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Chronology of computation of π

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Part of a series of articles on the
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.

As of July 2024, π has been calculated to 202,112,290,000,000 (approximately 202 trillion) decimal digits. The last 100 decimal digits of the latest world record computation are:[1] 

7034341087 5351110672 0525610978 1945263024 9604509887 5683914937 4658179610 2004394122 9823988073 3622511852
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

[edit]
Date Who Description/Computation method used Value Decimal places
(world records
in bold)
2000? BC Ancient Egyptians[2] 4 × (8⁄9)2 3.1605... 1
2000? BC Ancient Babylonians[2] 3 + 1⁄8 3.125 1
2000? BC Ancient Sumerians[3] 3 + 23/216 3.1065 1
1200? BC Ancient Chinese[2] 3 3 0
800–600 BC Shatapatha Brahmana – 7.1.1.18 [4] 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."[5]

25⁄8 = 3.125 1
800? BC Shulba Sutras[6]

[7][8]

(6⁄(2 + √2))2 3.088311 ... 0
550? BC 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 BC Anaxagoras attempted to square the circle[9] compass and straightedge Anaxagoras did not offer a solution 0
400 BC to AD 400 Vyasa[10]

verses: 6.12.40-45 of the Bhishma Parva of the Mahabharata offer:
"...
The Moon is handed down by memory to be eleven thousand yojanas in diameter. Its peripheral circle happens to be thirty three thousand yojanas when calculated.
...
The Sun is eight thousand yojanas and another two thousand yojanas in diameter. From that its peripheral circle comes to be equal to thirty thousand yojanas.
..."

3 0
c. 250 BC Archimedes[2] 223⁄71 < π < 22⁄7 3.140845... < π < 3.142857... 2
15 BC Vitruvius[7] 25⁄8 3.125 1
Between 1 BC and AD 5 Liu Xin[7][11][12] Unknown method giving a figure for a jialiang which implies a value for π ≈ 162⁄(√50+0.095)2. 3.1547... 1
AD 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[7] 111035⁄35329 3.142885... 2
480 Zu Chongzhi[2] 3.1415926 < π < 3.1415927
355⁄113 		3.1415926 7
499 Aryabhata[2] 62832⁄20000 3.1416 3
640 Brahmagupta[2] √10 3.162277... 1
800 Al Khwarizmi[2] 3.1416 3
1150 Bhāskara II[7] 3927⁄1250 and 754⁄240 3.1416 3
1220 Fibonacci[2] 3.141818 3
1320 Zhao Youqin[7] 3.141592 6

1400–1949

[edit]
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[13] 10
1424 Jamshīd al-Kāshī[14] 16
1573 Valentinus Otho 355⁄113 6
1579 François Viète[15] 9
1593 Adriaan van Roomen[16] 15
1596 Ludolph van Ceulen 20
1615 32
1621 Willebrord Snell (Snellius) Pupil of Van Ceulen 35
1630 Christoph Grienberger[17][18] 38
1654 Christiaan Huygens Used a geometrical method equivalent to Richardson extrapolation 10
1665 Isaac Newton[2] 16
1681 Takakazu Seki[19] 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
1721 Anonymous Calculation made in Philadelphia, Pennsylvania, giving the value of pi to 154 digits, 152 of which were correct. First discovered by F. X. von Zach in a library in Oxford, England in the 1780s, and reported to Jean-Étienne Montucla, who published an account of it.[20] 152
1722 Toshikiyo Kamata 24
1722 Katahiro Takebe 41
1739 Yoshisuke Matsunaga 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[21] Calculated 140 decimal places, but not all were correct 126
1794 Adrien-Marie Legendre Showed that π2 (and hence π) is irrational, and mentioned the possibility that π might be transcendental.
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
1853 William Shanks[22] Expanded his calculation to 707 decimal places in 1873, but an error introduced at the beginning of his new calculation rendered all of the subsequent digits invalid (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.[23] 0
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 D. F. Ferguson Made use of a desk calculator[24] 620
1947 Ivan Niven Gave a very elementary proof that π is irrational
January 1947 D. F. Ferguson Made use of a desk calculator[24] 710
September 1947 D. F. Ferguson Made use of a desk calculator[24] 808
1949 Levi B. Smith and John Wrench Made use of a desk calculator 1,120

