Question

prove that 1+√3 is irrational​

Answer 1

$let \: 1 + \sqrt{3} = \frac{p}{q}$

$\sqrt{3} = \frac{p}{q} - 1 \\ \sqrt{3 } = \frac{p - q}{q}$

LHS is an irrational number and RHS is a rational number. This is a contradiction araised due to our supposition that

$1 + \sqrt{3} \: is \: a \: rational \: number$

$so \: 1 + \sqrt{3} is \: irrational$