Cronologia do cálculo de π

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Cronologia do cálculo de

A tabela abaixo é uma breve cronologia dos valores numéricos computados ou limites da constante matemática pi ().

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.

Antes de 1400[editar | editar código-fonte]

Data Quem Formulação Valor de pi Dígitos decimais
(recordes mundiais
em negrito)
2000? a.C. Matemática do Antigo Egito[1] 4*(8/9)2 3,16045... 1
2000? a.C. Antigos babilônios[1] 3+1/8 3,125 1
1200? a.C. China[1] 3 1
550? a.C. Bíblia (1 Kings 7:23)[1] "...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 1
434 a.C. Anaxágoras tentou a quadratura do círculo[2] régua e compasso Anaxagoras não obteve solução 0
350? a.C. Shulba Sutras[3][4] (6/(2+2))2 3,088311 … 1
c. 250 a.C. Arquimedes[1] 223/71 < < 22/7 3,140845... <  < 3,142857...
3,1418 (ave.)
3
15 a.C. Vitrúvio[3] 25/8 3,125 1
5 Liu Xin[3] o método exato é desconhecido 3,1457 2
130 Zhang Heng (Book of the Later Han)[1] 10 = 3,162277...
730/232
3,146551... 1
150 Ptolemeu[1] 377/120 3,141666... 3
250 Wang Fan[1] 142/45 3,155555... 1
263 Liu Hui[1] 3,141024 < < 3,142074
3927/1250
3,14159 5
400 He Chengtian[3] 111035/35329 3,142885... 2
480 Zu Chongzhi[1] 3,1415926 < < 3,1415927
Zu's ratio 355/113
3,1415926 7
499 Aryabhata[1] 62832/20000 3,1416 4
640 Brahmagupta[1] 10 3,162277... 1
800 al-Khwārizmī[1] 3,1416 4
1150 Bhaskara II[3] 3927/1250 e 754/240 3,1416 3
1220 Leonardo Fibonacci[1] 3,141818 3
1320 Zhao Youqin[3] 3,1415926 7

A partir de 1400[editar | editar código-fonte]

Data Quem Nota Dígitos  decimais
(recordes mundiais em negrito)
All records from 1400 onwards are given as the number of correct decimal places.
1400 Madhava de Sangamagrama Descobriu provavelmente a série de potências infinita de , conhecida como Fórmula de Leibniz para π[5] 10
1424 Jamshīd al-Kāshī[6] 17
1573 Valentinus Otho 355/113 6
1579 François Viète[7] 9
1593 Adriaan van Roomen[8] 15
1596 Ludolph van Ceulen 20
1615 32
1621 Willebrord Snel van Royen Pupil of Van Ceulen 35
1630 Christoph Grienberger[9][10] 38
1665 Isaac Newton[1] 16
1681 Seki Takakazu[11] 11
16
1699 Abraham Sharp[1] Calculou pi com 72 dígitos, mas nem todos estavam corretos 71
1706 John Machin[1] 100
1706 William Jones Introduziu a letra grega ''
1719 Thomas Fantet de Lagny[1] Calculou 127 dígitos decimais, mas nem todos eram corretos 112
1722 Toshikiyo Kamata 24
1722 Katahiro Takebe 41
1739 Yoshisuke Matsunaga 51
1748 Leonhard Euler Usou a letra grega '' em seu livro Introductio in Analysin Infinitorum e assegorou sua popularidade.
1761 Johann Heinrich Lambert Provou que é irracional
1775 Euler Assinalou a possibilidade de poder ser um número transcendental
1789 Jurij Vega Calculou 143 dígitos decimais, mas nem todos eram corretos 126
1794 Jurij Vega[1] Calculou 140 dígitos decimais, mas nem todos eram corretos 136
1794 Adrien-Marie Legendre Mostrou que ² (e portanto ) é irracional, e mencionou a possibilidade que pode ser transcendente.
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
1841 William Rutherford[1] Calculou 208 dígitos decimais, mas nem todos eram corretos 152
1844 Zacharias Dase e Leopold Karl Schulz von Strassnitzky[1] Calculou 205 dígitos decimais, mas nem todos eram corretos 200
1847 Thomas Clausen[1] Calculou 250 dígitos decimais, mas nem todos eram corretos 248
1853 Lehmann[1] 261
1855 Richter 500
1874 William Shanks[1] Durante 15 anos calculou 707 dígitos decimais, mas nem todos eram corretos (os erros foram identificados por D. F. Ferguson em 1946) 527
1882 Ferdinand von Lindemann Provou que é transcendental (teorema de Lindemann–Weierstrass)
1897 The U.S. state of Indiana Came close to legislating the value 3,2 (among others) for . 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.[12] 1
1910 Srinivasa Ramanujan Found several rapidly converging infinite series of , which can compute 8 decimal places of with each term in the series. Since the 1980s, his series have become the basis for the fastest algorithms currently used by Yasumasa Kanada e the Chudnovsky brothers to compute .
1946 D. F. Ferguson Desk calculator 620
1947 Ivan Morton Niven Apresentou uma Gave a very elementary proof that is irrational
January 1947 D. F. Ferguson Desk calculator 710
September 1947 D. F. Ferguson Desk calculator 808
1949 D. F. Ferguson e John Wrench Desk calculator 1,120

