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## Pi is still wrong

In anti-honor of “Pi Day,” I’d like to direct your attention to Michael Hartl’s The Tau Manifesto. The Manifesto is inspired by Bob Palais’ article $\pi$ is wrong! and presents a list of simple, but compelling, reasons that $2 \pi$, not $\pi$, is the more fundamental constant.

These ideas have been discussed on the blathosphere before, e.g. on Bill Gasarch and Lance Fortnow’s blog Computational Complexity. There Terence Tao makes the following remark:

It may be that $2 \pi i$ is an even more fundamental constant than $2 \pi$ or $\pi$. It is, after all, the generator of $\log(1)$. The fact that so many formulae involving $\pi^n$ depend on the parity of $n$ is another clue in this regard.

The basic argument for this point of view can be summarized as follows: $e^z : \mathbb{C} \to \mathbb{C}$ is a special function because it is the unique eigenvector of eigenvalue $1$ of the derivative operator acting on, say, complex-analytic functions on $\mathbb{C}$, and this function has period $2 \pi i$. So we see that this constant pops directly out of a definition of $\mathbb{C}$ and a definition of the derivative of a complex-analytic function: no arbitrary choices were necessary. (The closest thing to an arbitrary choice here is the decision to identify the tangent space of a point in a vector space with the vector space itself, but this is completely invariant.)

The $2 \pi$ here is precisely the circumference of a unit circle, which is distinguished among all circles because in $\mathbb{C}$ it is the only circle of positive radius closed under multiplication. This is a fundamental number because of the relationship between the unit circle and Pontrjagin duality (which has the Fourier transform and Fourier series as special cases), and is responsible for all appearances of $2 \pi$ in mathematics that I know of.

For example, the reason there is a factor of $\sqrt{2\pi}$ in the definition of the Gaussian distribution (which is where the factor of $\sqrt{2\pi}$ comes from in Stirling’s formula) is that the Gaussian distribution is its own Fourier transform. This factor is commonly cited as an application of $\pi$ that has nothing to do with circles, but of course the Fourier transform has everything to do with circles.

Edit, 3/15/11: Vi Hart also explains the wrongness of $\pi$ in video form. I have to admit I think I read the title of her post and then promptly forgot I had done so when writing this post.

### 13 Responses

1. on October 10, 2013 at 1:00 am | Reply Scott

(okay, I’m a couple years late here) $2 \pi i$ has the drawback of being non-canonical, because it requires a choice of square root of minus one. Terry Tao’s claim that it is *the* generator of log(1) is unfortunate, since there are two generators. The more canonical object is $\mathbb{Z}(1)$ itself, i.e., the kernel of the exponential.

If you’re wondering just how far one can push mathematics in the complex world without choosing a square root of minus one, you should try to locate a copy of Brian Conrad’s unfinished book on the Ramanujan conjecture.

2. […] Pi is still wrong via Annoying Precision […]

3. on April 18, 2011 at 8:20 am | Reply George Jelliss

I favour pi/4 which I denote by kappa:
http://www.mayhematics.com/n/constants.htm

4. […] us A Ramanujan series for calculating pi, 360 has The Difference and Qiaochu Yuan counters with Pi is still wrong.  Finally, madkane brings us a Pi day […]

5. on April 9, 2011 at 1:30 pm | Reply Dai Yang

Can’t we compromise and celebrate both? There’s no reason to stop celebrating pi day given all the history behind it.

The more important reason is that my birthday is pi day =p

6. […] Qiaochu Yuan: Pi is still wrong […]

7. on March 23, 2011 at 4:39 am | Reply Mark

Qiaochu Yuan I really like your blog but I want to know which of your posts (you think) make the most interesting connections between (different) mathematical objects. Personally, I really like the one where you construct groups from category theory.

8. on March 17, 2011 at 3:32 am | Reply Mark

i was going to object with e^(i*pi) +1 = 0, but then i realised: “that thing you just said”.

9. on March 15, 2011 at 5:49 am | Reply Roger Witte

See ViHart’s blog at http://vihart.com/blog/pi-is-still-wrong/ for an amusing exposition of this

• on March 15, 2011 at 6:27 am | Reply Qiaochu Yuan

Thanks for the heads up! I think I saw this yesterday and then forgot that I had seen it.

10. on March 15, 2011 at 2:46 am | Reply Sputnik

2\pi = 6.28318…

By that reasoning Tau Day should be the 28th of June. Let’s make it happen! (or if you live in Europe and are logical about it, like me, it should be the 6th day of the 28th month).

11. […] It’s pi day, and there’s been yammerings that “pi is wrong”, e.g. tau day and Qiaochu Yuan’s statement. […]

12. […] don’t we celebrate 2 Pi i day […]