Chronology of computation of π

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mathematical constant π
3.1415926535897932384626433...
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The table below is a brief chronology of computed numerical values of, or bounds on, the mathematical constant pi (π). For more detailed explanations for some of these calculations, see Approximations of π.

The last few decimal digits of the latest 2021 world record computation are:[1] 

6845711163 0056651643 5011939273 3317931338 5175251446  :  62,831,853,071,750
0666164596 1766612754 8681024493 0164977817 924264
Graph showing how the record precision of numerical approximations to pi measured in decimal places (depicted on a logarithmic scale), evolved in human history. The time before 1400 is compressed.

Before 1400[]

Date Who Description/Computation method used Value Decimal places
(world records
in bold)
2000? BCЕ Ancient Egyptians[2] 4 × (8⁄9)2 3.1605... 1
2000? BCЕ Ancient Babylonians[2] 3 + 1⁄8 3.125 1
1200? BCЕ Ancient Chinese[2] 3 0
800–600 BCE Shatapatha Brahmana (Sanskrit:शतपथ ब्राह्मण) – 7.1.1.18 [3] Instructions on how to construct a circular altar from oblong bricks:

He puts on (the circular site) four (bricks) running eastwards 1; two behind running crosswise (from south to north), and two (such) in front. Now the four which he puts on running eastwards are the body; and as to there being four of these, it is because this body (of ours) consists, of four parts 2. The two at the back then are the thighs; and the two in front the arms; and where the body is that (includes) the head."[4]

(Sanskrit: "स चतस्रः प्राचीरुपदधाति | द्वे पश्चात्तिरश्च्यौ द्वे पुरस्तात्तद्याश्चतस्रः प्राचीरुपदधाति स आत्मा तद्यत्ताश्चतस्रो भवन्ति चतुर्विधो ह्ययमात्माथ ये पश्चात्ते सक्थ्यौ ये पुरस्तात्तौ बाहू यत्र वा आत्मा तदेव शिरः)

(Sanskrit transliteration: sa catasraḥ prācīrupadadhāti | dve paścāttiraścyau dve purastāttadyāścatasraḥ prācīrupadadhāti sa ātmā tadyattāścatasro bhavanti caturvidho hyayamātmātha ye paścātte sakthyau ye purastāttau bāhū yatra vā ātmā tadeva śiraḥ)

25⁄8 = 3.125 1
800? BCЕ Sulbasutras[5]

[6][7]

(6⁄(2 + √2))2 3.088311 ... 0
550? BCЕ Bible (1 Kings 7:23)[2] "...a molten sea, ten cubits from the one brim to the other: it was round all about,... a line of thirty cubits did compass it round about" 3 0
434 BCE Anaxagoras attempted to square the circle[8] compass and straightedge Anaxagoras didn't offer any solution 0
c. 250 BCE Archimedes[2] 223⁄71 < π < 22⁄7 3.140845... < π < 3.142857... 2
15 BCE Vitruvius[6] 25⁄8 3.125 1
between 1 and 5 Liu Xin[6][9][10] Unknown method giving a figure for a Jialiang which implies a value for π π ≈ 162⁄(√50+0.095)2. 3.1547... 1
130 Zhang Heng (Book of the Later Han)[2] √10 = 3.162277...
736⁄232 		3.1622... 1
150 Ptolemy[2] 377⁄120 3.141666... 3
250 Wang Fan[2] 142⁄45 3.155555... 1
263 Liu Hui[2] 3.141024 < π < 3.142074
3927⁄1250 		3.1416 3
400 He Chengtian[6] 111035⁄35329 3.142885... 2
480 Zu Chongzhi[2] 3.1415926 < π < 3.1415927
3.1415926 7
499 Aryabhata[2] 62832⁄20000 3.1416 4[11]
640 Brahmagupta[2] √10 3.162277... 1
800 Al Khwarizmi[2] 3.1416 4[11]
1150 Bhāskara II[6] 3927⁄1250 and 754⁄240 3.1416 4[11]
1220 Fibonacci[2] 3.141818 3
1320 Zhao Youqin[6] 3.141592 6

1400–1949[]

