Chronology of computation of π

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.

Date Who Value of pi
(world records in bold)
26th century BCEgyptian Great Pyramid of Giza and Meidum Pyramid[1]3+1/7 = 22/7 = 3.142...
434 BCAnaxagoras attempted to square the circle with compass and straightedge
c. 250 BCArchimedes223/71 < π < 22/7
(3.140845... < π < 3.142857...)
20 BCVitruvius25/8 = 3.125
5Liu Xin3.1457
130Zhang Heng√10 = 3.162277...
730/232 = 3.146551...
150Ptolemy377/120 = 3.141666...
250Wang Fan142/45 = 3.155555...
263Liu Hui3.141024 < π < 3.142074
3927/1250 = 3.1416
400He Chengtian111035/35329 = 3.142885...
480Zu Chongzhi3.1415926 < π < 3.1415927
Zu's ratio 355/113 = 3.1415929
499Aryabhata62832/20000 = 3.1416
640Brahmagupta√10 = 3.162277...
800Al Khwarizmi3.1416
1150Bhāskara II3.14156
1220Fibonacci3.141818
1320 Zhao Youqin3.141592+
All records from 1400 onwards are given as the number of correct decimal places.
1400Madhava of Sangamagrama probably discovered the infinite power series expansion of π, now known as the Leibniz formula for pi[2]10 decimal places
1424Jamshīd al-Kāshī[3] 17 decimal places
1573Valentinus Otho (355/113) 6 decimal places
1579François Viète[4] 9 decimal places
1593Adriaan van Roomen[5] 15 decimal places
1596Ludolph van Ceulen 20 decimal places
1615 32 decimal places
1621Willebrord Snell (Snellius), a pupil of Van Ceulen 35 decimal places
1630Christoph Grienberger[6][7] 38 decimal places
1665Isaac Newton 16 decimal places
1681Takakazu Seki[8] 11 decimal places
16 decimal places
1699Abraham Sharp calculated pi to 72 digits, but not all were correct 71 decimal places
1706John Machin 100 decimal places
1706William Jones introduced the Greek letter 'π'  
1719Thomas Fantet de Lagny calculated 127 decimal places, but not all were correct 112 decimal places
1722Toshikiyo Kamata 24 decimal places
1722Katahiro Takebe 41 decimal places
1739Yoshisuke Matsunaga 51 decimal places
1748Leonhard Euler used the Greek letter 'π' in his book Introductio in Analysin Infinitorum and assured its popularity. 
1761Johann Heinrich Lambert proved that π is irrational 
1775Euler pointed out the possibility that π might be transcendental 
1789Jurij Vega calculated 143 decimal places, but not all were correct 126 decimal places
1794Jurij Vega calculated 140 decimal places, but not all were correct 136 decimal places
1794Adrien-Marie Legendre showed that π² (and hence π) is irrational, and mentioned the possibility that π might be transcendental. 
Late 18th centuryAnonymous 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 decimal places
1841William Rutherford calculated 208 decimal places, but not all were correct 152 decimal places
1844Zacharias Dase and Strassnitzky calculated 205 decimal places, but not all were correct 200 decimal places
1847Thomas Clausen calculated 250 decimal places, but not all were correct 248 decimal places
1853Lehmann 261 decimal places
1855Richter 500 decimal places
1874William Shanks took 15 years to calculate 707 decimal places but not all were correct (the error was found by D. F. Ferguson in 1946) 527 decimal places
1882Ferdinand von Lindemann proved that π is transcendental (the Lindemann–Weierstrass theorem) 
1897The U.S. state of Indiana came close to legislating the value of 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.[9] 
1910Srinivasa 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 (using a desk calculator) 620 decimal places
1947 Ivan Niven gave a very elementary proof that π is irrational
January 1947 D. F. Ferguson (using a desk calculator) 710 decimal places
September 1947 D. F. Ferguson (using a desk calculator) 808 decimal places
1949 D. F. Ferguson and John Wrench, using a desk calculator 1,120 decimal places
All records from 1949 onwards were calculated with electronic computers.
1949 John Wrench, and L. R. Smith were the first to use an electronic computer (the ENIAC) to calculate π (it took 70 hours) (also attributed to Reitwiesner et al.) [10] 2,037 decimal places
1953Kurt Mahler showed that π is not a Liouville number 
1954 S. C. Nicholson & J. Jeenel, using the NORC (13 minutes) [11] 3,093 decimal places
1957 George E. Felton, using the Ferranti Pegasus computer (London), calculated 10,021 digits, but not all were correct [12] 7,480 decimal places
January 1958 Francois Genuys, using an IBM 704 (1.7 hours) [13] 10,000 decimal places
May 1958 George E. Felton, using the Pegasus computer (London) (33 hours) 10,021 decimal places
1959 Francois Genuys, using the IBM 704 (Paris) (4.3 hours) [14] 16,167 decimal places
1961 Daniel Shanks and John Wrench, using the IBM 7090 (New York) (8.7 hours) <ref name="Calculation of Pi to 100,000 Decimals" in the journal Mathematics of Computation, vol 16 (1962), issue 77, page 76-99."> "Calculation of Pi to 100,000 Decimals" in the journal Mathematics of Computation, vol 16 (1962), issue 77, pages 76–99.</ref> 100,265 decimal places
1961 J.M. Gerard, using the IBM 7090 (London) (39 minutes) 20,000 decimal places
1966 Jean Guilloud and J. Filliatre, using the IBM 7030 (Paris) (taking 28 hours??) 250,000 decimal places
