# Numerator and denominator of rational number

I have to prove that $$\sqrt5$$ is irrational.

I prove it by contradiction. I assume that $$p$$ is an integer and $$q$$ is a positive integer such that $$gcd(p,q)=1$$ and $$(\frac{p}{q})^2 = 5$$. And then proceed with the proof.

But my textbook suggests that, "assume that $$p$$ and $$q$$ are positive integers such that.... "

My question is, is it necessary that $$p$$ is a 'positive' integer, when $$\frac{p}{q}$$ is a rational number?

It is not - for example, $$\frac{-1}{2}$$ is rational number.
However, in this case we can assume $$p$$ is positive: if $$p$$ is negative, take $$\frac{-p}{q}$$ instead, as $$\left(\frac{p}{q}\right)^2 = \left(\frac{-p}{q}\right)^2$$.