Let us consider √5 is rational.

So,

√5 = p/q.

(where p and q are co-prime number and q ≠ 0)

Squaring on both sides give,

5 = p²/q²

5q² = p²

From this we can say that 5 divides p² so 5 will also divide p.So, 5 is one of the factor of p.

So we can write,p = 5a

Therefore,

5q² = (5a)²

5q² = 25a²

q² = 5a²

From this we can say that 5 divides q² so 5 will also divide q.So, 5 is one of the factor of q.

As, we know p and q are co-prime so it cannot have common factor. But here a contradiction arise that 5 is factor of both p and q.

So, by this we can say that √5 is not rational which means √5 is irrational.

Let us assume that√5 is a rational number.

We know that the rational numbers are in the form of p/q form where p,q are intezers.

so,√5 = p/q

p =√5q

we know that 'p' is a rational number. so√5 q must be rational since it equals to pbut it doesnt occurs with√5 since its not an intezer

Therefore, p =/=√5q

This contradicts the fact that√5 is an irrational number .

Hence our assumption is wrong and√5 is an irrational number.

asume √5 as rational first then solve

√5 is an irrational number we have to prove it so let us assume for a while that it is a rational number

then we look at the question once again and we found that it is written that we should prove it as irrational so or is actually an irrational number so no need to prove it as we now know that it is an irrational number