Question:
Prove that $\frac{1}{\sqrt{3}}$ is irrational.
Solution:
Let $\frac{1}{\sqrt{3}}$ be rational.
$\therefore \frac{1}{\sqrt{3}}=\frac{a}{b}$, where $a, b$ are positive integers having no common factor other than 1
$\therefore \sqrt{3}=\frac{b}{a}$ .................(1)
Since $a, b$ are non-zero integers, $\frac{b}{a}$ is rational.
Thus, equation (1) shows that $\sqrt{3}$ is rational.
This contradicts the fact that $\sqrt{3}$ is rational.
Hence, $\frac{1}{\sqrt{3}}$ is irrational.