Click here to Skip to main content
Click here to Skip to main content
Go to top

Calculate pi to one million decimal places

, 19 Sep 2005
Rate this:
Please Sign up or sign in to vote.
A simple program that should take a few hours to run.

pi

Introduction

Pi is one of the most important numbers in mathematics. It is defined as the ratio of a circle's circumference to its diameter, but it crops up in all sorts of places in mathematics. It is an infinitely long non-recurring decimal number.

I'm not going to try to write pi as the Greek letter in this article, because some browsers might show it incorrectly.

Calculating lots of digits of pi

There are many ways to do this. Some methods converge rapidly but are complicated to implement. Some are simple to implement but converge very slowly. I've chosen a method that is fairly simple and converges reasonably fast. It is based on the following formula:

pi / 4 = 4 * tan-1(1 / 5) - tan-1(1 / 239)

This is Machin's formula and is exact.

tan-1() is the Inverse Tangent function, and I use the Maclaurin series to calculate it:

tan-1(z) = z - z3 / 3 + z5 / 5 - z7 / 7 + ...

By including sufficiently many terms of this series, we can achieve any desired accuracy. To get 1,000,000 decimal places accuracy for pi, we need about 715,000 terms of the tan-1(1/5) series and about 210,000 terms of the tan-1(1/239) series, but this doesn't have to be worked out in advance, the attached program stops automatically when it determines that the required accuracy has been reached.

Multi-length arithmetic

A 32-bit integer only gives us about 9 significant digits. To get one million decimal places, I've done a simple implementation of the multi-length arithmetic operations that I need, using a big array of ints. The multi-length numbers are wrapped up inside the class CRHMultiLengthInteger. I've made full use of the operator notation in C++ to make the code look like we're just working with ordinary numbers. For example:

term5m /= n5 * 2 + 1;

looks like it's just dividing term5m by a number, but it's actually calling CRHMultiLengthInteger::operator /=().

The program

The attached program is a straightforward Win32 Console Application (it started out as the standard "Hello, World!" application with MFC support). You can adjust how many decimal places it calculates by changing #define NDecimalPlaces (1000100) to some other number. The answer is written to a file called resultNDecimalPlaces.txt.

Points of Interest

  • The algorithm described above has an O(n2) duration. So 10 times as many decimal places will take about 100 times as long. My 7,000 MIPs PC took about 15 hours to calculate the first 1,000,000 decimal places of pi. That works out at about 380 million million instructions in total.
  • To get 1,000,000 decimal places, each multi-length number is implemented as an array of 166685 integers, using a base of 1,000,000 so that each integer gives us 6 significant digits. This actually gives us 1,000,104 decimal places, but the last few might be incorrect because of the infinite number of terms of the Maclaurin series that we have to throw away. Comparison with this shows that the last three decimal places are wrong, so we have actually got pi correct to 1,000,101 decimal places.
  • The first 1,000,000 decimal places of pi were first calculated in 1973.
  • The 19th century English mathematician William Shanks spent over 15 years calculating the first 707 places of pi using Machin's formula. He published his results in 1873. In 1944 it was found that he had made a mistake in the 528th place, and all the following digits were wrong. By changing to #define NDecimalPlaces (1000), you can now achieve in a fraction of a second more than William Shanks failed to achieve in 15 years.

Why?

This article is just for fun. I know it's pretty straightforward but I thought people might be interested in seeing it.

Getting a computer to calculate lots of digits of pi does have its uses, it can be used to show that the computer, or at least its ALU, is working properly.

Acknowledgments

Thanks very much to:

History

  • 6th September 2005 - first submitted.

License

This article has no explicit license attached to it but may contain usage terms in the article text or the download files themselves. If in doubt please contact the author via the discussion board below.

A list of licenses authors might use can be found here

Share

About the Author

Chris Hills
Software Developer (Senior)
United Kingdom United Kingdom
I've been programming computers since about 1968. I started at school with Algol 60 on an Elliott 803. From there I progressed through the Z80 and other microprocessors to the PC, DOS and Windows, Pascal, C and C++.
 
