TITLE: Is $e^{i\pi}+1=0$ all it's cracked up to be?
QUESTION [27 upvotes]: While it is beautiful and elegant and all that, isn't it true that Euler's identity is really just an artifact of how we define the radian? I'm speaking of those who say that it's great because it contains five important constants in one elegant identity.
If we had stuck with degrees, it would be $e^{180^\circ i}+1=0$, not quite as breathtaking.
I understand that the radian is a natural and dimensionless unit to use for angles, but I feel like there is still some element of arbitrariness to it which, to me at least, somewhat tarnishes the beauty of this identity.
Am I missing something that makes this relationship between $e, \pi, i, 1,$ and $0$ built-in to our universe? Or do we as mathematicians just like to think poetically?

REPLY [2 votes]: I suppose you could say that the mystery/beauty is that the exponential function, which is monotone increasing as a function of a real variable, becomes periodic as a function of a complex variable, and has period $2 \pi i.$