A trick to eliminate 2π2\pi (sometimes)

Published on 2023-06-28 (Tau Day)

As a physicist, I’ve learned that the presence of 2π2\pi [Or rather τ\tau, as it should actually be called.] in a formula usually spells trouble. It’s kind of annoying to write by hand, easy to forget, and it’s dimensionless, so you can’t easily check if you forgot to divide or multiply by it. Problems appear especially often when translating between theory and experiments. [I remember one of the very first lectures of my “Introduction to University Physics” course, where the lecturer wrote a long and complicated equation on the blackboard, and then went “Oh, and maybe there should be a 2π2\pi here, I’m not exactly sure now. But you get the idea.” I was absolutely shocked. How could anyone do physics with such imprecision and carelessness?! Later of course I’ve learned that sometimes it’s OK to be off by an order of magnitude here or there. But I think this is where my frustration with 2π2\pi started.]

Recently, I came up with a trick that can get rid of 2π2\pi in many cases. It’s pretty simple, but it has some interesting implications. This is the trick: I just define a new derivative operator, like so: đdxf(x)12πddxf(x). \frac{\mathrm{đ}}{\mathrm{d} x} f(x) \equiv \frac{1}{2\pi} \cdot \frac{\mathrm{d}}{\mathrm{d} x} f(x). That’s all. You just take the derivative and then divide by 2π2\pi. I call this derivative operator with the bar on the upper d\mathrm{d} the reduced derivative. [I discuss this name and notation below.]

Now, why is this interesting? To start off, we’ll note that the unique function f(x)f(x) which satisfies đdxf(x)=f(x),f(0)=1 \frac{\mathrm{đ}}{\mathrm{d} x} f(x) = f(x), \quad f(0) = 1 is f(x)=e2πx=Θx, f(x) = e^{2\pi x} = \Theta^x, where Θ\Theta (capital Theta) is defined as Θe2π=535.491\Theta \equiv e^{2\pi} = 535.491\dots. [I have a few words about the choice of symbol below.] Let me show you why Θx\Theta^x is actually a pretty nice function.

Probably the nicest thing about Θx\Theta^x has to do with trigonometric functions. You may or may not know that sine and cosine can be defined using complex exponentials like this: sin(x)=eixeix2i,cos(x)=eix+eix2, \begin{align*} \sin(x) &= \frac{e^{ix} - e^{-ix}}{2i}, \\ \cos(x) &= \frac{e^{ix} + e^{-ix}}{2}, \end{align*} but using Θ\Theta, we can define new “trigonometric functions”: usin(x)=ΘixΘix2i,ucos(x)=Θix+Θix2. \begin{align*} \operatorname{usin}(x) &= \frac{\Theta^{ix} - \Theta^{-ix}}{2i}, \\ \operatorname{ucos}(x) &= \frac{\Theta^{ix} + \Theta^{-ix}}{2}. \end{align*} These are some names that I just came up with, but the “u\mathrm{u}” stands for “unit” — these functions have the nice property that their period is 11 instead of 2π2\pi: usin(x+1)=usin(x),ucos(x+1)=ucos(x). \begin{align*} \operatorname{usin}(x + 1) &= \operatorname{usin}(x), \\ \operatorname{ucos}(x + 1) &= \operatorname{ucos}(x). \end{align*} I like this, because it kind of eliminates the need for radians: the xx in usin(x)\operatorname{usin}(x) has the unit of “turns”. I think this is conceptually much simpler. [These Θ\Theta-based unit-period trigonometric functions are also nicer to calculate on a computer. For example, MATLAB and Julia have functions sinpi(x) and cospi(x) for calculating sin(pi*x) and cos(pi*x) with better precision (they are exactly zero at integer values of x), although they miss the mark by using π\pi instead of 2π2\pi. I’m no floating point expert, but I feel that in addition to precision, using such functions should be more efficient as well. The range 010\dots 1 can be represented exactly in binary (and range reduction amounts to just discarding the integer part!), so it’s much simpler to deal with than 02π0\dots 2\pi. This point is also brought up by Casey Muratori. It’s also interesting to note that dealing with radians is still a topic of research.] For the physicists out there, this is pretty cool because usin(ft)\operatorname{usin}(f t) is a periodic function with frequency ff, without the need for angular frequency. Instead of mushing 2πf2\pi f into the single symbol ω\omega, we meld 2π2\pi into the trig function. Yes, this is just an alternative way of sweeping 2π2\pi under the rug, but I claim it makes it easier to avoid mistakes.

