What's the deal with Euler's identity?
lcamtuf.substack.com26 points by surprisetalk 5 days ago
26 points by surprisetalk 5 days ago
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.
i^i isn't anything. Please don't write this. Of the two inputs to the function (w, z) -> w^z = exp(z ln(w)), only z is a complex number, so that bit is OK. The problem is that w is NOT a complex number but a point on a particular Riemann surface, namely: The natural domain of the function ln. That particular Riemann surface looks like an endless spiral staircase. The more grown-up term might be "a helix". When you write informally "w=i", that could mean any of ln(w) = i pi/2, i (2pi + pi/2), i(4pi + pi/2), etc. Incidentally, w^z is then always a real number. However, there's an infinite sequence of those numbers that it could equal.
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.
Is your argument that complex powers isn't anything?
Their argument is that ln(z) where z is a complex number is a multi-valued function, so the statement "Explore why i^i is real number" could be misinterpreted as i^i = a single well-defined real value.
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).
This!
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!
Oh, pi has its place: in engineering, for example, it's much easier to measure the diameter of a pipe than its radius: just put calipers around the widest point (outside or inside depending) and you have the diameter. In fact, you probably wouldn't ever measure the radius; in places where you need the radius, you'd just measure the diameter and divide by 2.
But for teaching trig? Explaining radians should definitely be tau-based.
I wonder how many places we have in modern math symbols which we use for historical reasons, rather than because it's most convenient overall. I guess we are balancing things here.
Arguably, base-10 counting vs base-12 counting is one such example
Which one of those is preferable? It seems to me that they are both historically based. 10 x 10 is also 100 in base-12 (it's only in base-10 that it looks like 144).
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).
Though the argument is technically correct, it is unnecessary at this point of time. Same as renaming cities and countries to "correct" history.
Disagree. This is not so much about epistemological correctness as it is about what's useful and convenient. math.tau is an easier and more intuitive constant to work with.
Math didactics is all about making math more digestible for the next generation, even if it breaks with history.
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.
Which would be e^(i*tau) - 1 = 0 if you wanted to honor the spirit of the Identity.
Here is the Euler's identity in my recent side project, equations visualised - https://p.migdal.pl/equations-explained-colorfully/#euler.
Never liked that form of the Euler's formula. I prefer the following:
(-1)ˣ = cos(πx) + i sin(πx)My objection to that is that there isn't a particularly natural reason not to say
(-1)ˣ = cos(πx) - i sin(πx)
That's not the point of the Identity. You exponentiated the beauty right out of it.
Beauty is in the eye of the beholder.
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?"