Date Who Note Decimal places
(world records in bold)
All records from 1400 onwards are given as the number of correct decimal places.
1400 Madhava of Sangamagrama Discovered the infinite power series expansion of π,
now known as the Leibniz formula for pi[12] 		10
1424 Jamshīd al-Kāshī[13] 16
1573 Valentinus Otho 355⁄113 6
1579 François Viète[14] 9
1593 Adriaan van Roomen[15] 15
1596 Ludolph van Ceulen 20
1615 32
1621 Willebrord Snell (Snellius) Pupil of Van Ceulen 35
1630 Christoph Grienberger[16][17] 38
1654 Christiaan Huygens Used a geometrical method equivalent to Richardson extrapolation 10
1665 Isaac Newton[2] 16
1681 Takakazu Seki[18] 11
16
1699 Abraham Sharp[2] Calculated pi to 72 digits, but not all were correct 71
1706 John Machin[2] 100
1706 William Jones Introduced the Greek letter 'π'
1719 Thomas Fantet de Lagny[2] Calculated 127 decimal places, but not all were correct 112
1722 24
1722 Katahiro Takebe 41
1739 51
1748 Leonhard Euler Used the Greek letter 'π' in his book Introductio in Analysin Infinitorum and assured its popularity.
1761 Johann Heinrich Lambert Proved that π is irrational
1775 Euler Pointed out the possibility that π might be transcendental
1789 Jurij Vega Calculated 143 decimal places, but not all were correct 126
1794 Jurij Vega[2] Calculated 140 decimal places, but not all were correct 136
1794 Adrien-Marie Legendre Showed that π2 (and hence π) is irrational, and mentioned the possibility that π might be transcendental.
Late 18th century Anonymous manuscript Turns up at Radcliffe Library, in Oxford, England, discovered by F. X. von Zach, giving the value of pi to 154 digits, 152 of which were correct 152
1824 William Rutherford[2] Calculated 208 decimal places, but not all were correct 152
1844 Zacharias Dase and Strassnitzky[2] Calculated 205 decimal places, but not all were correct 200
1847 Thomas Clausen[2] Calculated 250 decimal places, but not all were correct 248
1853 Lehmann[2] 261
1853 Rutherford[2] 440
1874 William Shanks[2] Took 15 years to calculate 707 decimal places, but not all were correct (the error was found by D. F. Ferguson in 1946) 527
1882 Ferdinand von Lindemann Proved that π is transcendental (the Lindemann–Weierstrass theorem)
1897 The U.S. state of Indiana Came close to legislating the value 3.2 (among others) for π. House Bill No. 246 passed unanimously. The bill stalled in the state Senate due to a suggestion of possible commercial motives involving publication of a textbook.[19] 1
1910 Srinivasa Ramanujan Found several rapidly converging infinite series of π, which can compute 8 decimal places of π with each term in the series. Since the 1980s, his series have become the basis for the fastest algorithms currently used by Yasumasa Kanada and the Chudnovsky brothers to compute π.
1946 Most digits ever calculated by hand. 620
1947 Ivan Niven Gave a very elementary proof that π is irrational
January 1947 Made use of a desk calculator 710
September 1947 Desk calculator 808
1949 and John Wrench Made use of a desk calculator 1,120

1949–2009[]