These unit trigonometric functions satisfy similar differential equations as their more well-known siblings, except in terms of the funny reduced derivative operator we defined above: đdxusin(x)=ucos(x),đdxucos(x)=usin(x). \begin{align*} \frac{\mathrm{đ}}{\mathrm{d} x} \operatorname{usin}(x) & = \hphantom{-} \operatorname{ucos}(x), \\ \frac{\mathrm{đ}}{\mathrm{d} x} \operatorname{ucos}(x) & = -\operatorname{usin}(x). \end{align*} Nice!

You might be aware that ee, 2π2\pi and trigonometric functions are pretty widespread in math. Because of this, there are actually quite many situations where Θx\Theta^x pops up and makes that pesky 2π2\pi go away. Here are some examples I came up with off the top of my head and with a quick skim through Wikipedia:

The Fourier transform f^\hat f of a function ff: f^(s)=e2πisxf(x)dx=Θisxf(x)dx. \begin{align*} \hat f(s) &= \int_{-\infty}^\infty e^{-2\pi i s x} f(x) \mathrm{d}x \\ &= \int_{-\infty}^\infty \Theta^{-i s x} f(x) \mathrm{d}x. \end{align*}

The inverse Fourier transform: f(x)=e2πisxf^(s)ds=Θisxf^(s)ds. \begin{align*} f(x) &= \int_{-\infty}^\infty e^{2\pi i s x} \hat f(s) \mathrm{d}s \\ &= \int_{-\infty}^\infty \Theta^{i s x} \hat f(s) \mathrm{d}s. \end{align*} Note how the forward and inverse transforms are symmetric without any prefactors.

Dirac delta function: δ(xa)=e2πis(xa)ds=Θis(xa)ds. \begin{align*} \delta(x-a) &= \int e^{2\pi i s (x-a)} \mathrm{d} s \\ &= \int \Theta^{i s (x-a)} \mathrm{d} s. \end{align*}

Kronecker delta: δnm=1Nk=1Ne2πikN(nm)=1Nk=1NΘikN(nm). \begin{align*} \delta_{nm} &= \frac{1}{N} \sum_{k=1}^N e^{2\pi i \frac{k}{N} (n-m)} \\ &= \frac{1}{N} \sum_{k=1}^N \Theta^{i \frac{k}{N} (n-m)} . \end{align*}

Euler’s identity: e2πi=Θi=1. e^{2\pi i} = \Theta^i = 1.

Roots of unity: zn=1z=e2πik/n=Θik/n, \begin{align*} z^n = 1 \Leftrightarrow z &= e^{2\pi i k / n} = \Theta^{ik/n} , \end{align*} where nN,k=0,1,,n1n \in \mathbb{N}, \, k = 0, 1, \dots, n-1.

Square of the nome: q2=(eiπτ)2=Θiτ, q^2 = \left(e^{i\pi \tau}\right)^2 = \Theta^{i\tau} , where τ\tau is the half-period ratio of an elliptic function.

Gaussian-like integrals: [This result is easy to see from e(xb)2/(2c2)dx=c2π\int_{-\infty}^\infty e^{-(x-b)^2/(2c^2)} \mathrm{d} x = |c| \sqrt{2\pi}, by substituting c=1/2πc = 1/\sqrt{2\pi}.] Θx2/2dx=1, \int_{-\infty}^\infty \Theta^{-x^2/2} \mathrm{d}x = 1 , or more generally, Θ(xb)2/(2c2)dx=c. \int_{-\infty}^\infty \Theta^{-(x-b)^2/(2c^2)} \mathrm{d}x = |c| . I admit that this may not be so useful for probability theory: the distribution with probability density function Θx2/2\Theta^{-x^2/2} has a standard deviation of 1/2π1/\sqrt{2\pi} instead of 11, so the ugliness has just been pushed elsewhere.

