5 Days to Tau Day 2011: why use radians?

23 June 2011

Ah, friends, Tau Day draws ever closer.

Last time I talked a little about why some conventions are better .. righter .. than others, in a sense more meaningful than mere personal preference.

Circle constants are not just found in geometry.  They are ubiquitous in mathematics.  Everywhere you find yourself tripping over 3.1415somethings and 6.2831and-so-ons, in applied and theoretical stuff.  You probably call those numbers \pi and 2\pi, and I’d rather call them \tau/2 and \tau. The Tau Day claim is that the latter choice is better than the former.

There are lots of ways to make the case for the primacy of \tau (and I’ll mention more as it gets closer), but the most concise is this: There are \tau radians in a full circle.

This begs a question, of course.  How many angle-units are in a full circle depends on what units we use.  There are 360 degrees, or 4 right-angleses (much early Euclid-style geometric writing measured angles relative to the right angle), or 400 gradians, or 4 million myriogrades, if that’s your thing.  So we’ve traded “Why \tau?” for “Why radians?”.

But that’s a relatively easy question, and certainly a well-known one.  There are lots of reasons why radians are the most natural/desirable, and lots of places to read about them.  Even the wikipedia article does a decent job.  So let me just give you a few concise points.

  1. The radian measure of an angle can be defined intrinsically as follows.  For any angle, draw a circle centered at the vertex, and compute the ratio of the arc length inside the angle to the radius of the circle.  (Hence the term radian ; the angle measure in radians tells you how many multiples of the radius are in the arc length contained in the angle.)  It seems like this might depend on how big you draw the circle, but a moment’s thought shows that it doesn’t, and you end up with a nonarbitrary angle measure.
  2. The calculus-related behavior of the trigonometric functions is best when you work in radians.  For example, if you work in radians, the derivative of \cos x is \sin x and the derivative of \sin x is -\cos x.  If you don’t work in radians, you get ugly correction factors.
  3. (This is actually very closely related to the previous.)  If we work in radians, sine and cosine have the simplest possible power series, and it is easy to see (at least formally) the celebrated Euler’s formula e^{ix} = \cos x + i\sin x, where that e is the base of the exponential function, the one we met yesterday.
  4. If we use radians, then the best first-order approximation to \sin x is, if x is small enough, simply x itself.  (That is, \sin x\approx x if x\approx 0.)  For any other choice of units, there’s a correction factor.

A parting math-Zen koan, especially for those of you who haven’t thought about this much.  I have asserted (and do deeply believe) that there is a canonical, nonarbitrary right unit for measuring angles.  What is the analogous best unit for measuring length?