* 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.
I suppose that by pure convention, "w=e" is understood as denoting a single unique point on the helix. But extending that convention to w=i starts to look like a recipe for confusion.
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).
I've been posting the manifesto to friends and colleagues every tau day for the past ten years. Let's keep chipping away at it and eventually we won't obfuscate radians for our kids anymore.
Friends don't let friends use pi!
But for teaching trig? Explaining radians should definitely be tau-based.
IMHO, in a modern setting base-16 would be the most convenient. Then I maybe wouldn't struggle to remember that the CIDR range C0.A8.0.0/18 (192.168.0.0/24) consists of 10 (16) blocks of size 10 (16).
For now, I’ve just explicitly written exp(2πiν) etc instead of exp(iπν) in my work; explicitly writing out 2π and treating it as effectively one symbol does have similar conceptual benefits as working with τ.
(-1)ˣ = cos(πx) + i sin(πx) (-1)ˣ = cos(πx) - i sin(πx)
Instead shoehorning it into an arbitrary symbol salad by gimping its generality, I prefer the one which makes a statement: "What does it mean to apply inversion partially?"