1949–2009

[edit]
Date Who Implementation Time Decimal places
(world records in bold)
All records from 1949 onwards were calculated with electronic computers.
September 1949 G. W. Reitwiesner et al. The first to use an electronic computer (the ENIAC) to calculate π [25] 70 hours 2,037
1953 Kurt Mahler Showed that π is not a Liouville number
1954 S. C. Nicholson & J. Jeenel Using the NORC[26] 13 minutes 3,093
1957 George E. Felton Ferranti Pegasus computer (London), calculated 10,021 digits, but not all were correct[27][28] 33 hours 7,480
January 1958 Francois Genuys IBM 704[29] 1.7 hours 10,000
May 1958 George E. Felton Pegasus computer (London) 33 hours 10,021
1959 Francois Genuys IBM 704 (Paris)[30] 4.3 hours 16,167
1961 Daniel Shanks and John Wrench IBM 7090 (New York)[31] 8.7 hours 100,265
1961 J.M. Gerard IBM 7090 (London) 39 minutes 20,000
February 1966 Jean Guilloud and J. Filliatre IBM 7030 (Paris)[28] 41.92 hours 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 Kazunori Miyoshi and Yasumasa Kanada FACOM M-200[28] 137.3 hours 2,000,036
1981 Jean Guilloud Not known 2,000,050
1982 Yoshiaki Tamura MELCOM 900II[28] 7.23 hours 2,097,144
1982 Yoshiaki Tamura and Yasumasa Kanada HITAC M-280H[28] 2.9 hours 4,194,288
1982 Yoshiaki Tamura and Yasumasa Kanada HITAC M-280H[28] 6.86 hours 8,388,576
1983 Yasumasa Kanada, Sayaka Yoshino and Yoshiaki Tamura HITAC M-280H[28] <30 hours 16,777,206
October 1983 Yasunori Ushiro and Yasumasa Kanada HITAC S-810/20 10,013,395
October 1985 Bill Gosper Symbolics 3670 17,526,200
January 1986 David H. Bailey CRAY-2[28] 28 hours 29,360,111
September 1986 Yasumasa Kanada, Yoshiaki Tamura HITAC S-810/20[28] 6.6 hours 33,554,414
October 1986 Yasumasa Kanada, Yoshiaki Tamura HITAC S-810/20[28] 23 hours 67,108,839
January 1987 Yasumasa Kanada, Yoshiaki Tamura, Yoshinobu Kubo and others NEC SX-2[28] 35.25 hours 134,214,700
January 1988 Yasumasa Kanada and Yoshiaki Tamura HITAC S-820/80[32] 5.95 hours 201,326,551
May 1989 Gregory V. Chudnovsky & David V. Chudnovsky CRAY-2 & IBM 3090/VF 480,000,000
June 1989 Gregory V. Chudnovsky & David V. Chudnovsky IBM 3090 535,339,270
July 1989 Yasumasa Kanada and Yoshiaki Tamura 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 Yoshiaki Tamura HITAC S-820/80[33] 1,073,740,799
August 1991 Gregory V. Chudnovsky & David V. Chudnovsky Homemade parallel computer (details unknown, not verified) [34][33] 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) [35] 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) [36][37] 56.74 hours? 4,294,960,000
11 October 1995 Yasumasa Kanada and Daisuke Takahashi HITAC S-3800/480 (dual CPU) [38][37] 116.63 hours 6,442,450,000
6 July 1997 Yasumasa Kanada and Daisuke Takahashi HITACHI SR2201 (1024 CPU) [39][40] 29.05 hours 51,539,600,000
5 April 1999 Yasumasa Kanada and Daisuke Takahashi HITACHI SR8000 (64 of 128 nodes) [41][42] 32.9 hours 68,719,470,000
20 September 1999 Yasumasa Kanada and Daisuke Takahashi HITACHI SR8000/MPP (128 nodes) [43][44] 37.35 hours 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[45] 600 hours 1,241,100,000,000
29 April 2009 Daisuke Takahashi et al. T2K Open Supercomputer (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[46] 29.09 hours 2,576,980,377,524

