Calculation of π to 100,000 Decimals

  • Daniel Shanks
  • John W. WrenchJr.


The following comparison of the previous calculations of π performed on electronic computers shows the rapid increase in computational speeds which has taken place.


Electronic Computer Machine Time Decimal Place Computational Speed Taylor Model 
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  1. 1.
    G. Reitwiesner, “An ENIAC determination of r and e to more than 2000 decimal places,” MTAC, v. 4, 1950, p. 11–15.MathSciNetGoogle Scholar
  2. 2.
    S. C. Nicholson & J. Jeenel, “Some comments on a NORC computation of jr, ” MTAC, v. 9, 1955, p. 162–164.MathSciNetzbMATHGoogle Scholar
  3. 3.
    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, 1967, p. 12–17, footnote p. 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only 7480D, as was established by Felton in a second calculation, using formula (5), completed in 1958 but apparently unpublished.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 sr see J. W. Wrench, Jr., “The evolution of extended decimal approximations to r,” The Mathematics Teacher, v. 53, 1960, p. 644–650.Google Scholar
  4. 4.
    F. Genuys, “Dix milles decimales de sr,” Chiffres, v. 1, 1958, p. 17–22.MathSciNetzbMATHGoogle Scholar
  5. 5.
    This unpublished value of 7 to 16167D was computed on an IBM 704 system at the Commissariat à l’Energie Atomique in Paris, by means of the program of Genuys.Google Scholar
  6. 6.
    C. Störmer, “Sur l’application de la théorie des nombres entiers complexes à la solution en nombres rationnels, z,, zs, • • •, x„, c,, c,, • •, c,,, k de l’équation c, arctg z, + ci arctg zs + • • • + c,, arctg x„ = ksr/4,” Archiv for Mathematik og Naturvidenskab, v. 19, 1896, p. 69.Google Scholar
  7. 7.
    C. F. Gauss, Werke, Göttingen, 1863; 2nd ed., 1876, v. 2, p. 499–502.Google Scholar
  8. 8.
    S. Ramanujan, “Modular equations and approximations to sr,” Quart. J. Pure Appl. Math., v. 45, 1914, p. 350–372; Collected papers of Srinivasa Ranw.nujan, Cambridge, 1927, p. 23–39.Google Scholar
  9. Note added in proof, December 1, 1961. J. M. Gerard of IBM United Kingdom Limited, who was then unaware of the computation described above, computed sr to 20,000 D on the 7090 in the London Data. Centre on July 31, 1961. His program used Machin’s formula, (1), and required 39 minutes running time. His result agrees with ours to that number of decimals.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Daniel Shanks
    • 1
  • John W. WrenchJr.
    • 1
  1. 1.Applied Mathematics LaboratoryDavid Taylor Model BasinUSA

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