Is π Wrong?

Several years ago, I came across an interesting article [pdf] by Bob Palais. The basic premise of the article is that the numerical constant π is wrong. Unlike many mathematical cranks, Palais is not arguing that the computed value of π is incorrect. Instead, his thesis is that π is defined in a way that is unnatural and unhelpful. Given that the idea of replacing π with another constant seems to be getting some traction (notably in Michael Hartl’s Tau Manifesto), I thought I might offer my own two cents.

First, let’s examine Palais’ argument. You probably remember that π is defined as the ratio between a circle’s circumference (the distance around the edge of a circle) and its diameter (the distance from one side of a circle to the other, passing through the center). That is,

\(\pi = C/D \approx 3.14\).

Circles, and circle-like objects, pop up in a lot of places in mathematics. This means that π appears in many places. In trigonometry, angles can be measured in radians, where 2π radians equal 360° (one full turn). The frequency of trigonometric functions is also related to π:

\(\sin(x+2\pi) = \sin(x)\).

In statistics, the normal distribution is given by

\({1\over\sqrt{2\pi\sigma^2}}{\rm exp}\left({-{(x-\mu)^2\over 2\sigma^2}}\right)\).

In calculus, integrals in polar space often have the form

\(\int_0^{2\pi}\int_0^\infty f(r,\theta)r\, dr\, d\theta\).

In each of these examples, as well as countless others from across all branches of mathematics, π is often preceded by the number 2. Like “q” and “u” in English, π and 2 are often seen together. Unfortunately, this means that almost any formula or equation involving π is also going to involve an extraneous 2. This means that there is one more thing to forget or mistype, adding an extra level of difficulty for students who are just being introduced to a new concept.

The reason for this, Palais argues, is that π is the wrong constant. Instead, we should be using a constant whose value is defined as the ratio between the circumference of a circle, and its radius (rather than its diameter). Hartl calls this constant τ (tau), and defines it as

\(\tau = C/r \approx 6.28\).

Note that \(\tau = 2\pi\). This means that where ever we see 2π, we are entitled to replace it with τ. It is fairly obvious that this would simplify any of the formulæ above, but it also has the advantage of simplifying many other operations.

For instance, consider the measurement of angles in radians. We are taught that there are 2π radians in a circle. A full turn (a 360° turn) is 2π radians, and a quarter turn (a right angle, or a 90° turn) is π/2 radians. This is a little clumsy: a full turn is two of something, and a 1/4 turn is actually 1/2 of something. If we use τ instead, we can define a full turn as τ radians, and a quarter turn as τ/4 radians (that is, a quarter turn is one quarter of τ).

From a pedagogical standpoint, this is huge. There is a direct relationship between τ and rotations on a circle. There are no extraneous twos running around, and students don’t have to worry about half a turn being a whole constant and a whole turn being two constants. This makes the relationship between radians and angles obvious, and removes a major source of error.

Unfortunately, π is almost certainly here to stay. While τ might be the simpler constant, mathematicians are human, and humans are generally resistant to change. The concept of π is almost as old as mathematics itself—the ancient Greek mathematician Archimedes used it over 2,000 years ago—which makes it even harder to displace. We may never see a day when τ displaces π. None the less, I am convinced that τ really is the more fundamental constant, and that π is wrong.

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