Let \(\frac{1}{\sqrt3}\) be rational.
∴ \(\frac{1}{\sqrt3}\) = \(\frac{a}{b}\), where a, b are positive integers having no common factor other than 1
∴ \(\sqrt3\) = \(\frac{b}{a}\).....(1)
Since a, b are non-zero integers, \(\frac{b}{a}\) is rational.
Thus, equation (1) shows that \(\sqrt3\) is rational.
This contradicts the fact that \(\sqrt3\) is rational.
The contradiction arises by assuming \(\sqrt3\) is rational.
Hence, \(\frac{1}{\sqrt3}\) is irrational.