*Published on 2023-06-28 (Tau Day)*

As a physicist, I’ve learned that the presence of $2\pi$ [Or rather $\tau$, 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 $2\pi$ 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 $2\pi$ started.]

Recently, I came up with a trick that can get rid of $2\pi$ 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: $\frac{\mathrm{đ}}{\mathrm{d} x} f(x)
\equiv
\frac{1}{2\pi}
\cdot
\frac{\mathrm{d}}{\mathrm{d} x} f(x).$ That’s all. You just take the derivative and then divide by $2\pi$. I call this derivative operator with the bar on the upper $\mathrm{d}$ 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 $f(x)$ which satisfies $\frac{\mathrm{đ}}{\mathrm{d} x} f(x) = f(x), \quad f(0) = 1$ is $f(x) = e^{2\pi x} = \Theta^x,$ where $\Theta$ (capital Theta) is defined as $\Theta \equiv e^{2\pi} = 535.491\dots$. [I have a few words about the choice of symbol below.] Let me show you why $\Theta^x$ is actually a pretty nice function.

Probably the nicest thing about $\Theta^x$ has to do with trigonometric functions. You may or may not know that sine and cosine can be defined using complex exponentials like this: $\begin{align*}
\sin(x) &= \frac{e^{ix} - e^{-ix}}{2i},
\\
\cos(x) &= \frac{e^{ix} + e^{-ix}}{2},
\end{align*}$ but using $\Theta$, we can define new “trigonometric functions”: $\begin{align*}
\operatorname{usin}(x) &= \frac{\Theta^{ix} - \Theta^{-ix}}{2i},
\\
\operatorname{ucos}(x) &= \frac{\Theta^{ix} + \Theta^{-ix}}{2}.
\end{align*}$ These are some names that I just came up with, but the “$\mathrm{u}$” stands for “unit” — these functions have the nice property that their period is $1$ instead of $2\pi$: $\begin{align*}
\operatorname{usin}(x + 1) &= \operatorname{usin}(x),
\\
\operatorname{ucos}(x + 1) &= \operatorname{ucos}(x).
\end{align*}$ I like this, because it kind of eliminates the need for radians: the $x$ in $\operatorname{usin}(x)$ has the unit of “turns”. I think this is conceptually much simpler. [These $\Theta$-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 $\pi$ instead of $2\pi$. 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 $0\dots 1$ 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 $0\dots 2\pi$. 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 $\operatorname{usin}(f t)$ is a periodic function with frequency $f$, without the need for angular frequency. Instead of mushing $2\pi f$ into the single symbol $\omega$, we meld $2\pi$ into the trig function. Yes, this is just an alternative way of sweeping $2\pi$ 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: $\begin{align*} \frac{\mathrm{đ}}{\mathrm{d} x} \operatorname{usin}(x) & = \hphantom{-} \operatorname{ucos}(x), \\ \frac{\mathrm{đ}}{\mathrm{d} x} \operatorname{ucos}(x) & = -\operatorname{usin}(x). \end{align*}$ Nice!

You might be aware that $e$, $2\pi$ and trigonometric functions are pretty widespread in math. Because of this, there are actually quite many situations where $\Theta^x$ pops up and makes that pesky $2\pi$ 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 $\hat f$ of a function $f$: $\begin{align*} \hat f(s) &= \int_{-\infty}^\infty e^{-2\pi i s x} f(x) \mathrm{d}x \\ &= \int_{-\infty}^\infty \Theta^{-i s x} f(x) \mathrm{d}x. \end{align*}$

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

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

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

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

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

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

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

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

Similarly, I dare to propose writing the Schrödinger equation like this: $ih \frac{\mathrm{đ}}{\mathrm{d} t} \ket{\Psi} = \hat H\ket{\Psi}.$ Note the use of the regular $h$ (which is really the fundmnental constant of nature) instead of $\hbar$. With this equation, some basic results of quantum mechanics can be rewritten without $2\pi$ and using just plain $h$, and sure enough, $\Theta^x$, $\operatorname{usin}(x)$ and $\operatorname{ucos}(x)$ 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 $\mathrm{đ}x = \frac{1}{2\pi}\mathrm{d}x$, which can then simplify some integral formulae, such as the Gauss-Bonnet theorem: $\begin{align*} \int_M K \mathrm{d}A + \int_{\partial M} k_g \mathrm{d}s &= 2 \pi \chi(M) \\ \Rightarrow \frac{1}{2\pi}\left(\int_M K \mathrm{d}A + \int_{\partial M} k_g \mathrm{d}s\right) &= \chi(M) \\ \Rightarrow \int_M K \mathrm{đ}A + \int_{\partial M} k_g \mathrm{đ}s &= \chi(M) \end{align*}$ or Cauchy’s integral formula: $\begin{align*} f(a) &= \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-a} \mathrm{d} z \\ &= \frac{1}{ i} \oint_C \frac{f(z)}{z-a} \mathrm{đ} z \\ &= i \oint_C \frac{f(z)}{a-z} \mathrm{đ} z . \end{align*}$ 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.