Apart from Θx\Theta^x, it’s interesting to see whether the reduced derivative could simplify some equations. I think it makes sense to try to rewrite differential equations in terms of đdx\frac{\mathrm{đ}}{\mathrm{d} x} whenever we expect some sort of periodic solution. For example, the equation of a mass mm attached to a damped harmonic oscillator is [This can be obtained by dividing Newton’s second law through by (2π)2m(2\pi)^2 m.] đ2dt2x(t)+2ζf0đdtx(t)+f02x(t)=0, \frac{\mathrm{đ}^2}{\mathrm{d} t^2} x(t) + 2 \zeta f_0 \frac{\mathrm{đ}}{\mathrm{d} t} x(t) + f_0^2 x(t) = 0, where f0=k(2π)2mf_0 = \sqrt{\frac{k}{(2\pi)^2 m}} is the oscillation frequency, ζ=c/(2mk)\zeta = c / (2\sqrt{mk}) is the damping ratio, kk is the spring constant and cc is the damping coefficient. This has the solution x(t)=AΘζf0tusin(1ζ2f0t+φ), x(t) = A \Theta^{-\zeta f_0 t} \operatorname{usin}\left(\sqrt{1 - \zeta^2} f_0 t + \varphi\right), which is exactly the same as the regular solution, except that we have linear frequency f0f_0 instead of angular frequency ω0\omega_0, and the phase offset φ\varphi is in units of turns. [I actually like how the time constant of the decay 1/(f0ζ)1/(f_0 \zeta) is defined in terms of the linear frequency f0f_0. With this definition, if the oscillation has decayed for “one time constant”, the amplitude is Θ10.2%\Theta^{-1} \approx 0.2\% of the original. Normally, at least I use “three time constants” as some sort of cutoff for “the oscillation has mostly decayed”, because e35%e^{-3} \approx 5\%, but that’s still pretty far from zero.] Admittedly, 2π2\pi shows up in the definition of f0f_0, [But notably not in that of ζ\zeta!] so we haven’t completely gotten rid of it. [Maybe it could be hidden by defining an “angular spring constant” k~=k/(2π)2\tilde k = k / (2\pi)^2 or something. Or, could we interpret k/m\sqrt{k/m} as the circumference of a circle with radius f0f_0? I’ll concede that I’m grasping at straws here.]

Similarly, I dare to propose writing the Schrödinger equation like this: ihđdtΨ=H^Ψ. ih \frac{\mathrm{đ}}{\mathrm{d} t} \ket{\Psi} = \hat H\ket{\Psi}. Note the use of the regular hh (which is really the fundmnental constant of nature) instead of \hbar. With this equation, some basic results of quantum mechanics can be rewritten without 2π2\pi and using just plain hh, and sure enough, Θx\Theta^x, usin(x)\operatorname{usin}(x) and ucos(x)\operatorname{ucos}(x) show up in there. These are a bit too much to include here, so I’ll leave that as an exercise to the reader for a separate blog post.

This notation could be abused even further by denoting đx=12πdx\mathrm{đ}x = \frac{1}{2\pi}\mathrm{d}x, which can then simplify some integral formulae, such as the Gauss-Bonnet theorem: MKdA+Mkgds=2πχ(M)12π(MKdA+Mkgds)=χ(M)MKđA+Mkgđs=χ(M) \begin{align*} \int_M K \mathrm{d}A + \int_{\partial M} k_g \mathrm{d}s &= 2 \pi \chi(M) \\ \Rightarrow \frac{1}{2\pi}\left(\int_M K \mathrm{d}A + \int_{\partial M} k_g \mathrm{d}s\right) &= \chi(M) \\ \Rightarrow \int_M K \mathrm{đ}A + \int_{\partial M} k_g \mathrm{đ}s &= \chi(M) \end{align*} or Cauchy’s integral formula: f(a)=12πiCf(z)zadz=1iCf(z)zađz=iCf(z)azđz. \begin{align*} f(a) &= \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-a} \mathrm{d} z \\ &= \frac{1}{ i} \oint_C \frac{f(z)}{z-a} \mathrm{đ} z \\ &= i \oint_C \frac{f(z)}{a-z} \mathrm{đ} z . \end{align*} But this is perhaps stretching it a bit. I’m a meager experimental physicist who never even took a course on measure theory, so maybe this is total nonsense.

What is the meaning of all this?

So that’s quite a few occurrences of Θ\Theta in the wild. I hope that I’ve convinced you at least a little bit that treating e2πe^{2\pi} as a single symbol might make sense. But is there some deeper meaning to this? Should we declare Θ\Theta as the new circle constant, and throw 2π2\pi and the teachings of The Tau Manifesto in the garbage? Or is this all just sweeping 2π2\pi under the rug to make some formulae superficially prettier?

