Prove that \(\frac{1}{\sqrt3}\) is irrational.

1.	Prove that \(\frac{1}{\sqrt3}\) is irrational.
Answer» 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.

Answer»

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.

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