## Pi Day Post Mortem

Dessert aside, long-time readers are probably already aware of my decidedly mixed feelings towards Pi Day (see, for example, here). Nevertheless, the holiday seems only to be growing in popularity, and so I feel compelled to take it to task once again.

In my earlier post, I complained about mathematical mistakes that frequently appeared in Pi Day articles aimed at a general audience; these errors still exist, but rather than nitpick, let me instead focus on the most bothersome activity of the day. I’m speaking, of course, about π recitation competitions.

Reciting the digits of π is, unfortunately, becoming a popular activity - dare I say even a tradition - on Pi Day. Competitors recite as many digits of π as they can, and the person who can recite the most digits is declared the winner. As I’ve said before, I fail to see the point of this exercise. From a mathematical standpoint, students aren’t learning anything while memorizing digits of π (although learning memorization techniques may be useful in its own right). Arguably more significant, though, is the fact that there’s nothing special about the decimal expansion as a way of expressing π.

This kind of summarizes how I feel.

After all, what does the decimal expansion of π tell us? It tells us that if we split [0,10] into 10 evenly sized pieces, π will lie in [3,4]. Then, if we split [3,4] into 10 evenly sized pieces, π will lie in the second piece — in other words, π will be in [3.1, 3.2]. If we then split [3.1, 3.2] into 10 evenly sized pieces, we will see that π lies in [3.14, 3.15] and so on. In other words, the decimal expansion tells us that

π = 3 × 10^{0} + 1 × 10^{-1} + 4 × 10^{-2} + 1 × 10^{-3} + 5 × 10^{-4} …

More generally, we have

π = *a*_{0} × 10^{0} + *a*_{1} × 10^{-1} + *a*_{2} × 10^{-2} + *a*_{3} × 10^{-3} + *a*_{4} × 10^{-4} …

where *a _{k}* denotes the

*k*

^{th}decimal in the decimal expansion of π.

The point is that there’s nothing sacred about the number 10. One could just as easily consider the expansion of π with respect to some other base, and would then obtain a perfectly good infinite sequence of digits representing the number π. For example, suppose we want to write π as sums of powers of 2 rather than sums of powers of 10 — this will give us the binary expansion of π, rather than the decimal expansion, and each digit will be 0 or 1, rather than an integer from 0 to 9 (in general, the digits in a base of n will be the integers from 0 to n). The first few digits of the binary expansion of π are 11.001001000011111… - in other words

π = 1 × 2^{1} + 1 × 2^{0} + 0 × 2^{-1} + 0 × 2^{-2} + 1 × 2^{-3} …

For another example, consider the following video (sent to me courtesy of my friend Nate). In this video, each digit from 1 to 9 is assigned a note, and the first 31 digits of pi are then used to define a melody, which is then played on various musical instruments. Presumably only the first 31 digits are used because the 32nd digit is 0. But, there’s really nothing stopping you from assigning each digit from 0 to 9 a note, and then using as many digits as you want to define a melody.

Again, though, there’s nothing special about the decimal expansion. So, if one is going to use digits of pi to construct a melody, why not look at the base-8 expansion if one uses a major/minor scale, or the base-13 expansion if one wants to use a chromatic scale? Or, one could use the base-6 expansion and use the pentatonic scale. Of course, there is no limit here - one could also use the base-88 expansion and assign one digit for each key on a piano.

Using technology, any one of these expansions is easy to calculate - in fact, Wolfram Alpha will give you as many digits as you please:

Base 8: pi =3.1103755242102…

Base 13: pi =3 . 1 10 12 1 0 4 9 0 5 2 10 2 12…

Base 6: pi = 3.0503300514151…

Base 88: pi = 3 . 12 40 43 37 64 73 60 72 49 38 64 86 23… .

This expansions tell us, that, for instance,

π = 3 × 13^{0} + 1 × 13^{-1} + 10 × 13^{-2} + 12 × 13^{-3} + 1 × 13^{-4} …

3 + 1/13 + 10/13^{2} + 12/13^{3} + 1/13^{4} + … .

Analogous statements hold for the other expansions.

The point is that from a mathematical standpoint, none of these expansions should be viewed any differently from the decimal expansion, and each one will give you a different sequence of digits corresponding to π. What’s more, there are many other ways to represent numbers that don’t involve positional notation at all. For example, one could represent π using its continued fraction decomposition. The continued fraction of a number is constructed in the following way: if *x* is a real number, the first digit in the continued fraction decomposition is the greatest integer less than *x* (for example, the first digit in the continued fraction decomposition of π is 3). If the difference between *x* and this integer is 0, stop; otherwise, take the reciprocal of this difference and calculate the greatest integer less than this value — this will be the second digit in the continued fraction decomposition. Keep doing this and you will get a sequence of digits which terminates if and only if *x* is rational.

Let’s see what happens in the case of π. The first digit in the continued fraction decomposition is 3. Since π - 3 is nonzero, take the reciprocal of this difference: 1/(π - 3) ≈ 7.062513305… . The greatest integer less than this value is 7. Subtracting this difference and taking the reciprocal again, we find that 1/(1/(π - 3) - 7) ≈ 15.9965944066… . So, the first three digits in the continued fraction expansion of π are 3, 7, and 15. If you keep going, you’ll find more digits. The first few digits in this decomposition are (using the usual notation for continued fractions): [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, …]. In other words,

π = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + 1/(1 + 1/(1 + 1/(1 + … ))))))).

In some sense, this sequence of digits is more “natural” than the decimal expansion, because it doesn’t depend on the choice of an arbitrary base. What’s more, the sequence of digits in the continued fraction decomposition give excellent rational approximations to π. For example, if we take only the first two digits in the continued fraction decomposition for π, we get an approximation of 3 + 1/7 = 22/7, a commonly used rational approximation to π (note 22/7 = 3.142857…). In fact, this is a better approximation to π than 3.14. Similarly, if we take the first three digits in the continued fraction, we get an approximation of 3 + 1/(7 + 1/15) = 333/106. This number agrees with the decimal expansion of π to 5 digits (333/106 = 3.14150943…).

Note that even though this decomposition is more natural, we still made a choice in the construction. Namely, we found our digits by always taking the greatest integer less than some number. We could have made other choices here - the smallest integer greater than some number, for example, or whichever integer is closest to the given number - and these choices would yield different expansions. What’s more, there are other ways to generalize the concept of a continued fraction — some of these generalizations yield expansions of pi which have a perfectly nice pattern (these are the types of expansions I’d be awesome at memorizing for a recitation competition).

The point here is that there’s nothing special about the decimal expansion of π — expressing the number in base 10 isn’t particularly natural, so it shouldn’t be surprising that the digits show no pattern. There are many nicer ways to express π, but then there would be no challenge in trying to memorize digits. But, since memorizing digits isn’t a particularly fruitful exercise, there’s really nothing lost.

In an age when the general population already has a poor enough understanding of what mathematicians do, advocating these incredibly boring recitation competitions isn’t helping — especially when there is so much beautiful mathematics at our fingertips. By all means enjoy Pi Day if you must (in particular, I have no objection to you gorging yourself on actual pie), but please, let’s table the digit recitations.

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