Question

Prove that $\frac{1}{\sqrt{3}}$ is irrational.

Answer 1

Let $\frac{1}{\sqrt{3}}$ be rational.
∴ $\frac{1}{\sqrt{3}}$ = $\frac{a}{b}$, where $a,b$ are positive integers having no common factor other than 1
∴ $\sqrt{3}=\frac{b}{a}$ ...(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.
The contradiction arises by assuming $\sqrt{3}$ is rational.
Hence, $\frac{1}{\sqrt{3}}$ is irrational.