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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Approximations of pi | 3/10 | https://en.wikipedia.org/wiki/Approximations_of_pi | reference | science, encyclopedia | 2026-05-05T16:19:48.727542+00:00 | kb-cron |
See Ramanujan–Sato series. From the mid-20th century onwards, all improvements in calculation of π have been done with the help of calculators or computers. In 1944−45, D. F. Ferguson, with the aid of a mechanical desk calculator, found that William Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect. In the early years of the computer, an expansion of π to 100000 decimal places was computed by Maryland mathematician Daniel Shanks (no relation to the aforementioned William Shanks) and his team at the United States Naval Research Laboratory in Washington, D.C. In 1961, Shanks and his team used two different power series for calculating the digits of π. For one, it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,265 digits of π were published in 1962. The authors outlined what would be needed to calculate π to 1 million decimal places and concluded that the task was beyond that day's technology, but would be possible in five to seven years. In 1989, the Chudnovsky brothers computed π to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of π:
1
π
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12
∑
k
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0
∞
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k
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6
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!
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13591409
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545140134
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!
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k
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3
640320
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3
/
2
.
{\displaystyle {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}.}
Records since then have all been accomplished using the Chudnovsky algorithm. In 1999, Yasumasa Kanada and his team at the University of Tokyo computed π to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of π. In November 2002, Yasumasa Kanada and a team of 9 others used the Hitachi SR8000, a 64-node supercomputer with 1 terabyte of main memory, to calculate π to roughly 1.24 trillion digits in around 600 hours (25 days).
=== Recent records === In August 2009, a Japanese supercomputer called the T2K Open Supercomputer more than doubled the previous record by calculating π to roughly 2.6 trillion digits in approximately 73 hours and 36 minutes. In December 2009, Fabrice Bellard used a home computer to compute 2.7 trillion decimal digits of π. Calculations were performed in base 2 (binary), then the result was converted to base 10 (decimal). The calculation, conversion, and verification steps took a total of 131 days. In August 2010, Shigeru Kondo used Alexander Yee's y-cruncher to calculate 5 trillion digits of π. This was the world record for any type of calculation, but significantly it was performed on a home computer built by Kondo. The calculation was done between 4 May and 3 August, with the primary and secondary verifications taking 64 and 66 hours respectively. In October 2011, Shigeru Kondo broke his own record by computing ten trillion (1013) and fifty digits using the same method but with better hardware. In December 2013, Kondo broke his own record for a second time when he computed 12.1 trillion digits of π. In October 2014, Sandon Van Ness, going by the pseudonym "houkouonchi" used y-cruncher to calculate 13.3 trillion digits of π. In November 2016, Peter Trueb and his sponsors computed on y-cruncher and fully verified 22.4 trillion digits of π (22,459,157,718,361 (πe × 1012)). The computation took (with three interruptions) 105 days to complete, the limitation of further expansion being primarily storage space. In March 2019, Emma Haruka Iwao, an employee at Google, computed 31.4 (approximately 10π) trillion digits of pi using y-cruncher and Google Cloud machines. This took 121 days to complete. In January 2020, Timothy Mullican announced the computation of 50 trillion digits over 303 days. On 14 August 2021, a team (DAViS) at the University of Applied Sciences of the Grisons announced completion of the computation of π to 62.8 (approximately 20π) trillion digits. On 8 June 2022, Emma Haruka Iwao announced on the Google Cloud Blog the computation of 100 trillion (1014) digits of π over 158 days using Alexander Yee's y-cruncher. On 14 March 2024, Jordan Ranous, Kevin O’Brien and Brian Beeler computed π to 105 trillion digits, also using y-cruncher. On 28 June 2024, the StorageReview Team computed π to 202 trillion digits, also using y-cruncher. On 2 April 2025, Linus Media Group and Kioxia computed π to 300 trillion digits, also using y-cruncher. On 11 December 2025, the record returned to the StorageReview Team, after they computed π to 314 trillion digits, again using y-cruncher.
== Practical approximations == Depending on the purpose of a calculation, π can be approximated by using fractions for ease of calculation. The most notable such approximations are 22⁄7 (relative error of about 4·10−4) and 355⁄113 (relative error of about 8·10−8). In Chinese mathematics, the fractions 22/7 and 355/113 are known as Yuelü (约率; yuēlǜ; 'approximate ratio') and Milü (密率; mìlǜ; 'close ratio').
== Non-mathematical "definitions" of π == Of some notability are legal or historical texts purportedly "defining π" to have some rational value, such as the "Indiana Pi Bill" of 1897, which stated "the ratio of the diameter and circumference is as five-fourths to four" (which would imply "π = 3.2") and a passage in the Hebrew Bible that implies that π = 3.
=== Indiana bill ===