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

Question

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.

Answer 1

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.

Answer 2

It is not known, since it is not known if $\pi/e$ is rational or not: if $a\pi$ and $ae$ were rational, then so would be $\pi/e$.

Answer 3

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