Date Who Implementation Time Decimal places
(world records in bold)
All records from 1949 onwards were calculated with electronic computers.
1949 G. W. Reitwiesner et al. The first to use an electronic computer (the ENIAC) to calculate π [20] 70 hours 2,037
1953 Kurt Mahler Showed that π is not a Liouville number
1954 S. C. Nicholson & J. Jeenel Using the NORC[21] 13 minutes 3,093
1957 George E. Felton Ferranti Pegasus computer (London), calculated 10,021 digits, but not all were correct[22] 7,480
January 1958 Francois Genuys IBM 704[23] 1.7 hours 10,000
May 1958 George E. Felton Pegasus computer (London) 33 hours 10,021
1959 Francois Genuys IBM 704 (Paris)[24] 4.3 hours 16,167
1961 Daniel Shanks and John Wrench IBM 7090 (New York)[25] 8.7 hours 100,265
1961 J.M. Gerard IBM 7090 (London) 39 minutes 20,000
1966 Jean Guilloud and J. Filliatre IBM 7030 (Paris) 28 hours[failed verification] 250,000
1967 Jean Guilloud and M. Dichampt CDC 6600 (Paris) 28 hours 500,000
1973 Jean Guilloud and Martine Bouyer CDC 7600 23.3 hours 1,001,250
1981 and Yasumasa Kanada 2,000,036
1981 Jean Guilloud Not known 2,000,050
1982 2,097,144
1982 and Yasumasa Kanada 2.9 hours 4,194,288
1982 and Yasumasa Kanada 8,388,576
1983 Yasumasa Kanada, Sayaka Yoshino and 16,777,206
October 1983 and Yasumasa Kanada HITAC S-810/20 10,013,395
October 1985 Bill Gosper 17,526,200
January 1986 David H. Bailey CRAY-2 29,360,111
September 1986 Yasumasa Kanada, HITAC S-810/20 33,554,414
October 1986 Yasumasa Kanada, HITAC S-810/20 67,108,839
January 1987 Yasumasa Kanada, , and others NEC SX-2 134,214,700
January 1988 Yasumasa Kanada and HITAC S-820/80 201,326,551
May 1989 Gregory V. Chudnovsky & David V. Chudnovsky CRAY-2 & 480,000,000
June 1989 Gregory V. Chudnovsky & David V. Chudnovsky IBM 3090 535,339,270
July 1989 Yasumasa Kanada and HITAC S-820/80 536,870,898
August 1989 Gregory V. Chudnovsky & David V. Chudnovsky IBM 3090 1,011,196,691
19 November 1989 Yasumasa Kanada and HITAC S-820/80 1,073,740,799
August 1991 Gregory V. Chudnovsky & David V. Chudnovsky Homemade parallel computer (details unknown, not verified) [26] 2,260,000,000
18 May 1994 Gregory V. Chudnovsky & David V. Chudnovsky New homemade parallel computer (details unknown, not verified) 4,044,000,000
26 June 1995 Yasumasa Kanada and Daisuke Takahashi HITAC S-3800/480 (dual CPU) [27] 3,221,220,000
1995 Simon Plouffe Finds a formula that allows the nth hexadecimal digit of pi to be calculated without calculating the preceding digits.
28 August 1995 Yasumasa Kanada and Daisuke Takahashi HITAC S-3800/480 (dual CPU) [28] 4,294,960,000
11 October 1995 Yasumasa Kanada and Daisuke Takahashi HITAC S-3800/480 (dual CPU) [29] 6,442,450,000
6 July 1997 Yasumasa Kanada and Daisuke Takahashi HITACHI SR2201 (1024 CPU) [30] 51,539,600,000
5 April 1999 Yasumasa Kanada and Daisuke Takahashi (64 of 128 nodes) [31] 68,719,470,000
20 September 1999 Yasumasa Kanada and Daisuke Takahashi HITACHI SR8000/MPP (128 nodes) [32] 206,158,430,000
24 November 2002 Yasumasa Kanada & 9 man team HITACHI SR8000/MPP (64 nodes), Department of Information Science at the University of Tokyo in Tokyo, Japan[33] 600 hours 1,241,100,000,000
29 April 2009 Daisuke Takahashi et al. (640 nodes), single node speed is 147.2 gigaflops, computer memory is 13.5 terabytes, Gauss–Legendre algorithm, Center for Computational Sciences at the University of Tsukuba in Tsukuba, Japan[34] 29.09 hours 2,576,980,377,524

2009–present[]

