The teaching argument (which by the way this article did not address) is for me the most powerful one. All the other arguments look secondary to me.

> That said, I don't think it's worth it to make the switch because of all the hassle.

It would be no hassle for pupils. I bet that it would be easier and faster to teach Tau first, then mention that Pi is half Tau. In my opinion, teaching should have the priority. I don't really care if the rest of the world use Pi, but I'll teach Tau first.

I agree with teaching Tau first. Personally, it took me at least a week to begin to understand radians, and almost a month to understand it pretty well. Tau is more intuitive than Pi in this case.

It only took me about a day, but I suspect that's because I was already used to cycles at the time (1 cycle=2pi radians). Getting used to cycles took me about a week.

To my mind, the problem is not with pi; the problem is with degrees. Everyone learns about degrees first, and then must "un-learn" these artificial numbers and begin thinking in fractions of a circle (with an extra constant thrown in there one way or the other, in the case of radians). If we began labeling globes, protractors, and the like in radians (or fractional cycles), this problem would go away.

"the problem is with degrees"

I partly disagree. For kids, defining something using irrational numbers would probably be very confusing. Using integers is much easier.

So, why 360? Because we talk about right angles a lot and we want to have a third, a half, etc... and (I'm guessing) they wanted it to be a multiple of 10.

I think the point is to be able to teach geometry to kids without worrying about them getting confused by fractions and/or irrational numbers.

So just use cycles instead of degrees. Simple fractions or a circle. Fractions are already taught to young children as "how many pieces a circle[1] can be divided into." It would seem to kill two birds with one stone.

I think most kids who even study pre-Calc could handle applying a conversion of "2pi" from there.

[1] Where circle="cake","pie","pizza",etc.

Good point. I forgot we learned about "slices of a pizza" that early.

By the way, the history of the degree is somewhat interesting:

http://en.wikipedia.org/wiki/Degree_(angle)

I don't think in radians, I think in cycles (radians/tau). I used to think in degrees, and could never get the hang of radians. Unfortunately, I cannot imagine the mathematical world moving to cycles, ever. The most I can hope for is that teachers introduce others to a single intermediary constant (tau) rather than two (2,pi).

The problem with using cycles as the primary unit of angle is that the wonderful trigonometric derivative symmetry only occurs when the functions are calibrated for radians.

I've never understood the particular argument of "single intermediary constant (tau) rather than two (2,pi)". 2pi is one constant that contains multiple glyphs, just as 1/2 is one constant, just as tau/2 would be one constant. That it is derived logically from other constants does not make it two separate constants. Even multi-digit numbers are derived logically from their component glyphs (12 = 10+2).

The two-glyph thing is a matter of description length and parametrization. 10+2 is an expression, whereas 10 is also a constant that is the base of our number system. Thus, 10 carries more meaning than 12, even though both are constant - 10 is potentially the aforementioned parameter, but what is the 2? With sufficient study, 12 becomes a number of important constants as well (such as the number of inches in a foot, or the integral of a centered quadratic function), but not ones considered fundamental.

Essentially, it's about the difference between a concept and a measurement, or the difference between (x+y)/x and 1+x/y. 2pi is an expression, pi is the concept.

I've read this a few times, and I don't think I really understand. Perhaps you could explain further?

10+2 is an expression, whereas 10 is also a constant that is the base of our number system.

There is no notion of an "expression" as distinct from a "number" (or "function" if it involves a variable) in any branch of math apart from computer science[1]. In algebraic terms, (12) and (10+2) and (6x2) and (0xC) and (2^4-2) and "twelve" are all literally the same thing. Well, technically they are all equivalent notations for the same abstract concept.

Thus, 10 carries more meaning than 12

Even if I accept this (which I'm not convinced I do), it's beside the point: 10 and 12 are not equal. Unlike with pi and a hypothetical tau, using one where the other is called for would be an error.

[1]There is the notion of the limit, which is subtly different: limits do care how a function behaves at other points. One could make the case that this makes a limit into a sort of expression, but to be honest I think that only obscures the idea.

Okay, I'll try to grasp at my thoughts again.

Algebraically, 12 is not the same as 10+2. 12 is an element of, say, ℤ, while 10+2 is one of <+,ℤ²>. To make them interchangeable, we need to establish an equivalence relation. Given that relation, we then have the opportunity to express useful, non-obvious equivalences using transitivity.

Now that we have X=10+2=12, we need to choose which one represent the equivalence class of X. 12 is certainly shorter, but a seemingly magic constant. 10+2 implies that in other number systems, X=b+1+1 may be also true. If the scribe subscribes to the principle of MDL, we can speculate that this is the reason he chose the longer version, and if that is accurate, we have gained more information. If we chose 9+3, we would arguably lose information, since this expression is (hypothetically) misleading.

This is all to say that expressions are more informative than their equivalence classes, since they have been hand-picked to be representative.

To represent the equivalence class of 6.28... with 2*3.14... implies that the equivalence class of 3.14... is more important, and that the prototype likely involves two separate instances of the concept of π. This is misleading.