Why is $\cos(i)-i \sin(i)=e$?

Question

When I was typing $\cos(i)-i \sin(i)$ into the calculator, I found out that it is equal to e (Euler's Constant). I was amazed by that "discovery" so I checked in on the internet and there was no results. Someone please explain the connection of imaginary numbers and Euler's Constant.

Answer 1

We have $e^{iz}=\cos z+i \sin z$ for all $z \in \mathbb C$. This can easily seen with power series. Now plug in $z=-i$.