Is there a (real) number which gives a rational number both when multiplied by $\pi$ and when multiplied by $e$? 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.

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 $x^2 - (e +\pi) x + e \pi$.
