Why is it that some mathematicians are continuing to calculate pi to an immense number of decimal places? What are they trying to achieve? Does anyone know how many decimal places have been found to this day?

I think that they're just doing it for fun! Actually, there are some interesting questions about what sort of random tests the digits pass (they pass most tests even though there is a fixed, non-random way in which they are generated). It is true that about 20 decimal places is more than enough to do any calculation in physics so it can't be that! Perhaps they're just looking for something to do with their big, fast computers?

Calculation of many decimal places, while perhaps not having any practical use, is now done to test the speed of computers. Prime factorisation is also done to test the same. But a more interesting thing which Jonathan touched on is that pi, as well as passing randomness tests, is now used to check if randomness tests work!

Neil M

Lots of information about the computation of Pi available at David Bailey's web page:

http://www.nersc.gov/~dhbailey/

David Bailey (and others) discovered the formula for the nth hexadecimal digit of pi, independent of the first n-1 digits. The formula for the nth hexadecimal digit is:

4/(8n+1)-2/(8n+4)-1/(8n+5)-1/(8n+6)

You can download his paper about this (and a history of the computation of pi) by clicking here:

http://www.nersc.gov/~dhbailey/dhbpapers/pi-quest.pdf

It's a PDF file, so you need to download Adobe Acrobat, which you can find at http://www.adobe.com .

According to this, the latest record is 6.4 billion digits by Yasumasa Kanada of the University of Tokyo in October 1995, this might have been superseded by now. It describes the algorithm used to find this (the algorithm quadruples the number of digits computed at each iteration).

He also gives several reasons about WHY anyone would want to do this. The most important is that it is a good test of computer hardware, apparently a pi calculating programs exposed errors in the Cray-2 supercomputers in 1986.

I recommend going to his webpage, and downloading the paper above, it's good stuff.

I'm not entirely sure how many decimal places they've got now. I guess they do it more to test the algorithms and/ or computers they're using than to find out what the 4 billionth digit is (although they're probably also hankering after a slot in the Guiness Book of Records). There isn't really any good mathematical reason for knowing any of the digits. And the most precision, say, engineers could possibly ever need is about 10 decimal places, and in practice they probably use about 3.

Far more interesting are the facts that it is irrational (not the ratio of two integers) and transcendental (not a root of any polynomial with integral coefficients). The fact that it is irrational means that there is no repeating pattern in its decimal expansion (can you prove this?) It's also an incredibly ubiquitous number in mathematics. You wouldn't think that the ratio of circumference to diameter of a circle would be that important, but it is. There are lots of cunning and elegant formulae that involve it, e.g.

Although apparently, there are some computations in theoretical physics (according to David Bailey) which require pi to be known to 60 or 100 decimal places, for intermediate calculations (because you lose precision at every step of a long calculation).

The JoyOfPi website has a large number of links , from which you may well be able to find the latest news on number of decimal places.