TITLE: How Do I Understand $e^i$, the Euler Form of Complex Number
QUESTION [34 upvotes]: 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 $e^i$ ?

REPLY [2 votes]: I think you have problem with understanding what exactly is meant by $e^{i}$ rather than formulas to compute.
In that case we have problems to intuitively understand what exactly is $a^x$ when the domain is $\mathbb{R}$ .This can be easily understood when the domain is integers. For eg:
$a^2=a*a$ multiplying by $n$ times the number $a$ which is called as the base. We can also define the same thing similarly when $n$ is a rational number. But how can we define what exactly is $a^{\sqrt{2}}$ or $a^{\sqrt{\sqrt[3]{3}+\sqrt[5]{5}}}$ The same problem like raising to  an imaginary number occurs.
But we can simply avoid such problems by defining logarithm and then using that to define what is $a^x$.
Once you get the definition of $a^x$ with domain real numbers . The extension to imaginary numbers is easy .