Question

Prove that √5 is an irrartional number

Answer 1

Let us assume that √5 is a rational number

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

(where p and q are coprime numbers and q is not equal to 0)

Squaring both sides

$( \sqrt{5) }^{2} = \frac{p ^{2} }{q ^{2} }$

$5q ^{2} = p ^{2}$

Therefore, p^2 is divisible by 5

and p is also divisible by 5

Now, p=5m

(where m is any integer)

$5q ^{2} = 25m ^{2}$

$q ^{2} = 5m ^{2}$

Therefore,q^2 is divisible by 5

q is also divisible by 5

But this contradicts our assumption because p and q are coprime number

therefore, our assumption is wrong.

Hence √5 is an irrational no.