the contrary that $3-\sqrt{5}$ is rational. Then, there exist co-prime positive integers a and b such that,
$3-\sqrt{5}=\frac{a}{b}$
$\Rightarrow 3-\frac{a}{b}=\sqrt{5}$
$\Rightarrow \frac{3b-a}{b}=\sqrt{5}$
$\Rightarrow \sqrt{5}$ is rational
$[\because$ ab, b are integer $\therefore \frac{3b-a}{b}$ is a rational number]
This contradicts the fact that $\sqrt{5}$ is irrational
Hence, $3-\sqrt{5}$ is an irrational number