# Pi or Tau? How should the circle constant be taught?

Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?

How can both be most effectively taught in order to maximize student's understanding of the use of the circle constant?

• "Tau has more merits in its application." What makes you think this? The question assumes that teaching both is advisable. I strongly disagree. Mar 19 '14 at 18:37
• I have read a quantity of arguments for $\tau$ over the years and all of those boil down to "It makes the formulae look nicer". That's a lousy argument and no where near sufficient to justify teaching $\tau$ in place of $\pi$. Mar 20 '14 at 9:18
• @AndrewStacey: The other argument is that with $\tau$ students don't have to muck around with as many factors of $2$, this can lower the cognitive load and make learning the mathematics easier. One rotation equals $\tau$ radians makes things a lot nicer to think about, not just to look at. Mar 27 '14 at 11:09
• I find it most confusing that $\tau$ looks like half of the $\pi$ symbol; $\tau\tau$ etc. This makes me think $2\tau = \pi$ or maybe even $\tau^2 = \pi$. Apr 16 '14 at 9:16
• I am slightly surprised that this debate is so vigorous, while other arguably more unfortunate conventions stay undisputed. No one seems to question the prefix notation for functions, while $f(x)$ is confusing when composing functions: in $f\circ g(x)$ one applies first $g$ then $f$. This is why the matrices of permutations, the composition and diagram, many group actions look wrong for having extra inverses. If we used a postfix notation, say $(x)f$, then the composition of $g$ then $f$ would be written $(x)f\circ g$, we would praise right actions of groups, and the sun would shine again. Mar 18 '15 at 12:32

(Disclaimer: I peronally really don't care if one uses $\tau$ or $\pi$, both are just numbers for me.)

But I would strongly recommend to use $\pi$. Why?

In every technical literature, in many popular literature, the people always use $\pi$ (even worse: $\tau$ is used for different things than $\tau=2\pi$ which would confuse when reading those literature). If $\pi$ is not taught, the students would not be able to read older literature. Plus, it is confusing when they wanted to talk about it with parents, frieds, etc. In terms of consistency, everyone should know $\pi$. However, if you want you can also talk about $\tau$, but due to my experience students will mostly react like "And now? It's just the double value of $\pi$?!".

A similar thing is there in physics, where the direction of electric currents was defined in an other way than the electron flow. Due to old technical literature and technical reports, manuals, etc. the definition is still valid (and will be for the future), compare https://physics.stackexchange.com/questions/17109/why-is-the-charge-naming-convention-wrong

• I know it isn't being seriously suggested, but it's interesting that the most popular response here involves a futuristic dystopia where current generation of students, these future readers of older literature, would be rendered incapable of looking up or understanding the meaning of one symbol as a consequence of learning another one. Mar 10 '15 at 17:46
• Until I clicked on this thread and read the question, I thought (based on the title) this question was something about pi and the golden ratio. Maybe someone has already mentioned this, but $\tau$ is often used for the golden ratio. Mar 11 '15 at 18:51
• To be fair, it's not like $\pi$ isn't overloaded as well. For instance, in AI and machine learning $\pi$ is the typical name for a policy function. Apr 14 '15 at 8:48
• $\pi$ is also used for the prime-counting function, as well as homotopy groups. Sep 27 '15 at 3:40
• @DanielR.Collins Since we're at it, $\pi$ is used for permutations and projections and planes too. Nov 11 '15 at 15:27

I feel that it is perhaps a little irresponsible to teach $\tau$ instead of $\pi$. As a first introduction, it is the norm which should be taught: teaching a rare alternative to $\pi$ only serves to confuse students, especially when almost all available resources use $\pi$ instead of $\tau$. Imagine a student's confusion when they see $\tau$ in class and $\pi$ everywhere else.

I personally don't even know why the $\tau$ vs $\pi$ topic is even debatable; any miniscule advantage (if there even is an advantage) in using $\tau$ over $\pi$ is heavily offset by the work needed to establish an entirely new cultural norm.

• Imagine watching the last 30-40 years of intelligent/sciency comedy and not knowing what $\pi$ was... Mar 20 '14 at 9:54
• "the work needed to establish an entirely new cultural norm" -- it's been done before; e.g., decimalization in money, metric system, stock prices in 2000-2001, etc. Sep 27 '15 at 3:43

One should teach $\pi$. One might discuss that there is a choice that is made that is somewhat arbitrary, and there are also reasons for a different choice but I do not see this as that relevant to make much ado about this.

It should perhaps also be noticed that there are two conflicting proposals for $\tau$. Eagle (1958) proposed $\pi/2$ and Palais (2001) proposed $2 \pi$ (for detailed references see the Wikipedia page on Tau) and chance are there are others (at least others that proposed the same if not still other values). This further illustrates that the situation is not that clear, though "$\pi$ is wrong!" certainly makes interesting points.

