Prove that $4 − 5\sqrt{2}$ is an irrational number.

Given: $4\ −\ 5\sqrt{2}$

To do: Here we have to prove that $4\ −\ 5\sqrt{2}$ is an irrational number.

Solution:

Let us assume, to the contrary, that $4\ −\ 5\sqrt{2}$ is rational.

So, we can find integers a and b ($≠$ 0) such that  $4\ −\ 5\sqrt{2}\ =\ \frac{a}{b}$.

Where a and b are co-prime.

Now,

$4\ −\ 5\sqrt{2}\ =\ \frac{a}{b}$

$4\ -\ \frac{a}{b}\ =\ 5\sqrt{2}$

$\frac{4b\ -\ a}{b}\ =\ 5\sqrt{2}$

$\frac{4b\ -\ a}{5b}\ =\ \sqrt{2}$

Here, $\frac{4b\ -\ a}{5b}$ is a rational number but $\sqrt{2}$ is irrational number. 

But, Rational number  $≠$  Irrational number.

This contradiction has arisen because of our incorrect assumption that $4\ −\ 5\sqrt{2}$ is rational.

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Updated on: 10-Oct-2022

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