## Tau, not Pi

It bothers me that I found this post so convincing (via Division by Zero). Now I feel like I should do something about it.

Basically, the constant $\pi$ normally associated with circles is not the constant we should be using; rather, we should be using a constant $\tau=2\pi$. This makes sense—we should use the circumference of an entire unit circle rather than half of it. The linked article explains many ways that $\tau$ simplifies formulas.

Do I make the switch in my upcoming calculus class? I need to think about it.

I would appreciate people’s arguments in favor of continuing to use $\pi$ (aside from the fact that everyone knows and uses it already) in the comments.

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### 22 Responses to “Tau, not Pi”

1. Joboo Says:

besides the circumference formula, really where would tau be more of use than pi? … the area of a circle would be (tau x r^2) /2 … I think it is a parallel concept to just use d instead of r in circle formulas … instead of using something equal to 2 x pi, use the 2 with the radius to make diameter (2r = d) in most equations PS – I couldn’t open the linked article, so maybe I am missing out on the argument

• bretbenesh Says:

Hi Joboo,

The article listed several examples (try this: http://tauday.com/). The two main ones were:

1. You would evaluate integrals that look like $\int_0^{\tau} \sin(x) \ dx$ instead of $\int_0^{2\pi} \sin(x) \ dx$.
2. Radians would be nicer. For instance, an angle that goes $\frac{3}{4}$ of the way around a circle would be $\frac{3}{4}\tau$ radians (very natural) instead of $\frac{3}{2}\pi$ (weird) radians.

There are others ( $e^{\tau i}=1$, whereas $e^{\pi i} = -1$), but it seems like it would simplify trigonometry. As far as the area formula—see my response to Alex’s comment.

2. Alex Says:

So circumference would be \tau r, area would be \tau r^2 / 2??? Not sure that is simpler than \pi r^2. I’m gonna keep using pi, but you do what ever you want to those young impressionable students. Soon we may come to find there is a movie titled “tau”. Obviously it would be twice as good as “pi”.

• bretbenesh Says:

it is true that the area formula would have an extra factor of $\frac{1}{2}$ in it. However, it should have a factor of $\frac{1}{2}$ in it.

Since Joboo could not read the article, I will assume that you could not, either. Basically, many quadratics that naturally occur have a factor of $\frac{1}{2}$ in them—it pops up naturally when integrating a linear variable. The examples that Hartl lists are:

1. $\frac{1}{2}gt^2$ is the distance an object falls in $t$ seconds
2. $\frac{1}{2}kx^2$ is the energy in a spring (by Hooke’s Law)
3. $\frac{1}{2}mv^2$ is the kinetic energy of an object

So it is not unreasonable for there to be a factor of $\frac{1}{2}$. If you think there is, then we should start taking, say, “ $2$-seconds” as the standard unit of time so that an object will fall $gt^2$ meters after $t$ $2$-seconds.”

3. Aaron Says:

Even though we’re not supposed to use this argument, I don’t think you can disregard the convention of π. If you want to disregard conventions because an alternative “makes more sense,” you’ll end up changing the way you spell many/most words in English (e.g. http://www.nytimes.com/2010/06/27/magazine/27FOB-onlanguage-t.html) and having your students use Reverse Polish Notation.

• bretbenesh Says:

True—I rarely speak Esperanto. And I am unlikely to start incorporating $\tau$ (although I will entertain the notion). At the same time, I do not think that we should blindly follow tradition just because it is tradition.

Actually, I think that $\tau$ might be really good for certain, targeted students. In particular, students who have had difficulty in mathematics before and are not likely to continue with math could benefit a lot; trigonometry would be much easier to understand if everything was in terms of a fraction of the entire distance around a circle (rather than just half).

• Aaron Says:

I’m not advocating for blindly following anything for any reason. In particular, I think it is dangerous to blindly follow our “intuition.” Instead I think we should follow conventions with wide-open eyes.

It is important to collect *evidence* that using tau would make trigonometry “much easier to understand.” If there is indeed evidence that this is the case, then it would be wise for us to slowly, deliberately, and systematically shift to using tau.

• bretbenesh Says:

Here, here. I am all for evidence. However, this starts with one class, I think.

4. Joboo Says:

I get it … like you said in the beginning, the constant regarding a circle should be based on the circumference for visualization … I like it.

