Mathematics Weblog

Pi

Sunday 14 March 2004 at 6:35 pm | In Articles | Post Comment

The digits in the decimal of have always intrigued mathematicans. In the computer age has been calculated to billion of places. You can see the first 30 million digits at The Inverse Symbolic Calculator.

is known to be irrational (so the decimal expansion is infinite and non-recurring) and transcendental (so it is not a root of a polynomial equation with rational coefficients). You can see a proof on my page here. Yet it isn’t known if the digits are random in the statistical sense.

One of the things that doesn’t help in examining the decimal digits of is that if you want to know, say, the 1000^th decimal digit then you will have to calculate the previous 999.

Note the word decimal in the last paragraph. If you go over to hex digits (base 16) then you can calculate any digit without knowing the previous ones. This truly remarkable result was discovered in 1996 by David Bailey, Peter Borwein and Simon Plouffe (their paper can be found here). What is astonishing is that the formula for used to calcuate the n^th is simple enough to be shown here:

A program listing implementing this formula can be found here.
See Pi on the Web for lots more about .