TITLE: Is $e^{i+\pi}$ irrational or not?
QUESTION [7 upvotes]: Since we know that the value of $e$, $i$, and $\pi$ are irrational reals, how about $$e^{i+\pi}\;?$$ Is it still irrational (that is, not a Gaussian rational)?
The problem make me curious until now.

REPLY [27 votes]: Building off of Clive's answer, if $e^\pi \cos 1$ and $e^\pi \sin 1$ were both rational numbers, then so would be their quotient $\tan 1$.  But $\tan 1$ is irrational as a consequence of the Lindemann–Weierstrass theorem, because that theorem implies that $e^{2i}$ is transcendental, and because $e^{2i}=\dfrac{i-\tan 1}{\tan 1+i}$.  Therefore $e^{\pi+i}$ cannot have both its real and imaginary parts rational.