Let us assume, $5-\sqrt{3}$ is a rational number
$\Rightarrow 5-\sqrt{3}=\frac{\mathrm{p}}{\mathrm{q}}$, where p, q ∈ z ,q ≠ 0
$5-\frac{\mathrm{p}}{\mathrm{q}}=\sqrt{3}$
$\Rightarrow \frac{5\mathrm{q}-\mathrm{p}}{\mathrm{q}}=\sqrt{3}$
$\Rightarrow \sqrt{3}$ is a rational number $\because \frac{5\mathrm{q}-\mathrm{p}}{\mathrm{q}}$ is rational
but $\sqrt{3}$ is not a rational number.
This gives us a contradiction.
∴ our assumption that $5-\sqrt{3}$ is a rational number is wrong
$\Rightarrow 5-\sqrt{3}$ is an irrational number.