Pi is Forever
September 23, 2010
In the past, there was just one type of record involving pi; namely, the number of digits calculated. Of course, such a contest can go on forever, since pi is a transcendental, irrational number with a number sequence that does not repeat. How many decimal places of pi are really needed? I've memorized eleven places, which is definitely overkill for all practical calculations. As I wrote in a previous article, (Another Piece of Pi, July 28, 2010), we only need twenty-seven digits of pi to calculate the circumference of the universe to better than a meter, and thirty-eight digits to get a value of the circumference closer than the size of an atom. The current record for calculating digits of pi is apparently five trillion digits (5,000,000,000,000).[1] This record was accomplished in August, 2010, by Shigeru Kondo, a Japanese systems engineer, on home-built hardware. The calculation took a little more than ninety days. Yukiko Kondo, Shigeru's wife seemed to be more impressed by the $240 monthly electrical bill than the calculation,[1] but I'm sure she's really proud of her husband.
There are other pi records, like the recitation of digits of pi from memory. The record for this appears to held by another Japanese, Akira Haraguchi, who recited 100,000 digits of pi from memory in a span of sixteen hours in October, 2006.[2] Because of an interesting discovery by David Bailey, Peter Borwein and Simon Plouffe (BBP), there's another way to set a record for pi calculation. They published a formula for finding any digit of pi without calculation of the preceding digits.[3] That this result is unusual is clearly an understatement, but it gives computer people another shot at the record books; namely, the largest calculated digit of pi.
That record now stands at 2,000,000,000,000,000 binary digits.[4] Nicholas Sze of Yahoo used more than a thousand cloud computers to find the 2,000,000,000,000,000th binary digit of pi in 23 days - It's zero.[5] Of course, as one Internet commentator put it, Sze had a fifty-fifty chance of getting it right from the onset. The effort was done on a cluster that varied in size from 1000 to 4000 machines, each machine of which had two quad-core CPUs with clock speeds of 1.8 GHz to 2.5 GHz.[5]
If you're computing that many digits of anything, you need to use libraries that allow calculations with large numbers. The one that I use is the GNU Multiple Precision Arithmetic Library. If you're just calculating a few thousand digits of pi, this isn't required. I've written a C program that uses the BBP technique to calculate 20,000 hex digits of pi (source code here). It uses code published by David Bailey,[6] and it calculates these 20,000 hex digits in just 35 seconds on my dual core, 2.66 GHz computer. Here are the first 300 hex digits of pi, as calculated by the program.
| 243F6A8885 | A308D31319 | 8A2E037073 |
| 44A4093822 | 299F31D008 | 2EFA98EC4E |
| 6C89452821 | E638D01377 | BE5466CF34 |
| E90C6CC0AC | 29B7C97C50 | DD3F84D5B5 |
| B547091792 | 16D5D98979 | FB1BD1310B |
| A698DFB5AC | 2FFD72DBD0 | 1ADFB7B8E1 |
| AFED6A267E | 96BA7C9045 | F12C7F9924 |
| A19947B391 | 6CF70801F2 | E2858EFC16 |
| 636920D871 | 574E69A458 | FEA3F4933D |
| 7E0D95748F | 728EB65871 | 8BCD588215 |
While researching this article, I found an interesting article published almost a decade ago on the randomness of pi digits.[7] I believe that the digits of pi are random, and many experimental tests indicate that this is probably the case. Mathematics, however, is founded on proof, and not experiment. That the first few trillion digits of pi appear random is merely suggestive, not conclusive. The BBP formula demonstrates something that wasn't previously known about pi. David Bailey asserts that the digits of pi appear to be random because they are chaotic.[8]
Mathematician, Clifford Pickover, always a source of interesting ideas, has proposed the following:[9]
"Somewhere inside the digits of pi is a representation for all of us -- the atomic coordinates of all our atoms, our genetic code, all our thoughts, all our memories. Given this fact, all of us are alive, and hopefully happy, in pi. Pi makes us live forever. We all lead virtual lives in pi. We are immortal."
You can try to find parts of yourself in pi, here.[10]
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