Question

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

Answer 1

Let us assume that $\surd 5$ is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are coprime numbers.

so, $\sqrt{5}=\frac{p}{q}$

$p=\sqrt{(}5)q$

we know that 'p' is a rational number. so $\sqrt{(}5)$q must be rational since it equals to p

but it doesnt occurs with $\sqrt{(}5)q$since its not an integer

therefore, p is not equal to $\sqrt{(}5)q$

this contradicts the fact that $\sqrt{(}5)$ is an irrational number

hence our assumption is wrong and $\sqrt{(}5)$ is an irrational number