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 $e^i$ ?

In this answer, I show that
$$
e^{ix}=\lim_{n\to\infty}\left(1+\frac{ix}{n}\right)^n=\cos(x)+i\sin(x)
$$
using the geometric properties of complex multiplication. It is argued there that, as shown in this answer, since
$$
\lim_{x\to0}\frac{\tan(x)}{x}=1
$$
and
$$
\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{n/2}=1
$$
that the product of $n$ copies of $1+ix/n$
$\hspace{5cm}$
that is,
$$
\lim_{n\to\infty}\left(1+\frac{ix}{n}\right)^n
$$
has length $1$ and argument $x$, which means that it equals
$$
\cos(x)+i\sin(x)
$$