1967 Jean Guilloud and M. Dichampt, using the CDC 6600 (Paris) (28 hours) 500,000 decimal places
1973 Jean Guilloud and Martin Bouyer, using the CDC 7600 (23.3 hours) 1,001,250 decimal places
1981 Kazunori Miyoshi and Yasumasa Kanada, FACOM M-200 2,000,036 decimal places
1981 Jean Guilloud, Not known 2,000,050 decimal places
1982 Yoshiaki Tamura, MELCOM 900II 2,097,144 decimal places
1982 Yoshiaki Tamura and Yasumasa Kanada, HITAC M-280H (2.9 hours) 4,194,288 decimal places
1982 Yoshiaki Tamura and Yasumasa Kanada, HITAC M-280H 8,388,576 decimal places
1983 Yasumasa Kanada, Sayaka Yoshino and Yoshiaki Tamura, HITAC M-280H 16,777,206 decimal places
October 1983 Yasunori Ushiro and Yasumasa Kanada, HITAC S-810/20 10,013,395 decimal places
October 1985 Bill Gosper, Symbolics 3670 17,526,200 decimal places
January 1986 David H. Bailey, CRAY-2 29,360,111 decimal places
September 1986 Yasumasa Kanada, Yoshiaki Tamura, HITAC S-810/20 33,554,414 decimal places
October 1986 Yasumasa Kanada, Yoshiaki Tamura, HITAC S-810/20 67,108,839 decimal places
January 1987 Yasumasa Kanada, Yoshiaki Tamura, Yoshinobu Kubo and others, NEC SX-2 134,214,700 decimal places
January 1988 Yasumasa Kanada and Yoshiaki Tamura, HITAC S-820/80 201,326,551 decimal places
May 1989 Gregory V. Chudnovsky & David V. Chudnovsky, CRAY-2 & IBM 3090/VF 480,000,000 decimal places
June 1989 Gregory V. Chudnovsky & David V. Chudnovsky, IBM 3090 535,339,270 decimal places
July 1989 Yasumasa Kanada and Yoshiaki Tamura, HITAC S-820/80 536,870,898 decimal places
August 1989 Gregory V. Chudnovsky & David V. Chudnovsky, IBM 3090 1,011,196,691 decimal places
19 November 1989 Yasumasa Kanada and Yoshiaki Tamura, HITAC S-820/80 1,073,740,799 decimal places
August 1991 Gregory V. Chudnovsky & David V. Chudnovsky, Homemade parallel computer (details unknown, not verified) [15] 2,260,000,000 decimal places
18 May 1994 Gregory V. Chudnovsky & David V. Chudnovsky, New homemade parallel computer (details unknown, not verified) 4,044,000,000 decimal places
26 June 1995 Yasumasa Kanada and Daisuke Takahashi, HITAC S-3800/480 (dual CPU) [16] 3,221,220,000 decimal places
28 August 1995 Yasumasa Kanada and Daisuke Takahashi, HITAC S-3800/480 (dual CPU) [17] 4,294,960,000 decimal places
11 October 1995 Yasumasa Kanada and Daisuke Takahashi, HITAC S-3800/480 (dual CPU) [18] 6,442,450,000 decimal places
6 July 1997 Yasumasa Kanada and Daisuke Takahashi, HITACHI SR2201 (1024 CPU) [19] 51,539,600,000 decimal places
5 April 1999 Yasumasa Kanada and Daisuke Takahashi, HITACHI SR8000 (64 of 128 nodes) [20] 68,719,470,000 decimal places
20 September 1999 Yasumasa Kanada and Daisuke Takahashi, HITACHI SR8000/MPP (128 nodes) [21] 206,158,430,000 decimal places
24 November 2002 Yasumasa Kanada & 9 man team, HITACHI SR8000/MPP (64 nodes), 600 hours, Department of Information Science at the University of Tokyo in Tokyo, Japan [22] 1,241,100,000,000 decimal places
29 April 2009 Daisuke Takahashi et al., T2K Open Supercomputer (640 nodes), single node speed is 147.2 gigaflops, 29.09 hours, computer memory is 13.5 terabytes, Gauss–Legendre algorithm, Center for Computational Sciences at the University of Tsukuba in Tsukuba, Japan[23] 2,576,980,377,524 decimal places
All records from Dec 2009 onwards are calculated on 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
  • 131 days in total – The verification of the binary digits used a network of 9 Desktop PCs during 34 hours, Chudnovsky algorithm, see [24] for Bellard's homepage.[25]
 2,699,999,990,000 decimal places
2 August 2010 Shigeru Kondo[26]
  • using y-cruncher[27] by Alexander Yee
  • the Chudnovsky formula 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 x Intel Xeon X5680 @ 3.33 GHz – (12 physical cores, 24 hyperthreaded)
  • 96 GB DDR3 @ 1066 MHz – (12 × 8 GB – 6 channels) – Samsung (M393B1K70BH1)
  • 1 TB SATA II (Boot drive) – Hitachi (HDS721010CLA332), 3 × 2 TB SATA II (Store Pi Output) – Seagate (ST32000542AS) 16 x 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)
  • Total Time: 90 days – The verification of the binary digits were done simultaneously on two separate computers during the main computation.[28]
 5,000,000,000,000 decimal places
17 October 2011 Shigeru Kondo[29]
  • using y-cruncher by Alexander Yee
  • Computation: 371 days
  • Verification: 1.86 days and 4.94 days
  • Total time: 371 days
 10,000,000,000,050 decimal places
28 December 2013 Shigeru Kondo[30]
  • using y-cruncher by Alexander Yee
  • with 2 x Intel Xeon E5-2690 @ 2.9 GHz - (16 physical cores, 32 hyperthreaded)
  • 128 GB DDR3 @ 1600 MHz - 8 x 16 GB - 8 channels
  • Windows Server 2012 x64
  • Computation: 94 days
  • Verification: 46 hours
  • Total time: 94 days
 12,100,000,000,050 decimal places
8 October 2014 "houkouonchi"[31]
  • using y-cruncher by Alexander Yee
  • with 2 x Xeon E5-4650L @ 2.6 GHz
  • 192 GB DDR3 @ 1333 MHz
  • 24 x 4 TB + 30 x 3 TB
  • Computation: 208 days
  • Verification: 182 hours
 13,300,000,000,000 decimal places
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.