Idade da computação eletrônica (a partir de 1949)[editar | editar código-fonte]

Data Quem Implementação Tempo Dígitos decimais
(recordes mundiais em negrito)
All records from 1949 onwards were calculated with electronic computers.
1949 John Wrench e L. R. Smith Were the first to use an electronic computer (the ENIAC) to calculate (also attributed to Reitwiesner et al.) [13] 70 horas 2,037
1953 Kurt Mahler Mostrou que não é um número de Liouville
1954 S. C. Nicholson & J. Jeenel Using the NORC [14] 13 minutes 3,093
1957 George E. Felton Ferranti Pegasus computer (London), calculated 10,021 digits, but not all were correct [15] 7.480
janeiro de 1958 Francois Genuys IBM 704 [16] 1,7 horas 10.000
maio de 1958 George E. Felton Pegasus computer (London) 33 horas 10,021
1959 François Genuys IBM 704 (Paris)[17] 4,3 horas 16,167
1961 Daniel Shanks e John Wrench IBM 7090 (New York)[18] 8,7 horas 100.265
1961 J.M. Gerard IBM 7090 (London) 39 minutes 20.000
1966 Jean Guilloud e J. Filliatre IBM 7030 (Paris) 28 horas {?) 250,000
1967 Jean Guilloud e M. Dichampt CDC 6600 (Paris) 28 horas 500,000
1973 Jean Guilloud e Martin Bouyer CDC 7600 23,3 horas 1.001.250
1981 Kazunori Miyoshi e Yasumasa Kanada FACOM M-200 2.000.036
1981 Jean Guilloud Not known 2.000.050
1982 Yoshiaki Tamura MELCOM 900II 2.097.144
1982 Yoshiaki Tamura e Yasumasa Kanada HITAC M-280H 2,9 horas 4.194.288
1982 Yoshiaki Tamura e Yasumasa Kanada HITAC M-280H 8.388.576
1983 Yasumasa Kanada, Sayaka Yoshino e Yoshiaki Tamura HITAC M-280H 16.777.206
outubro de 1983 Yasunori Ushiro e Yasumasa Kanada HITAC S-810/20 10.013.395
outubro de 1985 Bill Gosper Symbolics 3670 17.526.200
janeiro de 1986 David H. Bailey CRAY-2 29.360.111
setembro de 1986 Yasumasa Kanada, Yoshiaki Tamura HITAC S-810/20 33.554.414
outubro de 1986 Yasumasa Kanada, Yoshiaki Tamura HITAC S-810/20 67.108.839
janeiro de 1987 Yasumasa Kanada, Yoshiaki Tamura, Yoshinobu Kubo e outros NEC SX-2 134.214.700
janeiro de 1988 Yasumasa Kanada e Yoshiaki Tamura HITAC S-820/80 201.326.551
maio de 1989 Gregory V. Chudnovsky & David V. Chudnovsky CRAY-2 & IBM 3090/VF 480,000,000
junho de 1989 Gregory V. Chudnovsky & David V. Chudnovsky IBM 3090 535.339.270
julho de 1989 Yasumasa Kanada e Yoshiaki Tamura HITAC S-820/80 536.870.898
agosto de 1989 Gregory V. Chudnovsky & David V. Chudnovsky IBM 3090 1.011.196.691
19 de novembro de 1989 Yasumasa Kanada e Yoshiaki Tamura HITAC S-820/80 1.073.740.799
agosto de 1991 Gregory V. Chudnovsky & David V. Chudnovsky Homemade parallel computer (details unknown, not verified) [19] 2.260.000.000
18 de maio de 1994 Gregory V. Chudnovsky & David V. Chudnovsky New homemade parallel computer (details unknown, not verified) 4.044.000.000
