Show that the following number is irrational.
$3-\sqrt{5}$

Given: $3\ -\ \sqrt{5}$

To do: Here we have to prove that $3\ -\ \sqrt{5}$ is an irrational number.

Solution:

Let us assume, to the contrary, that $3\ -\ \sqrt{5}$ is rational.

So, we can find integers a and b ($≠$ 0) such that  $3\ -\ \sqrt{5}\ =\ \frac{a}{b}$.

Where a and b are co-prime.

Now,

$3\ -\ \sqrt{5}\ =\ \frac{a}{b}$

$\sqrt{5}\ =\ 3\ -\ \frac{a}{b}$

$\sqrt{5}\ =\ \frac{3b\ -\ a}{b}$

Here, $\frac{3b\ -\ a}{b}$ is a rational number but $\sqrt{5}$ is irrational number. 

But, Irrational number  $≠$  Rational number.

This contradiction has arisen because of our incorrect assumption that $3\ -\ \sqrt{5}$ is rational.

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Updated on: 10-Oct-2022

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