Moreover, it is not quite clear when a choice differing from the mainstream one should happen in the teaching. To have somewhat good reasons to motivate a choice different than $\pi$ one needs more advanced mathematics than one typically has available when first needing a cricle constant.

Purely 'geometrically' the standard choice is at least as good, in my opinion. On the one hand, one can make an argument that area is the more straight forward notion than length, leading rather to $\pi$; second; if one goes with length to compare to the diameter does also no harm in that context and again has some merit, it might even be more natural.

• To me, the argument in favor of τ is the fact that we learn the formulas $x^2 + y^2 = r^2$, $A = \pi r^2$, and $C = 2\pi r$ instead of $x^2 + y^2 = d^2/4$, $A = \pi d^2 /4$, and $C = \pi d$. All mathematical convention regarding circles treats the radius as being more fundamental than the diameter, except for the choice of the circle constant itself. Apr 30 '14 at 2:17
• I do not follow this argument. Indeed, it could be used to argue that $A/r^2$, ie $\pi$, is the natural choice for "the circle constant." And, as said in my answer to consider area is more natural than to consider length, in my opinion, since it is the more stable and intuitive notion in that context.
– quid
Apr 30 '14 at 7:54
• Actually, saying that $A = \frac{1}{2} \tau r^2$ is preferable to $A = \pi r^2$, and this is one of the primary arguments for $\tau$ in terms of elementary geometry. The factor of 1/2 there comes from the fact that $\int x \, dx = \frac{1}{2}x^2 + c$, so obscuring this factor of 1/2 by cancelling it with the 2 from $C = 2 \pi r$ is undesirable. Indeed, if we must use $\pi$ it would be better to write $A = \frac{1}{2} (2 \pi) r^2$. Jan 11 '16 at 7:13

There is a simple solution to this debate:

• This should be a comment on the main question. Mar 21 '14 at 20:03
• @confutus, as serious as the whole $\tau$ proposal Mar 22 '14 at 12:23
• @vonbrand invalid proof by induction! Mar 25 '14 at 8:41
• As a comment, the xkcd would have gotten an upvote from me; it's amusing and includes a bit of interesting mathematical trivia in the hover text. But it hardly seems a useful answer to the question. Please see my answer to The Subjectivity Problem with Questions Appropriate to Mathematics Educators. Mar 25 '14 at 15:53
• Considering that the factor of 2 is multiplicative, I'd think $\sqrt{2}\pi$ is better. May 5 '14 at 23:42

# $\tau$ should be taught in schools

There's plenty of material arguing why $\tau$ is a much more intuitive and easier to teach concept (some of my favorites: 1,2,3) and I don't want to rehash their arguments, but if you think that $\tau$ is just to make equations look nicer, please check out those resources.

The question at hand is whether math educators should teach $\tau$. I will assume that $\tau$ is a more intuitive and didactically nicer concept than $\pi$ for the sake of argument - if you disagree, you can stop reading here.

# Many arguments against $\tau$ are appeals to tradition

There seems to be a lot of:

$\pi$ is what we've always used, it is what we currently use, so we should continue using it.

In fact, many of the arguments are generic enough to be able to apply to any proposed change. For example, imagine when the first few countries were switching to the metric system, I'm sure exactly the same arguments were used.

Here is part of EuYu's answer, but with $\tau$ and $\pi$ replaced with "the metric system of units" and "the British system", respectively:

I feel that it is perhaps a little irresponsible to teach the metric system of units instead of British units. As a first introduction, it is the norm which should be taught: teaching a rare alternative to the British units only serves to confuse students, especially when almost all available resources use British units instead of metric units. Imagine a student's confusion when they see meters in class and feet everywhere else.

# Thinking long-term

Some of the arguments for the status-quo have seemed a bit short-sighted. Many changes have a switching cost, but pay perpetual benefits after the switch is over, as in switching to metric units. I think we should be asking ourselves: "Do we really want to be using a didactally and intuively inferior circle constant for the next 500 years?" instead of "Do we want to have to rewrite our textbooks?"

• Yes, textbooks would need to be rewritten, but textbooks get updated all the time with new, modern notation. This is a good thing.
• Yes, "the students would not be able to read older literature" as Markus Klein points out in his answer. True, but when I read a very old journal paper, I already have a hard time. Again, it's because notation changes, usually for the better. I'm glad that people are no longer using the same exact math/physics notation that was used 100 years ago.

# Educators can make this change happen

Perhaps more people would agree to switch to $\tau$ if everyone were to do it at once - say, starting in 2015, all professors, all publishers, all researchers, etc. would switch to $\tau$ in one fell swoop. I'm down with that, but that's not going to happen if I've understood human nature correctly.

It has to start somewhere and educators are the gatekeepers of knowledge - educators teach the youngsters and write the textbooks. If the change is going to happen, it will happen thanks to them.

