Pi-Eyed

A year ago we commented on the achievement by computer scientists in calculating pi to 29 million decimal places. This is a calculation without any practical value, as far as anyone knows. But it continues a long tradition that goes back to the Greeks, who calculated pi--the ratio of the circumference of a circle to its diameter--to two places, and it has engaged some of the most important mathematicians in the two millennia since.

By now the only thing that extending the value of pi is good for is demonstrating the swiftness of a supercomputer. The calculation requires lots of arithmetic, which is right up a supercomputer’s alley. Using an NEC SX-2 machine, Yasuma Kanada of the University of Tokyo has now calculated the value of pi to 134,217,700 digits--more than quadrupling last year’s record, which was achieved on a Cray-2 at NASA’s Ames Research Center at Moffett Field, near San Jose.

It has long been known that pi is both irrational and transcendental, which means that its decimal expansion never ends and that it never repeats. Mathematicians believe that the digits occur randomly and that each digit appears as frequently as any other. The latest expansion is consistent with this belief, though no one can predict what may happen next.

But this does raise an interesting and annoying question about reality itself, and about our ability to know. The method of calculating pi used by Kanada was developed by Peter and Jonathan Borwein of Halifax, N.S. It can be shown that it is nearly optimal, which is to say that it is almost the best method possible. This means that, given the fact that the universe will exist for a finite amount of time, there is apparently an upper limit to the decimal expansion of pi. In a finite amount of time only a finite number of digits can be calculated, no matter how fast the calculation is done.

Because there is no way to predict what the next digit will be, each digit must be calculated before the next one can be known. So if someone were to ask, “What digit would you find if you expanded pi to 10 to the 1,000th decimal places?” the answer is, “We do not know, and we can never know.” After all, 10 to the 1,000th power is a huge number--larger than the number of electrons in the universe; 10 to the second power has 100 digits; 10 to the third power has 1,000 digits; 10 to the sixth power has a million digits, and even the latest achievement, 134,000,000 digits of pi, is only a little more than 10 to the eighth power. There is quite a way to go until 10 to the 1,000th power. In fact, it can never be reached.

Which raises the question whether that number exists at all. If something in principle cannot be known, does it exist? This is quite different from asking whether there is a sound if a tree falls in a forest and no one hears it. That merely depends on the definition of sound, and there is no reason in principle why the falling tree could not be heard. If someone were there, he would hear it.

But the question about the digit of pi in the 10 to the 1,000th decimal place is less easily answered. We can describe that number--it’s the 10 to the 1,000th decimal digit of pi--but it cannot be known. Merely being able to describe something does not mean that it exists. We can describe mermaids and unicorns, but in the real world you’ll never find one. Nor can you ever know what that digit of pi is. Does it exist?