Question

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

Answer 1

Let $5-\sqrt{3}$ is a rational number.
We have to find out two integers a and b such as
$5-\sqrt{3}=\frac{a}{b}$
$-\sqrt{3}=\frac{a}{b}-5$
$\sqrt{3}=5-\frac{a}{b}$
$\sqrt{3}=\frac{5b-a}{b}$
$a,b$ and $5$ all are integers
$\frac{5b-a}{b}$ is a rational number.
$\sqrt{3}$ will be also rational number.
But this contradicts the fact that $\sqrt{3}$ is an irrational number.
So our hypothesis is wrong.
Hence, $5-\sqrt{3}$ is an irrational number.
Hence proved.