Gracious Living

December 11, 2010, 19:32
Filed under: Math, personal | Tags: , , ,

I’ve been really busy with finals and haven’t had time to finish a proper post, but wow is this amazing.  The suggestion (not new, but very eloquently expressed) is to stop using \pi, the ratio of a circle’s circumference to its diameter, and start using \tau=2\pi, the ratio of its circumference to its radius.  (Spelled “tau”, rhymes with “cow”: do you know your Greek alphabet?)  Pretty much every important use of \pi is actually a use of 2\pi: the circumference formula, integration in polar coordinates or around a circle, finding roots of unity, but also Gaussian/normal distribution stuff, Fourier transforms, and zeta function identities.  Where there are exceptions, the insistence on using \pi instead of 2\pi obscures the nature of the equation: writing the area of a circle as \frac{1}{2}\tau r^2 instead of \pi r^2 highlights the fact that it’s the output of an integral, and though e^{i\pi}+1=0 looks beautiful, it obscures its own meaning: complex exponentiation by \pi corresponds to a rotation by 180 degrees or \pi radians.  If you write e^{i\tau}=1 (or, as the author suggests, e^{i\tau}=1+0 to preserve the “beauty”), you immediately see what the formula’s getting at: x\mapsto e^{ix} is a periodic function with period \tau corresponding to a rotation of the complex plane.

And if that weren’t enough, look at this piece of beauty:

A diagram of important angles on the circle, measured in multiples of tau

See how easy things could be?

Although \pi has been around for a long time, changes in terminology do happen and math moves right along. I don’t know what we would do if we still had to do group theory with Galois’ original wordings of “substitutions” and “arrangements.” So I think that among mathematicians, this could catch on pretty easily.  It’s easy to spread the meme, too — when you’re presenting something, just start out with “let \tau=2\pi” and watch the daylight glimmer on people’s faces.

But I’m not so sure that this could catch on in schools, which is where it is most needed.  I’m a math major and I still take a few seconds to remember those godforsaken radian angle measures.  It’s annoying and breaks one’s train of thought.  Why do we expect school-age kids to have to jump through this hoop every time they do a trig calculation, which is pretty much all they’re graded on? (another stupid thing about education, but I digress.)  All it teaches them is that math is about memorization of arbitrary things.  With \tau, there’s nothing to memorize: one-third of the circle is \tau/3.

I just feel that there’s considerable inertia in changing math curricula, especially over something as basic as this.  The only reason we learn math at all is because Eisenhower and Kennedy decided to put a man on the moon.  And because Egypt was troubled by the horrible asp.  Putting \tau into the curriculum would require schools to pay for new versions of these stupid things called “curriculum planners” and “textbooks.”  Quite frankly, that’s not what American schools need to be spending their money on, and it’s by no means even the biggest problem with math education.  The author of the page I linked suggests that teachers introduce \tau to teach kids critical thinking and show them that the best way to do something isn’t always the way you’re first taught.  This sounds awesome — an actually interesting debate in a math classroom?! — but it seems like a teacher that did that would meet interference from higher up, on the grounds that it stifles kids’ ability to communicate effectively (all math education being about communication, to people as diverse as cashiers and accountants).

On the other hand, the textbook-publishing lobby would love an excuse to publish a new edition.  In addition, if \tau becomes accepted in the math community, it’s only a matter of time before education starts changing as well.  Especially if the curriculum started to change so that it had more to do with actual math… wouldn’t that be a treat.

3 Comments so far
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I am a historian trying to write a book on women scientists to inspire High School graduates to take up science as a career. I have two great mathematicians to write about and have got stuck with real numbers and Dedekind cuts. How do I simplify these things for fifteen-year-olds to inspire them to take up math? A tough job I am facing. Anyone willing to help?

Comment by C S Lakshmi

This was like a year ago and presumably you’ve already found an answer, but this was an interesting enough comment that I felt like I should respond to it anyway. In terms of female mathematicians, you really can’t do better than Emmy Noether. She was a German professor in the early 20th century who worked at Goettingen with Hilbert and Klein. For much of her early career, she was denied a title, and even pay, because she was a woman. One of the foundational principles of modern physics is Noether’s theorem, which says that physical conservation laws correspond to smooth symmetries of the space in which they occur — thus, both trivialities such as the fact that physics works the same way no matter your position in space or time and more interesting statements like the conservation of energy and momentum occur as results of the quite general setup of a Lie group action on a manifold! Together with David Hilbert, she also more or less created abstract algebra, a revolutionary idea that (in my opinion) defined the course of mathematics for the next century. Though it now seems commonplace, at the time the idea of studying ideals in abstract rings in order to derive results about the concrete places where they appear, such as number theory and algebraic geometry, was unheard of, to say nothing of converting ‘concrete’ results to ‘abstract’ ones and then back into ‘concrete’ ones in other parts of mathematics. The Noetherian condition on a ring, which she invented and studied, is a technical and unintuitive condition that turns out to be precisely the right way of saying a ring is small enough to be usable (usually allowing you to work with finite sets of elements at a time, for example). I believe she’s also responsible for the ‘modern’ point of view of representation theory, treating group representations as modules over the group ring.

In response to your actual question, I’d actually guess that the construction of the real numbers might not be the best thing to talk about with 15-year-olds. When I started this blog, I might have felt differently, but it’s since become clear to me that a lot of things I felt like blogging about then may not have been truly interesting enough to be worth the effort. The things that get people interested in math are not its technical constructions, tedious fact-checking, or eternal search for ‘correct’ definitions — these all have their place, but appreciating them requires to some extent thinking like a mathematician. Instead, people are captivated by its surprises (every polynomial over the complex numbers has a root) and its metaphors (the complex numbers behave like a plane, which can be translated by addition and dilated and rotated by multiplication).

I’d advise instead showing your fifteen-year-olds Cantor’s proof that the real numbers are uncountable. This is a beautiful argument with a lot going for it: they learn that we can talk about infinity rigorously, but that it doesn’t work the way we might expect it to; that the very way things are written down can have surprising consequences; that sometimes we have to put things in weird places or orderings to study them correctly; and that the real numbers are a lot more complicated than we might give them credit for. It’s also a good example of a proof by contradiction (explain carefully that the proof doesn’t give you a new real number to add to the list, but that it shows that no such list can exist), and of a non-constructive proof (it shows that irrational numbers exist without actually constructing any; as a bonus, if you show that the algebraic numbers are countable, the same argument shows that transcendental numbers exist, but actually finding them is much harder than finding algebraic irrational numbers). And as some additional cheap entertainment, you can throw in the gory details of Cantor’s life, how he was blackballed by his own advisor for his controversial work, went in and out of insane asylums, etc.

Your book sounds interesting and I’d love to read it when it comes out (has it already?). I don’t know about the other sciences, but math still seems to have a hard time attracting women, which is a true disservice to both math and the women who could be doing it. I’d be curious to hear if you have any ideas on how to fix this problem.

Comment by Paul VanKoughnett

I really liked this. But I wouldn’t denote 2\pi with \tau, but with an encircled \pi, it is, just a \pi with a mark: it would be obvious that it’s a variation of the same concept. But which mark? A circle surrounding the \pi with put enphasis on the relation of \pi with circles. I’m tempted to not use \pi again.

Comment by weirdo

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