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Chronology of computation of π

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This is an old revision of this page, as edited by ChandlerMinh (talk | contribs) at 10:16, 28 November 2022 (→Before 1400: Aryabhata’s solution is for exactly four decimal places. It is not rounded). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Part of a series of articles on the
mathematical constant π
3.1415926535897932384626433...
Uses
Properties
Value
People
History
In culture
Related topics

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 100 decimal digits[1] of the latest 2022 world record computation are:[2] 

  4658718895 1242883556 4671544483 9873493812 1206904813 2656719174 5255431487 2142102057 7077336434 3095295560
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[3] 4 × (8⁄9)2 3.1605... 1
2000? BCЕ Ancient Babylonians[3] 3 + 1⁄8 3.125 1
2000? BCЕ Ancient Sumerians[4] 3 + 23/216 3.1065 1
1200? BCЕ Ancient Chinese[3] 3 3 0
800–600 BCE Shatapatha Brahmana – 7.1.1.18 [5] 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."[6]

25⁄8 = 3.125 1
800? BCЕ Shulba Sutras[7]

[8][9]

(6⁄(2 + √2))2 3.088311 ... 0
550? BCЕ Bible (1 Kings 7:23)[3] "...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[10] compass and straightedge Anaxagoras didn't offer any solution 0
400 BCE to 400 CE Vyasa[11]

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 BCE Archimedes[3] 223⁄71 < π < 22⁄7 3.140845... < π < 3.142857... 2
15 BCE Vitruvius[8] 25⁄8 3.125 1
between 1 and 5 Liu Xin[8][12][13] 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)[3] √10 = 3.162277...
736⁄232 		3.1622... 1
150 Ptolemy[3] 377⁄120 3.141666... 3
250 Wang Fan[3] 142⁄45 3.155555... 1
263 Liu Hui[3] 3.141024 < π < 3.142074
3927⁄1250 		3.1416 3
400 He Chengtian[8] 111035⁄35329 3.142885... 2
480 Zu Chongzhi[3] 3.1415926 < π < 3.1415927
3.1415926 7
499 Aryabhata[3] 62832⁄20000 3.1416 4
640 Brahmagupta[3] √10 3.162277... 1
800 Al Khwarizmi[3] 3.1416 4[14]
1150 Bhāskara II[8] 3927⁄1250 and 754⁄240 3.1416 4[14]
1220 Fibonacci[3] 3.141818 3
1320 Zhao Youqin[8] 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[15] 		10
1424 Jamshīd al-Kāshī[16] 16
1573 Valentinus Otho 355⁄113 6
1579 François Viète[17] 9
1593 Adriaan van Roomen[18] 15
1596 Ludolph van Ceulen 20
1615 32
1621 Willebrord Snell (Snellius) Pupil of Van Ceulen 35
1630 Christoph Grienberger[19][20] 38
1654 Christiaan Huygens Used a geometrical method equivalent to Richardson extrapolation 10
1665 Isaac Newton[3] 16
1681 Takakazu Seki[21] 11
16
1699 Abraham Sharp[3] Calculated pi to 72 digits, but not all were correct 71
1706 John Machin[3] 100
1706 William Jones Introduced the Greek letter 'π'
1719 Thomas Fantet de Lagny[3] Calculated 127 decimal places, but not all were correct 112
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 Calculated 143 decimal places, but not all were correct 126
1794 Jurij Vega[3] 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[22] 152
1824 William Rutherford[3] Calculated 208 decimal places, but not all were correct 152
1844 Zacharias Dase and Strassnitzky[3] Calculated 205 decimal places, but not all were correct 200
1847 Thomas Clausen[3] Calculated 250 decimal places, but not all were correct 248
1853 Lehmann[3] 261
1853 Rutherford[3] 440
1853 William Shanks[23] 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 incorrect (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.[24] 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 D. F. Ferguson Most digits ever calculated by hand. 620
1947 Ivan Niven Gave a very elementary proof that π is irrational
January 1947 D. F. Ferguson Made use of a desk calculator 710
September 1947 D. F. Ferguson Desk calculator 808
1949 Levi B. Smith 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 [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
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 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 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 29,360,111
September 1986 Yasumasa Kanada, Yoshiaki Tamura HITAC S-810/20 33,554,414
October 1986 Yasumasa Kanada, Yoshiaki Tamura HITAC S-810/20 67,108,839
January 1987 Yasumasa Kanada, Yoshiaki Tamura, Yoshinobu Kubo and others NEC SX-2 134,214,700
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

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 [47] for Bellard's homepage.[48]
 131 days 2,699,999,990,000
2 August 2010 Shigeru Kondo[49]
  • using y-cruncher[50] by Alexander Yee
  • the Chudnovsky algorithm was used for main computation
  • verification used the Bellard & BBP (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.[51]
 90 days 5,000,000,000,000
17 October 2011 Shigeru Kondo[52]
  • using y-cruncher by Alexander Yee
  • the Chudnovsky algorithm was used for main computation
  • Verification using the Bellard & BBP (Plouffe) formulas: 1.86 days and 4.94 days
 371 days 10,000,000,000,050
28 December 2013 Shigeru Kondo[53]
  • 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
  • the Chudnovsky algorithm was used for main computation
  • Verification using Bellard's variant of the BBP formula: 46 hours
 94 days 12,100,000,000,050
8 October 2014 Sandon Nash Van Ness "houkouonchi"[54]
  • 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
  • the Chudnovsky algorithm was used for main computation
  • Verification using the BBP formula: 182 hours
 208 days 13,300,000,000,000
11 November 2016 Peter Trueb[55][56]
  • 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
  • the Chudnovsky algorithm was used for main computation
  • Verification using Bellard's variant of the BBP 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, 1.4TB) with 30TB of SSD
  • Storage: 24× n1-standard-16 (16 vCPU, 60GB) with 10TB of SSD
  • the Chudnovsky algorithm was used for main computation
  • 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[59][60]
  • 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)
  • the Chudnovsky algorithm was used for main computation
  • 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[61][62]
  • using y-cruncher v0.7.8
  • Computation: AMD Epyc 7542 @ 2.9 GHz
  • 1 TiB of memory
  • 38x 16 TB HDDs (Of those, 34 are used for swapping and 4 used for storage)
  • the Chudnovsky algorithm was used for main computation
  • Verification: 34 hours using Bellard's 4-term BBP formula
 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
  • the Chudnovsky algorithm was used for main computation
  • Verification: 12.6 hours using BBP formula
 158 days 100,000,000,000,000

See also

References

  1. ^ The last digit shown here is the 100,000,000,000,000th digit of π.
  2. ^ "Validation File". Retrieved 2022-06-09.{{cite web}}: CS1 maint: url-status (link)
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  4. ^ "Origins: 3.14159265…". Biblical Archaeology Society. 2022-03-14. Retrieved 2022-06-08.
  5. ^ 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)
  6. ^ 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.
  7. ^ "4 II. Sulba Sutras". www-history.mcs.st-and.ac.uk.
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  14. ^ a b Rounded to the nearest decimal.
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