# 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:

653728f1.

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

27169820432.

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:

It 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 cnn.com got into the $\pi$ Day spirit!

Fortunately pie eventually ends.