# pi

Randall Twede
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i am currently doing a little "math program" so i am researching. did you know:
The current record is held by Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo with 51 billion digits of pi (51,539,600,000 decimal digits to be precise).

man! it sure gets complicated from here:
There are essentially 3 different methods to calculate pi to many decimals.

One of the oldest is to use the power series expansion of atan(x) = x - x^3/3 + x^5/5 - ... together with formulas like pi = 16*atan(1/5) - 4*atan(1/239). This gives about 1.4 decimals per term.

A second is to use formulas coming from Arithmetic-Geometric mean computations. A beautiful compendium of such formulas is given in the book pi and the AGM, (see references). They have the advantage of converging quadratically, i.e. you double the number of decimals per iteration. For instance, to obtain 1 000 000 decimals, around 20 iterations are sufficient. The disadvantage is that you need FFT type multiplication to get a reasonable speed, and this is not so easy to program.

A third one comes from the theory of complex multiplication of elliptic curves, and was discovered by S. Ramanujan. This gives a number of beautiful formulas, but the most useful was missed by Ramanujan and discovered by the Chudnovsky's. It is the following (slightly modified for ease of programming):

Set k_1 = 545140134; k_2 = 13591409; k_3 = 640320; k_4 = 100100025; k_5 = 327843840; k_6 = 53360;

Then pi = (k_6 sqrt(k_3))/(S), where

S = sum_(n = 0)^oo (-1)^n ((6n)!(k_2 + nk_1))/(n!^3(3n)!(8k_4k_5)^n)

The great advantages of this formula are that

1) It converges linearly, but very fast (more than 14 decimal digits per term).

2) The way it is written, all operations to compute S can be programmed very simply. This is why the constant 8k_4k_5 appearing in the denominator has been written this way instead of 262537412640768000. This is how the Chudnovsky's have computed several billion decimals.

An interesting new method was recently proposed by David Bailey, Peter Borwein and Simon Plouffe. It can compute the Nth hexadecimal digit of Pi efficiently without the previous N-1 digits. The method is based on the formula:

pi = sum_(i = 0)^oo (1 16^i) ((4 8i + 1) - (2 8i + 4) - (1 8i + 5) - (1 8i + 6))

in O(N) time and O(log N) space. (See references.)

The following 160 character C program, written by Dik T. Winter at CWI, computes pi to 800 decimal digits.

int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5;
for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a,
f[b]=d%--g,d/=g--,--b;d*=b);}

Paul Clapham
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Oh... I thought you meant they memorized 51 billion digits of pi.

Randall Twede
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roflmao

Randall Twede
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my goal of course is to provide "calculate pi" as an option from my menu. i bet "calculate e" will probably be hard too

any help appreciated

good grief!
The formula for working out the value of e looks fun.

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + ....

Matthew Brown
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So, if you've just broken the world record for generating digits of pi....

...how do you prove it?

Randall Twede
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good question

Randall Twede
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arg! how do i do factorials in java? here i am worried about e when i still haven't figured out pi

Randall Twede
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now i find out about Pascal's triangle....this is going to be fun

Dave Trower
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I think you need to look at BigInteger and BigDecimal classes that are part of the java.math package. You will need these if you want to calculate e or pi to a large number of places.
I also think e is easier to calculate than pi based on what you listed here. It would be easy to write a function in java that returns a factorial.
If you like this type of thing, you should check out Project Euler at http://projecteuler.net/
Good luck.

Seetharaman Venkatasamy
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always big numbers represents as a String... not even BigInteger?

Jesper de Jong
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Dave Trower wrote:If you like this type of thing, you should check out Project Euler at http://projecteuler.net/

Yes, Project Euler is fun if you like playing with math puzzles and programming. They have a long list of math puzzles which you're supposed to solve by writing small programs. Some of the puzzles are very simple, and some are quite hard. I've used them to learn functional programming in Scala. You can for example compute the Fibonacci sequence in a very nifty way.

Jelle Klap
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Wait, not one pie joke? Really?

Akhilesh Trivedi
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Is this why my geometry and mensuration bad

Randall Twede
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project euler sound interesting right now. the first thing i did in this program was Fibonacci numbers. i agree calculating e will be easier. i almost have it just from thinking about it. the math for calculating pi looks much harder.

Jesper de Jong
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And today it's Pi Day! (Look at the moose!)

Mike Simmons
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Happy Half Tau Day!