# My Personal experience of teaching with $\tau$

I'm the TA for a Math Methods for Physics course at an American univeristy, and I use $\tau$ in my discussions. I wasn't sure how the students would react, but being nimble-minded, young students, they caught on pretty quickly.

I also felt it wasn't a huge burden on any students who were opposed to $\tau$ - I reminded them that they could always just write down "$2\pi$" in their notes every time they see me write "$\tau$". And I reminded them that they were allowed to ask questions using $\pi$ if they preferred (although most of them just asked questions using $\tau$).

Why don't you give $\tau$ a try in your class? See if your students like it. You may find yourself pleasantly surprised with how easily your students become fluent in $\tau$.

• I disagree with the imperial/metric example because metric is orders of magnitude better (base 10 vs arbitrary steps to go to larger/smaller units of measurement), and pi vs tau is just a matter of a constant factor. Apr 16 '14 at 11:35
• +1 for you approach! I love that you actually try to move forward and make things easier. Apr 16 '14 at 17:42
• @Jasper, that's true, but also consider that the costs of switching a country from imperial to metric is orders of magnitude higher than switching from $\pi$ to $\tau$. I think the analogy is OK, just that it's at a larger scale. Apr 16 '14 at 18:07
• The imperial/metric analogy is stretched somewhat. The conversion factor is just 2 for pi/tau but is rather more complicated from imperial to metric (feet to metres, inches to centimetres, and so on). However, my disagreement with this answer is more fundamental. The third reference you link to is brilliant: it undermines the whole rationale for tau in the final paragraph of Section 5.1 where it concludes that "the fundamental constant uniting the geometry of n-spheres is the measure of a right angle". Now that I can agree with and if we defined tau=pi/2 then I'd happily lead the revolution. Apr 16 '14 at 18:46
• I don't think that Base ten is arbitrary at all (fingers), but at least the metric system has a single base and not arbitrary factors between different units (12 inch = 1 foot, 3 feet=1 yard). Apr 30 '14 at 12:33

I cannot imagine a context where it would benefit a student to know about $\tau$ over and above knowing about $\pi$. Every text book a student will ever encounter will exclusively talk about $\pi$. Every calculator a student will ever use will have the constant $\pi$ pre-programmed in.

Although there are perfectly lovely arguments to show that $\tau$ is a nicer constant, it is just $2\pi$, so it really doesn't matter.

At most I would introduce $\tau$ to higher ability students as an interesting aside, but I cannot see the value in actually teaching it as a useful constant.

I feel as though a lot of this misses the point. Teaching the unit circle and trig functions in terms of "one turn" makes a lot more intuitive sense to young students than do arbitrary variables tau or pi. It just so happens that Tau is equivalent to one turn. Once students grasp the concept of the unit circle, then the pi conventions can be explored.

After coming back to this question, it strikes me that in fact the best thing might be to explicitly teach both. Not to accommodate the advantages of $\tau$ with the entrenchment of $\pi$, but simply because it makes clearer that pi is a convention, has a definition, and is not just some magical formula given to us. It is a choice. The real thing that is given as a fact is that a circle twice as large is twice as long, after that we name one ratio or the other (perimeter over diameter or over radius), but it does not matter that much which one.

I once had to teach some trigonometry using complex numbers, and asked myself for one quarter of an hour how damn to prove that $e^{i\theta}$ equals $1$ precisely at $\theta=2\pi$, until I realized that in that setup it was nothing else than the definition of $\pi$. I guess many students also need to be reminded that once in a while, and teaching $\tau$ is probably a good opportunity.

I only became aware of τ a few weeks ago. It's clever, and in theory, might simplify the teaching a bit. If one could wave a wand and rewrite every text book, and change every pi key on a calculator to tau, I'm in. Even though it would be tough for me to rewire my brain away from the pi related equations I've come to memorize. Not to mention the mnemonic poem that helped me memorize the first 20 digits of pi.

Short of that, the question is one of transition. There will be a period of time one needs to learn both numbers as a way to bridge the gap. I am still getting over the trauma of losing an entire planet. I wonder when the last textbook calling Pluto a planet will be retired. How do you propose to replace every math text?

I feel like $\tau$ is very useful, but of course everybody uses $\pi$ in 'the real world'.

I wouldn't hurt to mention $\tau$ one of two times though, and show to students that if they are having trouble with visualizing the length of circle segments or find $\tau$ convenient for other reasons, they can always use $\tau$ 'internally', as long as they formulate their final answers in $\pi$ (this is a trivial conversion and should not cause confusion to any student).

Or you could even allow answers formulated in $\tau$, but I don't know if it's a good idea to let students get used to $\tau$. Maybe I'm a little too sceptical: it may be very easy for students to use both $\tau$ and $\pi$ without getting confused, but I wouldn't risk it.