2009–present

[edit]
Date Who Implementation Time Decimal places
(world records in bold)
All records from Dec 2009 onwards are calculated and verified on commodity x86 computers with commercially available parts. All use the Chudnovsky algorithm for the main computation, and Bellard's formula, the Bailey–Borwein–Plouffe formula, or both for verification.
31 December 2009 Fabrice Bellard[47][48]
  • Computation: Intel Core i7 @ 2.93 GHz (4 cores, 6 GiB DDR3-1066 RAM)
  • Storage: 7.5 TB (5x 1.5 TB)
  • Red Hat Fedora 10 (x64)
  • Computation of the binary digits (Chudnovsky algorithm): 103 days
  • Verification of the binary digits (Bellard's formula): 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.
 131 days 2,699,999,990,000
= 2.7×1012 − 104
2 August 2010 Shigeru Kondo[49]
  • using y-cruncher[50] 0.5.4 by Alexander Yee
  • 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 (Bellard formula), 66 hours (BBP formula)
  • Verification of the binary digits were done simultaneously on two separate computers during the main computation. Both computed 32 hexadecimal digits ending with the 4,152,410,118,610th.[51]
 90 days 5,000,000,000,000
= 5×1012
17 October 2011 Shigeru Kondo[52]
  • using y-cruncher 0.5.5
  • 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), 5× 2 TB SATA II (Store Pi Output), 24× 2 TB SATA II (Computation)
  • Windows Server 2008 R2 Enterprise (x64)
  • Verification: 1.86 days (Bellard formula) and 4.94 days (BBP formula)
 371 days 10,000,000,000,050
= 1013 + 50
28 December 2013 Shigeru Kondo[53]
  • using y-cruncher 0.6.3
  • Computation: 2× Intel Xeon E5-2690 @ 2.9 GHz – (32 cores, 128 GiB DDR3-1600 RAM)
  • Storage: 97 TB (32x 3 TB, 1x 1 TB)
  • Windows Server 2012 (x64)
  • Verification using Bellard's formula: 46 hours
 94 days 12,100,000,000,050
= 1.21×1013 + 50
8 October 2014 Sandon Nash Van Ness "houkouonchi"[54]
  • using y-cruncher 0.6.3
  • Computation: 2× Xeon E5-4650L @ 2.6 GHz (16 cores, 192 GiB DDR3-1333 RAM)
  • Storage: 186 TB (24× 4 TB + 30× 3 TB)
  • Verification using Bellard's formula: 182 hours
 208 days 13,300,000,000,000
= 1.33×1013
11 November 2016 Peter Trueb[55][56]
  • using y-cruncher 0.7.1
  • Computation: 4× Xeon E7-8890 v3 @ 2.50 GHz (72 cores, 1.25 TiB DDR4 RAM)
  • Storage: 120 TB (20× 6 TB)
  • Linux (x64)
  • Verification using Bellard's formula: 28 hours[57]
 105 days 22,459,157,718,361
= ⌊πe×1012⌋
14 March 2019 Emma Haruka Iwao[58]
  • using y-cruncher v0.7.6
  • Computation: 1× n1-megamem-96 (96 vCPU, 1.4 TB) with 30 TB of SSD
  • Storage: 24× n1-standard-16 (16 vCPU, 60 GB) with 10 TB of SSD
  • Windows Server 2016 (x64)
  • Verification: 20 hours using Bellard's 7-term formula, and 28 hours using Plouffe's 4-term formula
 121 days 31,415,926,535,897
= ⌊π×1013⌋
29 January 2020 Timothy Mullican[59][60]
  • using y-cruncher v0.7.7
  • Computation: 4× Intel Xeon CPU E7-4880 v2 @ 2.5 GHz (60 cores, 320 GB DDR3-1066 RAM)
  • Storage: 406.5 TB – 48× 6 TB HDDs (Computation) + 47× LTO Ultrium 5 1.5 TB Tapes (Checkpoint Backups) + 12× 4 TB HDDs (Digit Storage)
  • Ubuntu 18.10 (x64)
  • Verification: 17 hours using Bellard's 7-term formula, 24 hours using Plouffe's 4-term formula
 303 days 50,000,000,000,000
= 5×1013
14 August 2021 Team DAViS of the University of Applied Sciences of the Grisons[61][62]
  • using y-cruncher v0.7.8
  • Computation: AMD Epyc 7542 @ 2.9 GHz (32 cores, 1 TiB RAM)
  • Storage: 608 TB (38× 16 TB HDDs, 34 are used for swapping and 4 used for storage)
  • Ubuntu 20.04 (x64)
  • Verification using the 4-term BBP formula: 34 hours
 108 days 62,831,853,071,796
= ⌈2π×1013⌉
21 March 2022 Emma Haruka Iwao[63][64]
  • using y-cruncher v0.7.8
  • Computation: n2-highmem-128 (128 vCPU and 864 GB RAM)
  • Storage: 663 TB
  • Debian Linux 11 (x64)
  • Verification: 12.6 hours using BBP formula
 158 days 100,000,000,000,000
= 1014
18 April 2023 Jordan Ranous[65][66]
  • using y-cruncher v0.7.10
  • Computation: 2 x AMD EPYC 9654 @ 2.4 GHz (96 cores, 1.5 TiB RAM)
  • Storage: 583 TB (19× 30.72 TB)
  • Windows Server 2022 (x64)
 59 days 100,000,000,000,000
= 1014
14 March 2024 Jordan Ranous, Kevin O’Brien and Brian Beeler[67][68]
  • using y-cruncher v0.8.3
  • Computation: 2 x AMD EPYC 9754 @ 2.25 GHz (128 cores, 1.5 TiB RAM)
  • Storage: 1,105 TB (36× 30.72 TB)
  • Windows Server 2022 (x64)
 75 days 105,000,000,000,000
= 1.05×1014
28 June 2024 Jordan Ranous, Kevin O’Brien and Brian Beeler[69][70]
  • using y-cruncher v0.8.3
  • Computation: 2 x Intel Xeon Platinum 8592+ @ 1.9 GHz (128 cores, 1.0 TiB DDR5 RAM)
  • Storage: 1.5 PB (28× 61.44 TB)
  • Windows 10 (x64)
 104 days 202,112,290,000,000
= 2.0211229×1014

See also

[edit]

References

[edit]
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  3. ^ "Origins: 3.14159265..." Biblical Archaeology Society. 2022-03-14. Retrieved 2022-06-08.
  4. ^ 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.{{cite book}}: CS1 maint: date and year (link)
  5. ^ 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.
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  14. ^ 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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    Sandifer, Ed (2006). "Why 140 Digits of Pi Matter" (PDF). Southern Connecticut State University. Archived from the original (PDF) on 2012-02-04.
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  39. ^ ftp://pi.super-computing.org/README.our_last_record_51b[permanent dead link]
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