TITLE: Is there a (real) number which gives a rational number both when multiplied by $\pi$ and when multiplied by $e$?
QUESTION [9 upvotes]: Besides $0$ of course. What about addition and exponentiation? I would think there's no such number, but I'm not sure if I could prove it.

REPLY [25 votes]: It is open whether there is such number. The reason is that we do not know whether $e/\pi$ is rational, but this is equivalent to your question (for a recent accessible reference, see for example here and the links provided there):
Note that if $xe$ and $x\pi$ are rational, then so is their quotient, $e/\pi$. Conversely, if $e/\pi$ is rational, then take $x=1/\pi$ and note that $xe$ and $x\pi$ are rational.

REPLY [4 votes]: However, it is known that at least one of
$e+\pi$ and $e \pi$ is not only irrational
but transcendental.
We just don't know which one.
Of course,
everybody believes that both are.
Proof: Consider $x^2 - (e +\pi) x + e \pi$.