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 normally associated with circles is *not* the constant we should be using; rather, we should be using a constant . 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 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 (aside from the fact that everyone knows and uses it already) in the comments.

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June 29, 2010 at 4:47 am |

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

June 29, 2010 at 5:01 pm |

Hi Joboo,

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

There are others (, whereas ), but it seems like it would simplify trigonometry. As far as the area formula—see my response to Alex’s comment.

June 29, 2010 at 6:16 am |

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”.

June 29, 2010 at 5:12 pm |

it is true that the area formula would have an extra factor of in it. However, it

shouldhave a factor of 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 in them—it pops up naturally when integrating a linear variable. The examples that Hartl lists are:

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

June 29, 2010 at 12:46 pm |

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.

June 29, 2010 at 4:54 pm |

True—I rarely speak Esperanto. And I am unlikely to start incorporating (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 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).

June 30, 2010 at 1:47 pm

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.

June 30, 2010 at 2:39 pm

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

June 30, 2010 at 2:57 am |

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

August 10, 2010 at 3:34 pm |

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.”

August 11, 2010 at 12:41 am |

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

January 2, 2011 at 2:48 pm |

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

July 22, 2011 at 1:30 pm |

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..

July 25, 2011 at 3:09 pm |

Hi Pradeep,

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

December 16, 2011 at 6:35 am |

no offense, but who cares?

December 16, 2011 at 6:44 pm |

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 , 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 , 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 .

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 ?

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

May 16, 2012 at 5:38 pm |

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.

May 16, 2012 at 5:46 pm |

Hi Bill,

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

Thanks for the suggestion!

Bret

December 31, 2013 at 7:10 pm |

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.

January 2, 2014 at 2:30 am |

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).