0
$\begingroup$

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?

These minor details are very important in real analysis. So, please help me clear this doubt. Thanks in advance.

$\endgroup$
  • $\begingroup$ It is necessary that p and q are both positive or both negative, so that their fraction becomes positive. It is necessary in this question because sqrt(5) is +ve and hence the fraction p/q should be positive $\endgroup$ – Ak19 Apr 26 at 14:40
  • $\begingroup$ @Ak19 Thanks. So p and q both are positive for this particular case. $\endgroup$ – user587389 Apr 26 at 14:52
  • $\begingroup$ Yeah you may take either both positive or both neagtive $\endgroup$ – Ak19 Apr 26 at 14:53
1
$\begingroup$

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 $\left(\frac{p}{q}\right)^2 = \left(\frac{-p}{q}\right)^2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.