My other interests include astronomy and classical music. All of my contributions to Code Project have arisen from programs I've written in these areas.

Comments and Discussions

 
SuggestionPi Pictures. PinmemberEduardo N Hering29-Aug-11 14:45 
GeneralArticle for review Pinmemberssnirgudkar21-Dec-09 15:15 
GeneralRe: Article for review Pinmemberssnirgudkar21-Dec-09 15:17 
GeneralEnough with the term "million million"! Pinmemberprom 197616-May-08 0:03 
GeneralRe: Enough with the term "million million"! PinmemberChris Hills17-May-08 3:39 
GeneralRe: Enough with the term "million million"! Pinmembercanozurdo25-Jun-08 8:37 
QuestionRadians or Degrees? PinmemberThmath17763-May-08 10:06 
NewsOptimization to the death Pinmemberpouyaebad9-Mar-08 1:01 
GeneralRe: Optimization to the death Pinmemberpouyaebad12-Mar-08 1:09 
GeneralRe: Optimization to the death PinmemberBrian Wiegand9-Feb-09 18:12 
GeneralRe: Optimization to the death Pinmemberhashme334-Nov-09 9:21 
QuestionWhy is this important? Pinmembercuriousone11-Jun-07 9:31 
AnswerRe: Why is this important? Pinmemberlancere25-Aug-07 14:12 
AnswerRe: Why is this important? PinmemberCrusiatus Black3-Nov-08 2:24 
GeneralRe: Why is this important? Pinmemberron real30-Jul-10 12:30 
QuestionMultilength arit. Pinmemberdiegusaldus2-Apr-07 23:05 
AnswerRe: Multilength arit. PinmemberChris Hills8-Apr-07 10:04 
GeneralIt's not Exact PinmemberZmurf7-May-06 23:55 
GeneralRe: It's not Exact PinmemberSuper Lloyd24-May-06 19:40 
GeneralRe: It's not Exact PinmemberZmurf25-May-06 2:24 
GeneralRe: It's not Exact PinmemberSuper Lloyd25-May-06 2:54 
Are you a kind of Troll?
Why do you contradict yourself and take it as a proof you are right?
 
Let me be sure I'm understanding you correctly, I will repeat what I understood you saying:
"The result will never be PI, if you graph the will find it follow a curve which tends asymptotically toward PI"?
Is it what you said?
 
One of us is very confused here ...
 
There is one more thing which puzzle me greatly, what do you means by: "No matter how long you compute PI using this method, even if you were to do it for an infinite period of time, your answer would not be exact"
While it is completely wrong, do you know of any method which compute "exact the decimal value" of pi in a finite amount of time?
(if you do: patent it, you will be the greates Mathematician of the 10 next milleniums!)
 
Because there many methods which compute the decimal value of PI, this is one them, and they all take an infinite amount of time....
The key is: name your error, that gives you the number of iteration.... (the smaller, the more iteration...)

GeneralBailey-Borwein-Plouffe formula (this is amazing) PinmemberDon Clugston16-Oct-05 23:28 
Generalsimple optimizations Pinmembercub4bear28-Sep-05 5:32 
GeneralAnother optimization ? PinmemberMr. Fox30-Sep-05 6:19 
GeneralRe: Another optimization ? Pinmembercub4bear30-Sep-05 8:31 

General General    News News    Suggestion Suggestion    Question Question    Bug Bug    Answer Answer    Joke Joke    Rant Rant    Admin Admin   

Use Ctrl+Left/Right to switch messages, Ctrl+Up/Down to switch threads, Ctrl+Shift+Left/Right to switch pages.

| Advertise | Privacy | Mobile
Web02 | 2.8.140926.1 | Last Updated 19 Sep 2005
Article Copyright 2005 by Chris Hills
Everything else Copyright © CodeProject, 1999-2014
Terms of Service
Layout: fixed | fluid