Any Way You Slice It, Pi Never Runs Out

Today, March 14, at 1:59 p.m., the Exploratorium in San Francisco will be celebrating one of the most important numbers of mathematics: pi. Unchanging, if hard to pin down, it expresses the ratio between the circumference of a circle and its diameter. As the date and time highlighted by the science museum reveals, its decimal rendition starts 3.14159. But the numbers don’t end there. Instead, pi continues on forever in a dizzy dance of digits.

When I tell people I am a mathematician, they tend to think I sit in my office obsessively calculating mysterious numbers like pi to billions of decimal places. Knowing the growing power of the computer, they look at me pityingly, realizing I will soon be redundant. Thankfully, math isn’t such a tedious trial, despite what you learned in school.

We are taught mathematics like piano students who are allowed to play only scales and arpeggios, without ever hearing the wonderful music they can aspire to once they master the technical exercises. Of course, students will give up the piano absent an encounter with Rachmaninoff, say, or Gershwin. And most pupils never realize there is wonderful mathematical music waiting to be played and composed, if only they can master the arithmetic of the classroom.

So what do I do all day in my office? I’m a pattern searcher. I listen to the mathematical noise around me and I try to find the hidden music that might explain the way things work.

Darren Aronofsky’s film “Pi” features a mathematician character, Max Cohen, who begins every day by quoting this mantra: “There are patterns everywhere in nature.” Max is convinced that he has found an important pattern in the decimal expansion of pi, a sequence that will help him to predict the stock market and that even contains Cabbalistic messages from God. In keeping with Hollywood’s stereotyping of mad scientists, Max gets crazier as the film progresses; still, his passion for finding patterns perfectly captures what drives mathematicians.

The patterns are not only important in pure math terms; they are central to many technological developments outside the world of mathematics. For example, a prime-number pattern that Pierre de Fermat found is the key to the invention of Internet cryptography. (What was the pattern? Multiply any number together a prime number of times, then divide by the prime, and the remainder will be the number you started with. So, if you start with 2, multiplied together 5 times (5 is a prime), you get 32. Divide by 5 and you get 6, with a remainder of ... 2.)

The interesting part about being a mathematician is that the patterns are often far from obvious. It turns out that the decimal expansion, for instance, is the wrong place to look for important patterns in pi. Mathematicians proved early in the 20th century that it is more than likely that a random number will contain whatever pattern you might want to find embedded somewhere in its decimal expansion. So Max could find the ASCII code for Melville’s “Moby-Dick” as well as Cabbalistic messages encoded in the never-ending decimal expansion of pi.

And are there significant patterns in pi? Indeed. Gottfried Wilhelm Leibniz came across a beautiful pattern in pi. He discovered that if you take the fractions corresponding to all the odd numbers, 1, 1/3, 1/5, 1/7 ... and alternately add and subtract each fraction, the answer is equal to pi/4: pi/4=1-1/3 +1/5-1/7+1/9-1/11 and so on.

People compete to remember the decimal expansion of pi to millions of digits. Although no one is trying for a record for reciting this expression to millions of terms, it is still a beautiful formula because it shows that two very different bits of the mathematical world are related. Pi, a number that explains something about the geometry of the circle, can be connected to numbers from arithmetic, with no obvious geometry involved. Mathematicians love to find these hidden tunnels from one end of the mathematical world to the other. They reveal that math is elegantly constructed, deeply interconnected.

As Liebniz’s formula shows, the relationships that pi expresses and reveals give the number its significance, not its decimal representation. The use of decimal numbers is, anyway, merely a human construct that depends on the fact that we have 10 fingers. If we had evolved with a different anatomy, say with eight fingers, pi’s written representation would look very different. It would still be the same number -- an expression of the unchanging ratio between the circumference and diameter of a circle. But using powers of eight rather than powers of 10 as our natural base would mean pi began as 3.110375.

If we had been born with eight fingers, we would have been celebrating pi day in the early hours of last Thursday the 11th, and not this afternoon just before 2.

Marcus du Sautoy is professor of mathematics at the University of Oxford and author of “The Music of the Primes” (HarperCollins, 2003).