Question

Prove that $(4-5\sqrt{2})$ is an irrational number.

Answer 1

Let us assume that $(4-5\sqrt{2})$ is rational .

Subtract given number from $4,$ considering $4$ is rational number. as we know difference of two rational numbers is rational.

$4-(4-5\sqrt{2})$is rational

$\Rightarrow 5\sqrt{2}$ is rational
which is only possible if $5$ is rational and $\sqrt{2}$ is rational.
As we know prouduct of two rational number s rational
But the fact is $\sqrt{2}$ is an irrational.
Which is contradictory to our assumption.
Hence, $4-5\sqrt{2}$ is irrational . hence proved.