Question

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

Answer 1

Let us assume the given number be rational and we will write the given number in p/q form

$\Rightarrow 5-\sqrt{3}=\frac{p}{q}$

$\Rightarrow \sqrt{3}=\frac{5q-p}{q}$

We observe that LHS is irrational and RHS is rational, which is not possible.

This is contradiction.

Hence our assumption that given number is rational is false

$\Rightarrow 5-\sqrt{3}$ is irrational