Is Tau Better Than Pi? Irrational Arguments

Is Tau Better Than Pi? Irrational Arguments

Happy Tau Day, the most exciting math holiday you’ve yet to discover! Today, June 28, is 6/28, which contains in order the first three digits of tau (τ), the rival of math’s most popular irrational number, pi (π). In 2001, Bob Palais wrote an article for the Mathematical Investigator called ,“π is wrong!” In it, he insists that the choice of using π in our mathematical formulas for hundreds of years is no good. He argues that the use of τ would simplify many formulas and that its derivation is much more intuitive. (Notice that the symbol resembles that for pi, but with one "leg" instead of two.)

The significance of our beloved irrational number π is that it is equal to the ratio of the circumference of any circle to its diameter--in notation, π = C/d. However, the most defining characteristic of a circle is not its diameter but its radius. A circle is defined as the collection of points on a plane that are exactly the same distance, its radius, from a point, its center. Palais argues that intuition should direct us to the use of a more elegant Circle Constant, tau, where τ is the ratio of the circumference of a circle to its radius--in notation, τ = C/r.

Self-described “notorious mathematical propagandist” Michael Hartl takes the argument even further in his now-famous “The Tau Manifesto,” which he published on Tau Day of 2010, exactly one year ago. He demonstrates with many adapted formulas that the factor of 2 is unnecessary if we incorporate it into the ratio itself. For instance, the periods of basic trigonometric functions f(x) = sin(x), and f(x) = cos(x), are in both cases 2π. Why not change them to tau instead? Palais and Hartl each list numerous other examples from calculus and physics, in which the factor of 2 is rendered obsolete by replacing 2π with τ.

The really intuitive part is revealed if you think of angle measure. How things are done now with π, a half turn of the circle is π radians, and a full turn is 2π radians. Should we adopt τ instead, τ radians would be a full turn, τ/2 radians a half turn, τ/4 radians a quarter turn, and so on. There are, of course, instances where π appears un-doubled. For instance, the formula for area of a circle: A = πr². Hartl shows, in a mathematically sophisticated way, that the replacement of π by τ even in this instance is the more sound choice, since it is analogous to similar formulas in physics. An article in today’s BBC News paints the issue as a violent conflict, with pi detractors up in arms over a lifetime of educational betrayal, which seems to this mathematician something of a manufactured controversy. (I can imagine you'd be upset if you are the sort of mathematician that has memorized pi to the nth digit. If you are one of these folks, here's the start for your new parlor trick: reciting tau, 6.283185307...)

Is it worthwhile to switch to tau use, and abandon pi? Unless your school has the budget to replace all of its textbooks, calculators, posters, etc., Hartl’s most convincing argument is that “we can embrace the situation as a teaching opportunity: the idea that π might be wrong is interesting, and students can engage with the material by converting the equations in their textbooks to see for themselves which choice is better.” Certainly π needs to be treated as a ratio, something that implies a relationship, as opposed to a discrete number. Engagement with this controversy will allow students to develop and question ideas about the significance of π, and exploration of its uses will reveal truths about mathematical relationships. This makes the debate worth mentioning in your classroom.

But what about all the jokes, the T-shirts, the tattoos that pi has inspired? I, for one, will not miss the pie puns one bit[e], but Hartl’s article concedes the word playfulness of pi is one of its assets, and he offers some initial tau jokes to entice skeptics. The inevitable conclusion: tau is twice as great as pi.