Physicists use angular frequency quite often, because 2π2\pi is genuinely annoying to deal with. It’s a major pain when numerical simulations don’t quite give the expected results, and you have to hunt for a possibly missing factor of 2π2\pi or 1/(2π)1/(2\pi) or 1/2π1/\sqrt{2\pi} (shudder). I do believe that using Θix\Theta^{ix}, usin\operatorname{usin}, and ucos\operatorname{ucos} really presents a new concept that lies somewhere between angular and linear frequency, and I think it can reduce mistakes quite a lot. It could be argued that remembering to put the bar on đ\mathrm{đ} is as much mental effort as remembering to keep 2π2\pi around, but I say this is not the case: differentials appear less often than e2πe^{2\pi}, and they are more of a special object, so they are handled with more care by default. [Except when physicists treat dydx\frac{\mathrm{d}y}{\mathrm{d}x} as a fraction.] Overall, I’ve had some moderate success with simplifying code by using turns instead of radians or degrees.

I’m not entirely sure about the intuitive meaning of “taking the derivative and dividing by 2π2\pi”. Is there some sort of fundamental connection to periodic functions? Maybe it helps with some dimensional analysis of functions on a circle, if we give angles units like in this paper. But I have yet to do any serious math using the reduced derivative, so this needs some more fleshing out. Originally, I only thought about Θ\Theta, usin\operatorname{usin}, and ucos\operatorname{ucos}, and the reduced derivative came up as a post-hoc argument against sin\sin and cos\cos being fundamental functions that satisfy ddxf(x)=g(x)\frac{\mathrm{d}}{\mathrm{d} x} f(x) = g(x) and ddxg(x)=f(x)\frac{\mathrm{d}}{\mathrm{d} x} g(x) = -f(x).

One thing I can say for sure is that this trick doesn’t eliminate 2π2\pi from everywhere for all eternity. You could express the circumference of a circle as ln(Θ)r\ln(\Theta) r, but I don’t expect this to catch on. Euler’s formula is also a fairly advanced concept, so the right place for Θix\Theta^{i x} might be an undergraduate physics or complex analysis course rather than in every textbook. But I’m pretty sure that working with just fractions of a turn instead of degrees makes trigonometry easier to grasp even at the high-school level, as The Tau Manifesto argues.

In general, I think that even though this kind of fiddling with notation may seem superficial, it’s actually important. Not having to worry about 2π2\pi reduces friction, so that you can focus on the problem you are trying to solve. If you reduce friction enough, even by small steps like this one, some untractable problems can become tractable.

So, even if it’s a bit unclear if there really is something to this idea, for now I plan to try and see if Θx\Theta^x makes my life easier. I encourage you to do the same. If you find something interesting while doing so, feel free to drop me an email!

Some additional notes

Who came up with this?

I could not really find any references online or in the literature discussing a concept like the reduced derivative I’ve introduced here. Searching for “derivative divided by 2pi” brings up mostly high-school level homework help of course, but that’s basically all I could find with my search-engine-fu. As far as I can tell, e2πe^{2\pi} as a single, explicit constant worth considering has not been discussed anywhere either. The closest I could find was Gelfond’s constant, which is eπ=Θe^{\pi} = \sqrt{\Theta}. For example, OEIS describes e2πe^{2\pi} as just “Gelfond’s constant squared”. If you know of any reference where this is discussed before, please let me know! I’d hate to take undeserved credit.

Naming and notation

I’m not sure what “the derivative divided by 2π2\pi” should really be called. I picked “reduced derivative”, like how the Planck constant divided by 2π2\pi is called the reduced Planck constant. This is also the inspiration for the notation đ\mathrm{đ} (\text{\dj} in LaTeX\LaTeX), which I propose to be pronounced as “d-bar”. [I put the bar only on the d\mathrm{d} in the numerator of đdx\frac{\mathrm{đ}}{\mathrm{d}x}, so that the division works out if you think of it as “đ=d/(2π)\mathrm{đ} = \mathrm{d} / (2\pi)”. Of course someone who is more versed in differentials (or is a mathematician or something) may object to this.] Of course, “reduced derivative” already means something, so maybe it’s not the best possible name.

Other options I came up with are “angular derivative” or “rotational derivative”, but I’m not so happy with these either. Angular derivative in particular seems already to be a thing in complex analysis, see for example here, here [PDF], or here. Suggestions are welcome!

I chose Θ\Theta as the symbol for e2πe^{2\pi}, because it doesn’t seem to be that widely used in math and physics (I checked here, here, and here), it’s kind of like a circle, which hints at rotations and angles, and Θx\Theta^x also kind of resembles exe^x, which is a reminder of its definition. Apart from these extremely compelling aesthetic arguments, I should note that Θ\Theta might be easy to confuse with the lowercase θ\theta or ee in handwriting, so I’m also open to suggestions for the symbol.

