Prove that the following number is irrational.
$\frac{2}{\sqrt{7}}$

Given: $\frac{2}{\sqrt{7}}$

To do: Here we have to prove that $\frac{2}{\sqrt{7}}$ is an irrational number.

Solution:

Let us assume, to the contrary, that $\frac{2}{\sqrt{7}}$ is rational.

So, we can find integers a and b ($≠$ 0) such that  $\frac{2}{\sqrt{7}}\ =\ \frac{a}{b}$.

Where a and b are co-prime.

Now,

$\frac{2}{\sqrt{7}}\ =\ \frac{a}{b}$

$\frac{2b}{a}\ =\ \sqrt{7}$

Here, $\frac{2b}{a}$ is a rational number but $\sqrt{7}$ is irrational number. 

But, Rational number  $≠$  Irrational number.

This contradiction has arisen because of our incorrect assumption that $\frac{2}{\sqrt{7}}$ is rational.

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Updated on: 10-Oct-2022

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