Why does $e^{i\pi}=-1$? I will first say that I fully understand how to prove this equation from the use of power series, what I am interested in though is why $e$ and $\pi$ should be linked like they are.

As far as I know $\pi$ comes from geometry (although it does have an equivalent analytical definition), and $e$ comes from calculus. 

I cannot see any reason why they should be linked and the proof doesn't really give any insights as to why the equation works. 

Is there some nice way of explaining this?
Euler's formula describes two equivalent ways to move in a circle.


Starting at the number $1$, see multiplication as a transformation that changes the number $1 \cdot e^{i\pi}$.
Regular exponential growth continuously increases $1$ by some rate; imaginary exponential growth continuously rotates a number in the complex plane.
Growing for $\pi$ units of time means going $\pi\,\rm radians$ around a circle
Therefore, $e^{i\pi}$ means starting at 1 and rotating $\pi$ (halfway around a circle) to get to $-1$.


For more details explaining each step, read this article.
$$e^{ix} = \cos x + i\sin x$$

the $\pi$ is to some extent an arbitrary definition of the angle.