Assume that $5-\sqrt{3}$ is a rational number. Then it can be expressed as a fraction,

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

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

Where p and q are co-prime numbers and q ≠ 0.

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

It can be observed that $\frac{5\mathrm{q}-\mathrm{p}}{\mathrm{q}}$ is a rational number because adding or subtracting two rational numbers gives a rational number but $\sqrt{3}$ is an irrational number which cannot be possible.

Therefore, the assumption is wrong.

Hence, $5-\sqrt{3}$ is an irrational number.