I used pi often … I bet I am one of the only non-academics that uses pi on a daily basis … you are right, we should not let 1/2 scare us in equations.

Hope you are doing well Brett …

-Joboo

5. Matthew Leingang Says:

This reminds me of a discussion I had with a colleague who said “the fundamental unit of trigonometry is not the radian, it’s the revolution.”

6. bretbenesh Says:

This makes complete sense to me. However, there is the thing about “common language” that holds me back. This is the same reason why I still spell “cough” the way I do.
Bret

7. 2010 in review « Solvable by Radicals Says:

[…] Tau, not Pi June 201011 comments 4 […]

8. Pradeep Tiawari Says:

please can any one tell me what is the area of circle?
it’s pi*r*r or tau*r*r if second one is correct give reason..

• bretbenesh Says:

The area of a circle is still pi*r*r. With tau, you would get a different formula. Note that tau=2*pi, so pi=(1/2)*tau. Then the area of a circle is:

pi*r*r=((1/2)*tau)*r*r = (1/2)*tau*r*r

So the formula becomes slightly messier (because of the 1/2), but still correct.
Bret

9. jon Says:

no offense, but who cares?

• bretbenesh Says:

Hi Jon,

No offense taken. I will give you three reasons:

1. No one cares. We aren’t actually going to change how we think of $\pi$, so this is more of an amusing thought than anything that we should actually care about.

2. We all might care. Mathematicians might actually change how we think of $\pi$, since mathematicians eventually embrace superior ideas (even thought it may take a long time). We started with the Cauchy integral, then changed to the Riemann integral, then changed to the Lebesgue integral, and will probably eventually move to the Henstock–Kurzweil integral. We always allow for definitions to be improved upon, even if the definition is as old as $\pi$.

3. We all might care. Trigonometry is a very difficult subject for many students. If removing one layer of fractions from the process allows more students to succeed, why wouldn’t we do it? In a world where mathematical skills are so important, why would we risk filtering out students by a stupid factor of $\frac{1}{2}$?

Notice that the last two were “‘might’ care.” We are merely evaluating the idea right now, a process which might take 100 years. I doubt that we will change, but it is worth discussing—at least for one blog post.
Bret

10. Bill Fahle Says:

Certainly there are valid reasons for both. From a pedagogical point of view, you can teach both and likely get more class participation that you normally would. That’s reason enough to teach it right there.

• bretbenesh Says:

Hi Bill,

I hadn’t thought of that. That would be a great classroom controversy! Students could debate.

Thanks for the suggestion!
Bret

11. Pinky Says:

When I first heard of using tau over pi, I did some digging into the different arguments. I think the issue is that one works best for radians the other for angles, or something similar. Each side focuses on what works best for their symbol and ignores the other, leading to a never-ending struggle of apples and oranges.

• bretbenesh Says:

I actually think that both are mostly related to measuring angles, and the debate essentially comes down to how to define “radians.” So I think that it is not comparing apples to oranges, but it is seemingly similar to the (not-so-raging) debate in America about whether we should measure distances in feet or meters.

Both have advantages. Meters make converting to other units (centimeters, millimeters, etc.) really easy, and the rest of the world uses it (this is kind of like Tau. It seems—to me, anyway— that it is more natural to measure angles in terms of “how far around the _whole_ circle have you gone” versus “how far around _half of_ the circle have you gone?”).

Similarly, feet has the advantage that we have been using it for a long time (like Pi), and. . .um. . .maybe that it is it.

Just as Tau is twice Pi, meters are just (roughly) 3.3 times the number of feet. So we are really just talking about scaling our units. This whole debate (it is not really a debate—I don’t know anyone who really thinks that Tau is going to win) is simply about how you want to deal with a factor of 2 (do you want to include it in the units, or do you want to include it in the number of radians).

• Bill Says:

Since you seem to be following this thread over the years, and I happened back on it through a different wordpress site, I will add a justification for English measures such as feet over metric. Metric measures are base-10 based, whereas English measures often involve halves, fourths, eighths, etc. The value 0.1 cannot be precisely represented in floating point binary in a computer, whereas 1/8, 1/16, etc., can be.

• bretbenesh Says:

Good point—the British were 100 years ahead of their time!