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.
But yes, if the world switched to tau then you wouldn't need pi anymore, you'd just write tau/2 in the rare cases where having the circumference/diameter ratio handy is useful.
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).
A number theorist would probably want a prime base, so that N (mod 10) would be a field.
A power-of-two base wouldn’t be particularly convenient to anyone except a small minority consisting mostly of hardware and software engineers.
That would mean 1/5=0.(3)₁₆ would be an infinite fraction as well. A more convenient would be 6 or 12 because it allows to represent 1/3 exactly.
This happens to be the most common situation in which I measure angles.
Edit: or, when you can't do actual math, you complain about notation.
The first time pupils encounter pi isn't when measuring angles. At least over here, that's still done in degrees, which is much easier to explain, and also latches onto common cultural practice (e.g. a turn of 180 degrees). So I suppose that already makes them good engineers.
But the first time pupils encounter pi is when computing the circumference and surface of a circle. While the former would look easier with the radius (tau * r), it looks just as weird when using diameter or when using it for the surface.
If there's more data available, I don't yet know where to find it.
P.S. Yes, angles are first presented in degrees in most contexts, and understanding sines and cosines is easier when given the degree units you're familiar with. But radians do need to get introduced at some point during trig, and it's exactly the study of radians which should be done using tau (the equivalent of 360°) rather than pi (180°). Because a right angle, 90°, is a quarter of the way around the circle, and that's tau/4. A 45° angle is tau/8, one-eighth of the way around the circle. There's no need to memorize formulas when you do it this way, it's just straight-up intuitive (whereas 45° = pi/4 is not intuitive the same way).
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.
* 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.
Quaternions: not profound, C is complete, quirky but useful representation of SO(3)
Inverses: fun fact coincidence
1+i/1-i: not sure what to experiment with here
i^i: gateway to riemann surfaces.
Adding angles: comes out like this, that's the point of exp(i phi)
Unit circle: roots of unity?
Riemann sphere: cool stuff!
Quantum stuff: mathematical physicist here, no need to sell this one!
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.
Riemann surfaces are the only way to fix this. And they're not even that hard to understand, but I'm not sure if you do.
Stop making people confused.
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.
Now, I like to think of exponentiation as a kind of integral over infinitesimal generators; and $i$ just happens to be a generator for rotation about a circle in the plane (aka $\mathrm{U}(1)$ aka $x \mapsto e^{it}x$).
Things get so much more fun once you embrace spinors.
(-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?"
To be honest, this equation completely fails to represent this.
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).