See also

References

  1. ↑ Petrie, W.M.F. Surveys of the Great Pyramids. Nature Journal: 942–943. 1925
  2. ↑ Bag, A. K. (1980). "Indian Literature on Mathematics During 1400–1800 A.D." (PDF). Indian Journal of History of Science 15 (1): 86. — Madhava gave π ≈ 2,827,433,388,233/9×10−11 = 3.14159 26535 92222…, good to 10 decimal places.
  3. ↑ 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.
  4. ↑ Viète, François (1579). Canon mathematicus seu ad triangula : cum adpendicibus (in Latin).
  5. ↑ Romanus, Adrianus (1593). Ideae mathematicae pars prima, sive methodus polygonorum (in Latin).
  6. ↑ Grienbergerus, Christophorus (1630). Elementa Trigonometrica (PDF) (in Latin).
  7. ↑ Hobson, Ernest William (1913). "Squaring the Circle": a History of the Problem (PDF). p. 27.
  8. ↑ Yoshio, Mikami; Eugene Smith, David (April 2004) [January 1914]. A History of Japanese Mathematics (paperback ed.). Dover Publications. ISBN 0-486-43482-6.
  9. ↑ 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. Retrieved 2009-02-01.
  10. ↑ G. Reitwiesner, "An ENIAC determination of Pi and e to more than 2000 decimal places," MTAC, v. 4, 1950, pp. 11–15"
  11. ↑ S. C, Nicholson & J. Jeenel, "Some comments on a NORC computation of x," MTAC, v. 9, 1955, pp. 162–164
  12. ↑ 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 x see J. W. Wrench, Jr., "The evolution of extended decimal approximations to x," The Mathematics Teacher, v. 53, 1960, pp. 644–650
  13. ↑ F. Genuys, "Dix milles decimales de x," Chiffres, v. 1, 1958, pp. 17–22.
  14. ↑ This unpublished value of x to 16167D was computed on an IBM 704 system at the Commissariat à l'Energie Atomique in Paris, by means of the program of Genuys
  15. ↑ 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
  16. ↑ ftp://pi.super-computing.org/README.our_last_record_3b
  17. ↑ ftp://pi.super-computing.org/README.our_last_record_4b
  18. ↑ ftp://pi.super-computing.org/README.our_last_record_6b
  19. ↑ ftp://pi.super-computing.org/README.our_last_record_51b
  20. ↑ ftp://pi.super-computing.org/README.our_last_record_68b
  21. ↑ ftp://pi.super-computing.org/README.our_latest_record_206b
  22. ↑ http://www.super-computing.org/pi_current.html
  23. ↑ http://www.hpcs.is.tsukuba.ac.jp/~daisuke/pi.html
  24. ↑ http://bellard.org
  25. ↑ http://bellard.org/pi/pi2700e9/pipcrecord.pdf
  26. ↑ Shigeru Kondo
  27. ↑ y-cruncher – A Multi-Threaded Pi-Program
  28. ↑ Kondo's 2010 record, A. Yee's program y-cruncher
  29. ↑ Pi – 10 Trillion Digits
  30. ↑ Pi - 12.1 Trillion Digits
  31. ↑ Pi - 13.3 Trillion Digits

External links