Question

Show that $5-\sqrt{3}$ is an irrational number.

Answer 1

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

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

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

Where $p$ and $q$ are co-prime numbers and $q\ne 0$.

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

It can be observed that $\frac{5q-p}{q}$ 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.