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

Physicists use angular frequency quite often, because $2\pi$ 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 $2\pi$ or $1/(2\pi)$ or $1/\sqrt{2\pi}$ (*shudder*). I do believe that using $\Theta^{ix}$, $\operatorname{usin}$, and $\operatorname{ucos}$ 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 $\mathrm{đ}$ is as much mental effort as remembering to keep $2\pi$ around, but I say this is not the case: differentials appear less often than $e^{2\pi}$, and they are more of a special object, so they are handled with more care by default. [Except when physicists treat $\frac{\mathrm{d}y}{\mathrm{d}x}$ 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 $2\pi$”. 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 $\Theta$, $\operatorname{usin}$, and $\operatorname{ucos}$, and the reduced derivative came up as a post-hoc argument against $\sin$ and $\cos$ being fundamental functions that satisfy $\frac{\mathrm{d}}{\mathrm{d} x} f(x) = g(x)$ and $\frac{\mathrm{d}}{\mathrm{d} x} g(x) = -f(x)$.

One thing I can say for sure is that this trick doesn’t eliminate $2\pi$ from everywhere for all eternity. You *could* express the circumference of a circle as $\ln(\Theta) r$, but I don’t expect this to catch on. Euler’s formula is also a fairly advanced concept, so the right place for $\Theta^{i x}$ 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 $2\pi$ 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 $\Theta^x$ 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!

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, $e^{2\pi}$ as a single, explicit constant worth considering has not been discussed anywhere either. The closest I could find was Gelfond’s constant, which is $e^{\pi} = \sqrt{\Theta}$. For example, OEIS describes $e^{2\pi}$ 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.

I’m not sure what “the derivative divided by $2\pi$” should really be called. I picked “reduced derivative”, like how the Planck constant divided by $2\pi$ is called the reduced Planck constant. This is also the inspiration for the notation $\mathrm{đ}$ (`\text{\dj}`

in $\LaTeX$), which I propose to be pronounced as “d-bar”. [I put the bar only on the $\mathrm{d}$ in the numerator of $\frac{\mathrm{đ}}{\mathrm{d}x}$, so that the division works out if you think of it as “$\mathrm{đ} = \mathrm{d} / (2\pi)$”. 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 $\Theta$ as the symbol for $e^{2\pi}$, 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 $\Theta^x$ also kind of resembles $e^x$, which is a reminder of its definition. Apart from these *extremely compelling* aesthetic arguments, I should note that $\Theta$ might be easy to confuse with the lowercase $\theta$ or $e$ in handwriting, so I’m also open to suggestions for the symbol.

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 $\begin{align*} \frac{\mathrm{đ}}{\mathrm{d} x} \left(f(x) g(x)\right) &= \frac{1}{2\pi} \frac{\mathrm{d}}{\mathrm{d} x} \left(f(x) g(x)\right) \\ &= \frac{1}{2\pi} \left(\!\! \left(\frac{\mathrm{d}}{\mathrm{d} x} f(x)\right) g(x) + f(x) \left(\frac{\mathrm{d}}{\mathrm{d} x} g(x)\right) \!\!\right) \\ &= \left(\frac{\mathrm{đ}}{\mathrm{d} x} f(x)\right) g(x) + f(x) \left(\frac{\mathrm{đ}}{\mathrm{d} x} g(x)\right) , \end{align*}$ and the chain rule is $\begin{align*} \frac{\mathrm{đ}}{\mathrm{d} x} f(g(x)) &= \frac{1}{2\pi} \frac{\mathrm{d}}{\mathrm{d} x} f(g(x)) \\ &= \frac{1}{2\pi} \, \frac{\mathrm{d}}{\mathrm{d} y} f(y) \Big|_{y = g(x)} \cdot \frac{\mathrm{d}}{\mathrm{d} x} g(x) \\ &= \frac{\mathrm{đ}}{\mathrm{d} y} f(y) \Big|_{y = g(x)} \cdot \frac{\mathrm{d}}{\mathrm{d} x} g(x) \\ &= \frac{\mathrm{d}}{\mathrm{d} y} f(y) \Big|_{y = g(x)} \cdot \frac{\mathrm{đ}}{\mathrm{d} x} g(x) , \end{align*}$ 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 $f(x)$ and $g(x)$.

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

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

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

- The Tau Manifesto by Michael Hartl of course.
- Al-Kāshī’s Constant [PDF] by Peter Harremoës lists lots of formulae where $2\pi$ shows up.
- Trig Rerigged [PDF] by Thomas Colignatus discusses the functions I called $\operatorname{usin}$ and $\operatorname{ucos}$.
- This post on the Wolfram blog by Georgia Fortuna does some interesting quantitative analysis of whether using $\tau$ instead of $\pi$ would make formulae simpler. The conclusion is that in many cases it would not. While I’m not entirely convinced of the methodology (they look at formulae found in math papers on arXiv and determine their complexity by looking at the size of expression trees produced by the Wolfram Language, which in my opinion doesn’t consider enough of the surrounding context), an interesting point is brought up at the end: Treating $2\pi i$ as a single constant instead of just $2\pi$ would simplify formulae a lot more than changing from $\pi$ to $\tau$. However, I suspect what is actually happening is that formulae are simplified because they involve $e^{2\pi i}$! [This seems to be the case at least based on the word cloud of the simplified formulae they show, although there are a couple of examples where $2\pi i$ appears outside of an exponential.] This would be another argument in favor of using $\Theta^{ix}$.

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