26 de junho de 1995 Yasumasa Kanada e Daisuke Takahashi HITAC S-3800/480 (dual CPU) [20] 3.221.220.000
1995 Simon Plouffe Encontrou uma fórmula que permite calcular o n-ésimo dígito hexadecimal de pi sem calcular os dígitos precedentes.
28 de agosto de 1995 Yasumasa Kanada e Daisuke Takahashi HITAC S-3800/480 (dual CPU) [21] 4.294.960.000
11 de outubro de 1995 Yasumasa Kanada e Daisuke Takahashi HITAC S-3800/480 (dual CPU) [22] 6.442.450.000
6 de julho de 1997 Yasumasa Kanada e Daisuke Takahashi HITACHI SR2201 (1024 CPU) [23] 51.539.600.000
5 de abril de 1999 Yasumasa Kanada e Daisuke Takahashi HITACHI SR8000 (64 of 128 nodes) [24] 68.719.470.000
20 de setembro de 1999 Yasumasa Kanada e Daisuke Takahashi HITACHI SR8000/MPP (128 nodes) [25] 206.158.430.000
24 de novembro de 2002 Yasumasa Kanada & equipe de 9 pessoas HITACHI SR8000/MPP (64 nodes), Department of Information Science at the University of Tokyo in Tokyo, Japan [26] 600 horas 1.241.100.000.000
29 de abril de 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[27] 29,09 horas 2.576.980.377.524
Data Quem Implementação Tempo Dígitos decimais
(recordes mundiais em negrito)
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 dias
  • Verification of the binary digits: 13 dias
  • Conversion to base 10: 12 dias
  • Verification of the conversion: 3 dias
  • Verification of the binary digits used a network of 9 Desktop PCs during 34 horas, Chudnovsky algorithm, see [28] for Bellard's homepage.[29]
131 dias 2.699.999.990.000
2 de agosto de 2010 Shigeru Kondo[30]
  • using y-cruncher[31] 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 4.152.410.11.610th.
  • 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 dias
  • Conversion to base 10: 8,2 dias
  • Verification of the conversion: 45,6 horas
  • Verification of the binary digits: 64 horas (primary), 66 horas (secondary)
  • Verification of the binary digits were done simultaneously on two separate computers during the main computation.[32]
90 dias 5.000.000.000.000
17 de outubro de 2011 Shigeru Kondo[33]
  • using y-cruncher by Alexander Yee
  • Verification: 1,86 dias e 4,94 dias
371 dias 10.000.000.000.050
28 December 2013 Shigeru Kondo[34]
  • 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
  • Verification: 46 horas
94 dias 12.100.000.000.050
8 de outubro de 2014 "houkouonchi"[35]
  • 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
  • Verification: 182 horas
208 dias 13.300.000.000.000
11 de novembro de 2016 Peter Trueb[36][37]
  • using y-cruncher by Alexander Yee
  • with 4 x Xeon E7-8890 v3 @ 2,50 GHz (72 cores, 144 threads)
  • 1,25 TB DDR4
  • 20 x 6 TB
  • Verification: 28 horas[38]
105 dias 22.459.157.718.361[39]


