Question

Prove that $5-\sqrt{3}$ is irrational

Answer 1

Let us assume that $5-\sqrt{3}$ is a rational number.

So, it can be written in the form $\frac{a}{b}$
$5-\surd 3=\frac{a}{b}$

Here $a$ and $b$ are coprime numbers and $b\ne 0$

Solving $5-\surd 3=\frac{a}{b}$ we get,
$\Rightarrow \surd 3=5-\frac{a}{b}$
$\Rightarrow \surd 3=\frac{5b-a}{b}$
This shows $\frac{5b-a}{b}$ is a rational number. But we know that $\surd 3$ is an irrational number.

So, it contradicts our assumption. Our assumption of $5-\sqrt{3}$ is a rational number is incorrect.

$5-\sqrt{3}$ is an irrational number

Hence, it is proved that $5-\sqrt{3}$ is irrational.