What's the deal with Euler's identity?

Personally, I prefer the version with tau (2 times pi) in it rather than the one with pi:

e^(i*tau) = 1

I won't reproduce https://www.tauday.com/tau-manifesto here, but I'll just mention one part of it. I very much prefer doing radian math using tau rather than pi: tau/4 radians is just one-fourth of a "turn", one-fourth of the way around the circle, i.e. 90°. Which is a lot easier to remember than pi/2, and would have made high-school trig so much easier for me. (I never had trouble with radians, and even so I would have had a much easier time grasping them had I been taught them using tau rather than pi as the key value).

It's obviously of more interest as a piece of outreach than as a piece of mathematics. Nevertheless I've always wondered about the e^ipi + 1 = 0 formulation. It seems ugly and ad hoc, and the connection between the "5 constants" is not all that meaningful.

That e^ipi = -1 is related to the much more profound observation that the complex numbers represent a sort of rotation into a previously unknown dimension of numbers.

This is just scratch on the surface.

* Enter quaternions; things get more profound.

* Investigate why multiplicative inverse of i is same as its additive inverse.

* Experiment with (1+i)/(1-i).

* Explore why i^i is real number.

* Ask why multiplication should become an addition for angles.

* Inquire the significance of the unit circle in the complex plane.

* Think bout Riemann's sphere.

* Understand how all this adds helps wave functions and quantum amplitudes.

Here is the Euler's identity in my recent side project, equations visualised - https://p.migdal.pl/equations-explained-colorfully/#euler.

Nobody ever considers the spinorial version. e^iπ is a 360° rotation on a spinor, and + is averaging spinors rotationally. so e^iπ + 1 = 0 means there is no way to interpolate between the identity and a twist in the spinor, because the axis of a 360° rotation is undefined.

Things get so much more fun once you embrace spinors.

I do like this explanation better. I learned the Maclaurin series explanation in school, where you can show that the series approximations line up, but I never felt that explained why it worked. The idea of starting with -1 as a half rotation and then taking fractions of that really appeals to the intuition.

Never liked that form of the Euler's formula. I prefer the following:

    (-1)ˣ = cos(πx) + i sin(πx)

The real magic is that complex exponentials are describing rotations in R^2.

To be honest, this equation completely fails to represent this.