Referências

  1. a b c d e f g h i j k l m n o p q r s t u v w x David H. Bailey, Jonathan M. Borwein, Peter B. Borwein & Simon Plouffe (1997). «The quest for pi» (PDF). Mathematical Intelligencer. 19 (1): 50–57 
  2. https://www.math.rutgers.edu/~cherlin/History/Papers2000/wilson.html
  3. 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 
  4. https://books.google.com/books?id=DHvThPNp9yMC&lpg=PA18&ots=Aoy0T2r3Qz&dq=Shulba%20Sutras%20date%20of%20creation&hl=de&pg=PA18#v=onepage&q=Shulba%20Sutras%20date%20of%20creation&f=false
  5. Bag, A. K. (1980). «Indian Literature on Mathematics During 1400–1800 A.D.» (PDF). Indian Journal of History of Science. 15 (1): 86. ≈ 2.827.433.388.233/9×10−11 = 3,14159 26535 92222…, good to 10 decimal places. 
  6. approximated 2π to 9 sexagesimal digits. Al-Kashi, author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256 Predefinição:MacTutor. Azarian, Mohammad K. (2010), "al-Risāla al-muhītīyya: A Summary", Missouri Journal of Mathematical Sciences 22 (2): 64–85.
  7. Viète, François (1579). Canon mathematicus seu ad triangula : cum adpendicibus (em Latin). [S.l.: s.n.] 
  8. Romanus, Adrianus (1593). Ideae mathematicae pars prima, sive methodus polygonorum (em Latin). [S.l.: s.n.] 
  9. Grienbergerus, Christophorus (1630). Elementa Trigonometrica (PDF) (em Latin). [S.l.: s.n.] 
  10. Hobson, Ernest William (1913). "Squaring the Circle": a History of the Problem (PDF). [S.l.: s.n.] p. 27 
  11. Yoshio, Mikami; Eugene Smith, David (abril de 2004) [January 1914]. A History of Japanese Mathematics paperback ed. [S.l.]: Dover Publications. ISBN 0-486-43482-6 
  12. Lopez-Ortiz, Alex (20 de fevereiro de 1998). «Indiana Bill sets value of Pi to 3». the news.answers WWW archive. Department of Information and Computing Sciences, Utrecht University. Consultado em 1 de fevereiro de 2009. 
  13. G. Reitwiesner, "An ENIAC determination of Pi and e to more than 2000 decimal places," MTAC, v. 4, 1950, pp. 11–15"
  14. S. C, Nicholson & J. Jeenel, "Some comments on a NORC computation of x," MTAC, v. 9, 1955, pp. 162–164
  15. 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
  16. F. Genuys, "Dix milles decimales de x," Chiffres, v. 1, 1958, pp. 17–22.
  17. 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
  18. [1] "Calculation of Pi to 100,000 Decimals" in the journal Mathematics of Computation, vol 16 (1962), issue 77, pages 76–99.
  19. 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
  20. ftp://pi.super-computing.org/README.our_last_record_3b
  21. ftp://pi.super-computing.org/README.our_last_record_4b
  22. ftp://pi.super-computing.org/README.our_last_record_6b
  23. ftp://pi.super-computing.org/README.our_last_record_51b
  24. ftp://pi.super-computing.org/README.our_last_record_68b
  25. ftp://pi.super-computing.org/README.our_latest_record_206b
  26. http://www.super-computing.org/pi_current.html
  27. http://www.hpcs.is.tsukuba.ac.jp/~daisuke/pi.html
  28. «Fabrice Bellard's Home Page». bellard.org. Consultado em 28 de agosto de 2015. 
  29. http://bellard.org/pi/pi2700e9/pipcrecord.pdf
  30. «PI-world». calico.jp. Consultado em 28 de agosto de 2015. 
  31. «y-cruncher - A Multi-Threaded Pi Program». numberworld.org. Consultado em 28 de agosto de 2015. 
  32. «Pi - 5 Trillion Digits». numberworld.org. Consultado em 28 de agosto de 2015. 
  33. «Pi - 10 Trillion Digits». numberworld.org. Consultado em 28 de agosto de 2015. 
  34. «Pi - 12,1 Trillion Digits». numberworld.org. Consultado em 28 de agosto de 2015. 
  35. «y-cruncher - A Multi-Threaded Pi Program». numberworld.org. Consultado em 28 de agosto de 2015. 
  36. «pi2e». pi2e.ch. Consultado em 15 de novembro de 2016. 
  37. «y-cruncher - A Multi-Threaded Pi Program». numberworld.org. Consultado em 15 de novembro de 2016. 
  38. «Hexadecimal Digits are Correct! - pi2e trillion digits of pi». pi2e.ch. Consultado em 15 de novembro de 2016. 
  39. 22.459.157.718.361 is e*1012 rounded down.

Ligações externas[editar | editar código-fonte]