$e^{\pi i}$ is a silly notation for $(grow\ 1\ \pi i)$. Growth by $i$ is perpendicular growth. Perpendicular growth is circular motion on the complex plane. $\pi$ is the number of radian-lengths in half a circle.

Once you know all that, $e^{\pi i}$ translates to “rotate half a turn on the complex plane”. When you start at $1$ and you rotate half a turn you end up at $\text{–}1$.

You now understand why $e^{\pi i} = \text{–}1$. Congratulations! This is often considered advanced and counter-intuitive, but once you understand the definitions it’s almost tautological. You can now calculate $e^{2\pi i}$ without much trouble.

This still isn’t how math should be.

It’s easy to say that you now have a better intuition for $e$, $i$, and $\pi$. It’s easy to say that you’re now better at mathematics. It’s easy to say “see, $e$ and $i$ aren’t that hard”.

It’s harder to notice that they never should have been so hard to learn in the first place. Euler’s identity never should have existed. We shouldn’t have used maddening syntax and imaginary exponentiation to talk about walking around circles.

It’s interesting that perpendicular growth is the same as circular motion, but you don’t need to know that in order to know that walking halfway around the unit circle takes you from one to negative one. The syntax further clouds the revelation: Why do we have special unit for half a turn? What does $2.718$ have to do with it?

You’re beginning to see the problems with mainstream mathematics. The first problem is that common math education doesn’t teach you things the easy way. Most people have had to memorize $i$ and learn that $i = \sqrt{-1}$ without ever being taught that the complex number plane is just another add-on to the number line that’s particularly useful for talking about rotation.

The second problem is that the things you’re taught the hard way are themselves contrived. It would be nice if someone had taught you explicitly that $\pi$ is the number of radiuses it takes to go halfway around a circle. It would be better if you never learned $\pi$ at all and just learned everything in turns. Why do you need a special symbol for “half a turn”?

This is what’s wrong with mathematics. The symbols are needlessly complicated and you’re taught them in a counter-intuitive manner.

This is why we need simplifience.