What's the deal with Euler's identity?(lcamtuf.substack.com)
lcamtuf.substack.com
What's the deal with Euler's identity?
https://lcamtuf.substack.com/p/whats-the-deal-with-eulers-identity
47 comments
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!
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.
But for teaching trig? Explaining radians should definitely be tau-based.
Do you mean the advantage of writing pi*d for the circumference instead of tau*r or tau*d/2? I wouldn't keep pi around just for this...
Yes, though more broadly my point was that the radius is the natural measurement of the circle for most things since most things are center-based. But for some physical measurements, mostly based around pipes, "what is the width of this pipe" is the question you need answering, and that is diameter-based. And pi is circumference/diameter, while tau is circumference/radius.
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.
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.
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).
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).
There’s nothing particularly convenient about base-ten; for real-world uses base-twelve would be preferable thanks to its large number of divisors (and even larger number of divisors of its multiples like 60). Which is exactly why 12 and 60 historically appear in many contexts.
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.
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.
> base-16 would be the most convenient
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.
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.
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Ah, one of these battles that are very hard to fight to gain essentially nothing.
Edit: or, when you can't do actual math, you complain about notation.
Edit: or, when you can't do actual math, you complain about notation.
The gain is pedagogical: giving kids a good intuition about angles is so much easier when the constant you're working with represents an entire turn around the circle (360°) rather than a half-turn of 180°. The advantage of using tau instead of pi is much smaller in other situations, but when it comes to measuring angles in radians, it's huge. And kids who have a better understanding of angles and trigonometry are just a little bit more likely to become good engineers. So persuading math teachers that there's a better way to teach trig is an investment in the future whose potential payoff is 20-30 years (or more) down the road.
I'd really be curious to see any substantial proof for that claim.
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.
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.
I don't know of any studies yet comparing the two approaches, but https://www.tauday.com/a-tau-testimonial is the story of one student who finally "got it" when using tau instead of pi. I strongly suspect she's not unique.
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).
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).
PI is to clever by half.
The one place where radians are more convenient is when you are at the centre of the circle. Then something which is as wide (or tall) as it is far away subtends one radian in your view. (And correspondingly, if it subtends half a radian it is half as wide as it is far away, etc.)
This happens to be the most common situation in which I measure angles.
This happens to be the most common situation in which I measure angles.
More convenient than degrees. This is unrelated to pi vs tau (using tau or pi doesn't change the meaning of radians, the properties you mention are not affected). What OP is getting at is that the same number of radians, e.g. 1.57 (quarter turn) is more naturally expressed as tau/4 than pi/2.
Which would be e^(i*tau) - 1 = 0 if you wanted to honor the spirit of the Identity.
tau might make things easier and be a better basis for everything but it is not good for Euler's as you then miss out -
535.491…^i = 1
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.
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.
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.
* 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.
Any hints towards the answers? I've spent a lot of time with complex numbers, and my answers would be
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!
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!
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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.
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.
Yes, but it seems strange to claim that i^i isn't anything. That just completely ignores what's interesting, namely that i(π/2 + 2πk) is real for all k ∈ Z.
In maths, an expression only ever equals a single number. You can't say i^i = e^(-pi/2) and then also say that i^i = e^(3 pi / 2), because if i^i equals two things, then those two things are also equal to each other, and then we get that e^(-pi/2) = e^(3 pi / 2), which is wrong.
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.
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.
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Apologies if this is pedantic but "multi-valued function" is not a thing. The function doesn't have multiple values here. Saying "multi-valued function" is not just a way of misleading people about what's really happening, but it's almost the perfect way to stop people from finding out. Do people who say "multi-valued function" know what's really happening? Do you know what a Riemann surface is?
Do you know what a Riemann surface is? Because if you don't, then you don't know what you're talking about - and you should stop getting people confused.
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.
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.
Here is the Euler's identity in my recent side project, equations visualised - https://p.migdal.pl/equations-explained-colorfully/#euler.
I do not think I appreciated the formula until I had been exposed to the exponential map in its generalized forms: The so-called 'matrix exponential' and the exponential map of Lie algebra. They place Euler's humble formula into a grander and rather beautiful setting.
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 $x \mapsto e^{it}x$).
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 $x \mapsto e^{it}x$).
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.
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)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?"
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?"
The real magic is that complex exponentials are describing rotations in R^2.
To be honest, this equation completely fails to represent this.
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).