How Do I Understand $e^i$, the Euler Form of Complex Number Raising something to an imaginary number is weird, I have a hard time wrapping my head around that.
And $e$ seems even more common and comes up in many situations, such as:


*

*the non-geometric definition of sin, 

*the fourier transform, 

*$e^{\pi i} = -1$ !?! (see, for instance, here)


I'd really like to have some light shed on the matter.
How do I begin to form an intuitive grasp of $e^i$ ?

Well do you know the power series of the exponential function? 
$$\exp(x)=\sum_{k=0}^\infty \frac{x^k}{k!}=1+x+\frac{x^2}{2}+\frac{x^3}{3\cdot 2}+\dots$$
And in Special 
$$\exp(ix)=\cos(x) + i \sin(x)$$
The $e$ function is so common because it is the only function which holds $$f'(x)=f(x) \quad f(0)=1$$ for  all $x$ 