Product rule, chain rule

How do the basic rules of differentiation work with the reduced derivative? In short: they work just fine. Just by using the definition, the product rule is đdx(f(x)g(x))=12πddx(f(x)g(x))=12π( ⁣ ⁣(ddxf(x))g(x)+f(x)(ddxg(x)) ⁣ ⁣)=(đdxf(x))g(x)+f(x)(đdxg(x)), \begin{align*} \frac{\mathrm{đ}}{\mathrm{d} x} \left(f(x) g(x)\right) &= \frac{1}{2\pi} \frac{\mathrm{d}}{\mathrm{d} x} \left(f(x) g(x)\right) \\ &= \frac{1}{2\pi} \left(\!\! \left(\frac{\mathrm{d}}{\mathrm{d} x} f(x)\right) g(x) + f(x) \left(\frac{\mathrm{d}}{\mathrm{d} x} g(x)\right) \!\!\right) \\ &= \left(\frac{\mathrm{đ}}{\mathrm{d} x} f(x)\right) g(x) + f(x) \left(\frac{\mathrm{đ}}{\mathrm{d} x} g(x)\right) , \end{align*} and the chain rule is đdxf(g(x))=12πddxf(g(x))=12πddyf(y)y=g(x)ddxg(x)=đdyf(y)y=g(x)ddxg(x)=ddyf(y)y=g(x)đdxg(x), \begin{align*} \frac{\mathrm{đ}}{\mathrm{d} x} f(g(x)) &= \frac{1}{2\pi} \frac{\mathrm{d}}{\mathrm{d} x} f(g(x)) \\ &= \frac{1}{2\pi} \, \frac{\mathrm{d}}{\mathrm{d} y} f(y) \Big|_{y = g(x)} \cdot \frac{\mathrm{d}}{\mathrm{d} x} g(x) \\ &= \frac{\mathrm{đ}}{\mathrm{d} y} f(y) \Big|_{y = g(x)} \cdot \frac{\mathrm{d}}{\mathrm{d} x} g(x) \\ &= \frac{\mathrm{d}}{\mathrm{d} y} f(y) \Big|_{y = g(x)} \cdot \frac{\mathrm{đ}}{\mathrm{d} x} g(x) , \end{align*} where the last two rows are to show that you are free to choose which derivative you put the bar on. I suppose the desirable choice will depend on the specific functions f(x)f(x) and g(x)g(x).

Note that the product rule doesn’t produce very pretty results if one of the functions is not defined in terms of Θx\Theta^x, for example the reduced derivative of xusin(x)x \cdot \operatorname{usin}(x) (or regular sin(x)\sin(x) for that matter) has an extra factor of 1/2π1/2\pi in the đdxx\frac{\mathrm{đ}}{\mathrm{d} x} x term.

Why not also include the ii?

In most of the examples I showed, Θx\Theta^x actually appears in the form of Θix\Theta^{ix}. So why not define Θe2πi\Theta \equiv e^{2\pi i} and get even cleaner formulae? The problem is of course that e2πi=1e^{2\pi i} = 1, so (e2πi)x=1x=1(e^{2\pi i})^x = 1^x = 1 for any xx. But why does Θix\Theta^{i x} work then? This was a little bit surprising to me initially, but the answer is that in general, (ez)wezw(e^z)^w \neq e^{z w} for complex zz and ww. The correct formula is instead (ez)w=ewln(ez)(e^{z})^w = e^{w \ln(e^z)} [See for example here.], so that Θix=(e2π)ix=eixln(e2π)=e2πix,\Theta^{i x} = (e^{2\pi})^{ix} = e^{ix \ln(e^{2\pi})} = e^{2\pi ix}, but (e2πi)x=exln(e2πi),(e^{2\pi i})^{x} = e^{x \ln\left(e^{2\pi i}\right)}, where ln(e2πi)=ln1=0\ln\left(e^{2\pi i}\right) = \ln 1 = 0, so we get ex0=1e^{x \cdot 0} = 1. Note also that in general ln(z)\ln(z) is multi-valued for complex numbers.

Further reading

As I mentioned, there doesn’t seem to be much discussion on treating e2πe^{2\pi} as a single symbol, so these are mostly about π\pi vs τ\tau.

Thanks to Jami Kinnunen, Elsa Mannila, Viljami Pirttimaa, Aashish Sah, Lauri Seppäläinen, and Vasilii Vadimov for comments and feedback.