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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Approximations of pi | 10/10 | https://en.wikipedia.org/wiki/Approximations_of_pi | reference | science, encyclopedia | 2026-05-05T16:19:48.727542+00:00 | kb-cron |
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{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}}
Srinivasa Ramanujan. This converges extraordinarily rapidly. Ramanujan's work is the basis for the fastest algorithms used, as of the turn of the millennium, to calculate π. In 1988, David Chudnovsky and Gregory Chudnovsky found an even faster-converging series (the Chudnovsky algorithm):
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{\displaystyle {\frac {1}{\pi }}={\frac {1}{426880{\sqrt {10005}}}}\sum _{k=0}^{\infty }{\frac {(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}(-640320)^{3k}}}}
. The speed of various algorithms for computing pi to n correct digits is shown below in descending order of asymptotic complexity. M(n) is the complexity of the multiplication algorithm employed.
== Projects ==
=== Pi Hex === Pi Hex was a project to compute three specific binary digits of π using a distributed network of several hundred computers. In 2000, after two years, the project finished computing the five trillionth (51012), the forty trillionth (401012), and the quadrillionth (1015) bits. All three of them turned out to be 0.
== Software for calculating π == Over the years, several programs have been written for calculating π to many digits on personal computers.
=== General purpose === Most computer algebra systems can calculate π and other common mathematical constants to any desired precision. Functions for calculating π are also included in many general libraries for arbitrary-precision arithmetic, for instance Class Library for Numbers, MPFR and SymPy.
=== Special purpose === Programs designed for calculating π may have better performance than general-purpose mathematical software. They typically implement checkpointing and efficient disk swapping to facilitate extremely long-running and memory-expensive computations.
TachusPi by Fabrice Bellard is the program used by himself to compute world record number of digits of pi in 2009. y-cruncher by Alexander Yee is the program which every world record holder since Shigeru Kondo in 2010 has used to compute world record numbers of digits. y-cruncher can also be used to calculate other constants and holds world records for several of them. PiFast by Xavier Gourdon was the fastest program for Microsoft Windows in 2003. According to its author, it can compute one million digits in 3.5 seconds on a 2.4 GHz Pentium 4. PiFast can also compute other irrational numbers like e and √2. It can also work at lesser efficiency with very little memory (down to a few tens of megabytes to compute well over a billion (109) digits). This tool is a popular benchmark in the overclocking community. PiFast 4.4 is available from Stu's Pi page. PiFast 4.3 is available from Gourdon's page. QuickPi by Steve Pagliarulo for Windows is faster than PiFast for runs of under 400 million digits. Version 4.5 is available on Stu's Pi Page below. Like PiFast, QuickPi can also compute other irrational numbers like e, √2, and √3. The software may be obtained from the Pi-Hacks Yahoo! forum, or from Stu's Pi page. Super PI by Kanada Laboratory in the University of Tokyo is the program for Microsoft Windows for runs from 16,000 to 33,550,000 digits. It can compute one million digits in 40 minutes, two million digits in 90 minutes and four million digits in 220 minutes on a Pentium 90 MHz. Super PI version 1.9 is available from Super PI 1.9 page.
== See also == Diophantine approximation Milü Madhava's correction term Pi is 3
== Notes ==
== References == Bailey, David H.; Borwein, Peter B. & Plouffe, Simon (April 1997). "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation. 66 (218): 903–913. Bibcode:1997MaCom..66..903B. doi:10.1090/S0025-5718-97-00856-9. Beckmann, Petr (1971). A History of π. New York: St. Martin's Press. ISBN 978-0-88029-418-8. MR 0449960. Eves, Howard (1992). An Introduction to the History of Mathematics (6th ed.). Saunders College Publishing. ISBN 978-0-03-029558-4. Joseph, George G. (2000). The Crest of the Peacock: Non-European Roots of Mathematics (New ed., London : Penguin ed.). London: Penguin. ISBN 978-0-14-027778-4. Jackson, K; Stamp, J. (2002). Pyramid: Beyond Imagination. Inside the Great Pyramid of Giza. London: BBC. ISBN 9780563488033. Berggren, Lennart; Borwein, Jonathan M.; Borwein, Peter B. (2004). Pi: a source book (3rd ed.). New York: Springer Science + Business Media LLC. ISBN 978-1-4757-4217-6.