Post Pi Day

Apropos for Pi Day, Ed Karrels announced that he had computed new digits of \pi.  He does the calculations in hexadecimal (aka base 16).  Since computers compute things in bits (0’s and 1’s, or base 2), it is more natural for them to work base 2^n for some n, and for some reason computer folks settled on hexadecimal as the standard.

Anyway, he started at the 2,000,000,000,000,000th hexidecimal digit of \pi and computed that the next few digits (in hexidecimal) are:


Or, to convert them into decimal, they’re:


Since each hexadecimal digit encodes a number between 0 and 15, that means, on average, a string of hexadecimal digits converts into a string of decimal digits approximately 4/3 times as long.  So Ed Karrels computed decimal digits of \pi ’round about the 2,600,000,000,000,000th decimal digit.

But wait!, you say, I remember reading on this here blog that the current world record is  \pi computed out to the 10 trillionth (10,000,000,000,000) decimal digit.   And that going from the 5 trillionth to the 10 trillionth digit was a major computation that took a year!  How did Ed Karrels manage to go from the 10 trillionth digit to the 2,600 trillionth digit in only two years?!??!

The world record by Alexander J. Yee & Shigeru Kondo of computing \pi to the 10 trillionth digit is still the world record for digits computed from beginning (3.1415….) to end.

The secret is a formula for \pi discovered by Bailey, Borwein and Plouffe in 1995.  Here it is:

48f7653d58f4ad747327d271ed789415It is a simple infinite series like in Calc III, but the beauty of it is that you can use it to obtain an algorithm which allows you to compute hexadecimal digits of \pi without having to know all the proceeding digits!  It’s like having a mathematical wormhole which lets you make FTL jumps to anywhere in the digits of \pi!

Even got into the \pi Day spirit!

Fortunately pie eventually ends.

Fortunately pie eventually ends.