# Is $\large \frac {\pi}{e}$ rational, irrational, or trandescendal?

Is there an argument for why $\large \frac {\pi}{e}$ is rational, irrational, or trandescendal? Can the quotient of any two transcendental numbers (which are not rational multiples of each other) be rational, or at least irrational? Thanks.

#### Solutions Collecting From Web of "Is $\large \frac {\pi}{e}$ rational, irrational, or trandescendal?"

With $e$ also $\frac{1}{e}$ is transcendental. The polynomial $(x-\pi)(x-\frac{1}{e})=x^2-(\frac{1}{e}+\pi)+\frac{\pi}{e}$ cannot have rational coefficients, hence either $\frac{1}{e}+\pi$ or $\frac{\pi}{e}$ is irrational.
But indeed, the question is whether $π$ and $e$ are algebraically independent. See also https://mathoverflow.net/questions/33817/work-on-independence-of-pi-and-e.