Date Who Implementation Time Decimal places
(world records in bold)
All records from Dec 2009 onwards are calculated and verified on servers and/or home computers with commercially available parts.
31 December 2009 Fabrice Bellard
  • Core i7 CPU at 2.93 GHz
  • 6 GiB (1) of RAM
  • 7.5 TB of disk storage using five 1.5 TB hard disks (Seagate Barracuda 7200.11 model)
  • 64 bit Red Hat Fedora 10 distribution
  • Computation of the binary digits: 103 days
  • Verification of the binary digits: 13 days
  • Conversion to base 10: 12 days
  • Verification of the conversion: 3 days
  • Verification of the binary digits used a network of 9 Desktop PCs during 34 hours, Chudnovsky algorithm, see [35] for Bellard's homepage.[36]
 131 days 2,699,999,990,000
2 August 2010 Shigeru Kondo[37]
  • using y-cruncher[38] by Alexander Yee
  • the Chudnovsky algorithm was used for main computation
  • verification used the Bellard & Plouffe formulas on different computers, both computed 32 hexadecimal digits ending with the 4,152,410,118,610th.
  • with 2× Intel Xeon X5680 @ 3.33 GHz – (12 physical cores, 24 hyperthreaded)
  • 96 GiB DDR3 @ 1066 MHz – (12× 8 GiB – 6 channels) – Samsung (M393B1K70BH1)
  • 1 TB SATA II (Boot drive) – Hitachi (HDS721010CLA332), 3× 2 TB SATA II (Store Pi Output) – Seagate (ST32000542AS) 16× 2 TB SATA II (Computation) – Seagate (ST32000641AS)
  • Windows Server 2008 R2 Enterprise x64
  • Computation of binary digits: 80 days
  • Conversion to base 10: 8.2 days
  • Verification of the conversion: 45.6 hours
  • Verification of the binary digits: 64 hours (primary), 66 hours (secondary)
  • Verification of the binary digits were done simultaneously on two separate computers during the main computation.[39]
 90 days 5,000,000,000,000
17 October 2011 Shigeru Kondo[40]
  • using y-cruncher by Alexander Yee
  • Verification: 1.86 days and 4.94 days
 371 days 10,000,000,000,050
28 December 2013 Shigeru Kondo[41]
  • using y-cruncher by Alexander Yee
  • with 2× Intel Xeon E5-2690 @ 2.9 GHz – (16 physical cores, 32 hyperthreaded)
  • 128 GiB DDR3 @ 1600 MHz – 8× 16 GiB – 8 channels
  • Windows Server 2012 x64
  • Verification: 46 hours
 94 days 12,100,000,000,050
8 October 2014 Sandon Nash Van Ness "houkouonchi"[42]
  • using y-cruncher by Alexander Yee
  • with 2× Xeon E5-4650L @ 2.6 GHz
  • 192 GiB DDR3 @ 1333 MHz
  • 24× 4 TB + 30× 3 TB
  • Verification: 182 hours
 208 days 13,300,000,000,000
11 November 2016 Peter Trueb[43][44]
  • using y-cruncher by Alexander Yee
  • with 4× Xeon E7-8890 v3 @ 2.50 GHz (72 cores, 144 threads)
  • 1.25 TiB DDR4
  • 20× 6 TB
  • Verification: 28 hours[45]
 105 days 22,459,157,718,361
= ⌊πe × 1012⌋
14 March 2019 Emma Haruka Iwao[46]
  • using y-cruncher v0.7.6
  • Computation: 1× n1-megamem-96 (96 vCPU, 1.4TB) with 30TB of SSD
  • Storage: 24× n1-standard-16 (16 vCPU, 60GB) with 10TB of SSD
  • Verification: 20 hours using Bellard's 7-term BBP formula, and 28 hours using Plouffe's 4-term BBP formula
 121 days 31,415,926,535,897
= ⌊π × 1013⌋
29 January 2020 Timothy Mullican[47][48]
  • using y-cruncher v0.7.7
  • Computation: 4x Intel Xeon CPU E7-4880 v2 @ 2.50 GHz
  • 320GB DDR3 PC3-8500R ECC RAM
  • 48× 6TB HDDs (Computation) + 47× LTO Ultrium 5 1.5TB Tapes (Checkpoint Backups) + 12× 4TB HDDs (Digit Storage)
  • Verification: 17 hours using Bellard's 7-term BBP formula, 24 hours using Plouffe's 4-term BBP formula
 303 days 50,000,000,000,000
14 August 2021 Team DAViS of the University of Applied Sciences of the Grisons[49][50]
  • using y-cruncher v0.7.8
  • Computation: AMD Epyc 7542 @ 2.9 GHz
  • 1 TiB of memory
  • 38x 16 TB HDDs (Of those, 24 are used for swapping and 4 used for storage)
  • Verification: 34 hours using Bellard's 4-term BBP formula
 108 days 62,831,853,071,796
= ⌊2π × 1013⌋

See also[]

References[]

