1
$\begingroup$

I've been puzzling over this question about number theory.

Ignoring complex numbers, and working only with positive numbers:

Can one say that a positive rational number raised to a positive irrational exponent is always irrational?

Thank you, Hein

$\endgroup$
3
$\begingroup$

No. For example $x:=\log_2(3)$ is irrational but $2^x=3$.

$\endgroup$
  • $\begingroup$ Thanks so much, much appreciated $\endgroup$ – Hein Vogel Mar 4 '18 at 18:20
  • 3
    $\begingroup$ And the reason it's irrational: if we had $\log_2 3 = \frac{p}{q}$ with $p, q$ positive integers, then we would have $2^p = 3^q$, contradicting the fundamental theorem of arithmetic (unique factorization into primes). $\endgroup$ – Daniel Schepler Mar 4 '18 at 18:23
  • $\begingroup$ Thanks @DanielSchepler $\endgroup$ – Hein Vogel Mar 4 '18 at 18:26

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.