Given that $\sqrt{3}$ is an irrational number, prove that $(2+\sqrt{3})$ is an irrational number.

Given $\sqrt{3}$ is irrational number

Let $2+\sqrt{3}$ is rational number.

then $2+\sqrt{3}=\frac{p}{q}$

$\sqrt{3}=\frac{p-2q}{q}$

Since $\sqrt{3}$ is irrational number but $\frac{p-2q}{q}$ is rational it mean

irrational = rational

which contradict,

$\therefore$ our assumption is wrong and

$2+\sqrt{3}$ is irrational number.