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  3. ^ Eggeling, Julius (1882–1900). The Satapatha-brahmana, according to the text of the Madhyandina school. Princeton Theological Seminary Library. Oxford, The Clarendon Press. pp. 302–303.CS1 maint: date and year (link)
  4. ^ The Sacred Books of the East: The Satapatha-Brahmana, pt. 3. Clarendon Press. 1894. p. 303. Public Domain This article incorporates text from this source, which is in the public domain.
  5. ^ "4 II. Sulba Sutras". www-history.mcs.st-and.ac.uk.
  6. ^ a b c d e f Ravi P. Agarwal, Hans Agarwal & Syamal K. Sen (2013). "Birth, growth and computation of pi to ten trillion digits". Advances in Difference Equations. 2013: 100. doi:10.1186/1687-1847-2013-100.CS1 maint: uses authors parameter (link)
  7. ^ Plofker, Kim (2009). Mathematics in India. Princeton University Press. p. 18. ISBN 978-0691120676.
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  11. ^ a b c Rounded to the nearest decimal.
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  13. ^ approximated 2π to 9 sexagesimal digits. Al-Kashi, author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256 O'Connor, John J.; Robertson, Edmund F., "Ghiyath al-Din Jamshid Mas'ud al-Kashi", MacTutor History of Mathematics archive, University of St Andrews Azarian, Mohammad K. (2010). "Al-Risāla Al-Muhītīyya: A Summary". Missouri Journal of Mathematical Sciences. 22 (2): 64–85. doi:10.35834/mjms/1312233136.
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  15. ^ Romanus, Adrianus (1593). Ideae mathematicae pars prima, sive methodus polygonorum (in Latin). apud Ioannem Keerbergium. hdl:2027/ucm.5320258006.
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  19. ^ Lopez-Ortiz, Alex (February 20, 1998). "Indiana Bill sets value of Pi to 3". the news.answers WWW archive. Department of Information and Computing Sciences, Utrecht University. Archived from the original on 2005-01-09. Retrieved 2009-02-01.
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  22. ^ G. E. Felton, "Electronic computers and mathematicians," Abbreviated Proceedings of the Oxford Mathematical Conference for Schoolteachers and Industrialists at Trinity College, Oxford, April 8–18, 1957, pp. 12–17, footnote pp. 12–53. This published result is correct to only 7480D, as was established by Felton in a second calculation, using formula (5), completed in 1958 but apparently unpublished. For a detailed account of calculations of π see Wrench, J. W. Jr. (1960). "The evolution of extended decimal approximations to π". The Mathematics Teacher. 53 (8): 644–650. doi:10.5951/MT.53.8.0644. JSTOR 27956272.
  23. ^ Genuys, F. (1958). "Dix milles decimales de π". Chiffres. 1: 17–22.
  24. ^ This unpublished value of x to 16167D was computed on an IBM 704 system at the French Alternative Energies and Atomic Energy Commission in Paris, by means of the program of Genuys
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  26. ^ Bigger slices of Pi (determination of the numerical value of pi reaches 2.16 billion decimal digits) Science News 24 August 1991 http://www.encyclopedia.com/doc/1G1-11235156.html
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  37. ^ "PI-world". calico.jp. Archived from the original on 31 August 2015. Retrieved 28 August 2015.
  38. ^ "y-cruncher – A Multi-Threaded Pi Program". numberworld.org. Retrieved 28 August 2015.
  39. ^ "Pi – 5 Trillion Digits". numberworld.org. Retrieved 28 August 2015.
  40. ^ "Pi – 10 Trillion Digits". numberworld.org. Retrieved 28 August 2015.
  41. ^ "Pi – 12.1 Trillion Digits". numberworld.org. Retrieved 28 August 2015.
  42. ^ "y-cruncher – A Multi-Threaded Pi Program". numberworld.org. Retrieved 14 March 2018.
  43. ^ "pi2e". pi2e.ch. Retrieved 15 November 2016.
  44. ^ Alexander J. Yee. "y-cruncher – A Multi-Threaded Pi Program". numberworld.org. Retrieved 15 November 2016.
  45. ^ "Hexadecimal Digits are Correct! – pi2e trillion digits of pi". pi2e.ch. 31 October 2016. Retrieved 15 November 2016.
  46. ^ "Google Cloud Topples the Pi Record". Retrieved 14 March 2019.
  47. ^ "The Pi Record Returns to the Personal Computer". Retrieved 30 January 2020.
  48. ^ "Calculating Pi: My attempt at breaking the Pi World Record". 26 June 2019. Retrieved 30 January 2020.
  49. ^ "Pi-Challenge - world record attempt by UAS Grisons - University of Applied Sciences of the Grisons". www.fhgr.ch. 2021-08-14. Retrieved 2021-08-17.
  50. ^ "Die FH Graubünden kennt Pi am genauesten – Weltrekord! - News - FH Graubünden". www.fhgr.ch (in German). 2021-08-16. Retrieved 2021-08-17.

External links[]

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