Question

Show that $(5-\sqrt{3})$ is irrational.

Answer 1

Let us assume, to the contrary, that $(5-\sqrt{3})$ is rational.

So, we can find co-primes $a$ and $b$ $(b\ne 0)$ such that $(5-\sqrt{3})=\frac{a}{b}$

Therefore, $5-\frac{a}{b}=\frac{5b-a}{a}$

Rearranging this equation, we get: $\sqrt{3}=5-\frac{a}{b}=\frac{5b-a}{a}$

Since ‘a’ and ‘b’ are integers, $\frac{5b-a}{a}$ is a rational number.

This contradicts that $\sqrt{3}$ is irrational.

So, $(5-\sqrt{3})$ must be an irrational number.