##
**The quest for pi.**
*(English)*
Zbl 0878.11002

This article gives a brief history of the computation of the mathematical constant \(\pi=3.14159 \dots \) with emphasis on the most recent developments.

Various approximations were known to the ancients. In the Bible the value 3 occurs. Archimedes presented the first rigorous mathematical method for the calculation of \(\pi\). After the discovery of the calculus by Newton and Leibniz many formulas for \(\pi\) were derived from the power series for the inverse trigonometric functions.

The development of computers after the war increased the computing power. At the same time several new and unexpected new formulas connected with \(\pi\) were discovered. First progress was made with Ramanujan’s fast converging series. A second improvement was the quadratically convergent algorithm of Salamin and Brent (1976) based on the arithmetic-geometric mean. With related methods Kanada calculated in 1995 over 6 billion decimal digits of \(\pi\). These methods still require high precision arithmetic.

The spigot algorithm of Rabinowitz and Wagon calculates successive digits from previous digits recursively without the need of multiply precision computation software. Recently, a new algorithm has been discovered for computing individual hexadecimal digits of \(\pi\) without the need of calculation any previous digits. By the way, the 100-billion’th hexadecimal digit of \(\pi\) is 9.

The headline of the final section is “Why?”. There are some applications: Test computer hardware, advance computational techniques and theoretical questions, such as the equidistribution of digits. But the most fundamental motivation for computing \(\pi\) is: “Because it is there”.

Various approximations were known to the ancients. In the Bible the value 3 occurs. Archimedes presented the first rigorous mathematical method for the calculation of \(\pi\). After the discovery of the calculus by Newton and Leibniz many formulas for \(\pi\) were derived from the power series for the inverse trigonometric functions.

The development of computers after the war increased the computing power. At the same time several new and unexpected new formulas connected with \(\pi\) were discovered. First progress was made with Ramanujan’s fast converging series. A second improvement was the quadratically convergent algorithm of Salamin and Brent (1976) based on the arithmetic-geometric mean. With related methods Kanada calculated in 1995 over 6 billion decimal digits of \(\pi\). These methods still require high precision arithmetic.

The spigot algorithm of Rabinowitz and Wagon calculates successive digits from previous digits recursively without the need of multiply precision computation software. Recently, a new algorithm has been discovered for computing individual hexadecimal digits of \(\pi\) without the need of calculation any previous digits. By the way, the 100-billion’th hexadecimal digit of \(\pi\) is 9.

The headline of the final section is “Why?”. There are some applications: Test computer hardware, advance computational techniques and theoretical questions, such as the equidistribution of digits. But the most fundamental motivation for computing \(\pi\) is: “Because it is there”.

Reviewer: R.Wegmann (Garching)

### MSC:

11-03 | History of number theory |

11Y60 | Evaluation of number-theoretic constants |

65D20 | Computation of special functions and constants, construction of tables |

65-03 | History of numerical analysis |

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\textit{D. H. Bailey} et al., Math. Intell. 19, No. 1, 50--57 (1997; Zbl 0878.11002)

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### Online Encyclopedia of Integer Sequences:

a(n) = floor(log_10(1/error(n))), where error(n) is the error in the n-th iteration of the Salamin-Brent algorithm for computing Pi.a(n) = floor(log_10(1/error(n))), where error(n) is the error in the n-th iteration of the cubic Borwein-Borwein algorithm for computing 1/Pi.

a(n) = floor(log_10(1/error(n))), where error(n) is the error in the n-th iteration of the quartic Borwein-Borwein algorithm for computing 1/Pi.

### References:

[1] | Bailey, D. H., The computation of pi to 29,360,000 decimal digits using Borweins’ quartically convergent algorithm, Mathematics of Computation, 42, 283-296 (1988) · Zbl 0641.10002 |

[2] | D. H. Bailey, P. B. Borwein, and S. Plouffe, On the rapid computation of various polylogarithmic constants, (to appear in Mathematics of Computation). Available fromhttp://www.cecm.sfu/personal/pborwein/. · Zbl 0879.11073 |

[3] | Beckmann, P., A History of Pi (1971), New York: St. Martin’s Press, New York |

[4] | L. Berggren, J. M. Borwein, and P. B. Borwein,A Sourcebook on Pi, New York: Springer-Verlag (to appear). · Zbl 1054.11001 |

[5] | Borwein, J. M.; Borwein, P. B., Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (1987), New York: Wiley, New York · Zbl 0611.10001 |

[6] | J. M. Borwein and P. B. Borwein, Ramanujan and pi,Scientific American (February 1987), 112-117. |

[7] | Borwein, J. M.; Borwein, P. B.; Bailey, D. H., Ramanujan, modular equations, and approximations to pi, or how to compute one billion digits of pi, American Mathematical Monthly, 96, 201-219 (1989) · Zbl 0672.10017 |

[8] | Brent, R. P., Fast multiple-precision evaluation of elementary functions, Journal of the ACM, 23, 242-251 (1976) · Zbl 0324.65018 |

[9] | D. Chudnovsky and C. Chudnovsky, personal communication (1995). |

[10] | H. R. P. Ferguson and D. H. Bailey, Analysis of PSLQ, an integer relation algorithm, unpublished, 1996. |

[11] | T. L. Heath (trans.), The works of Archimedes, inGreat Books of the Western World (Robert M. Hutchins, ed.), Encyclopedia Britannica (1952), Vol. 1, pp. 447-451. |

[12] | Y. Kanada, personal communication (1996). See also Kanada’s book (in Japanese),Story of Pi, Tokyo: Tokyo-Toshyo Co. Ltd. (1991). |

[13] | Knuth, D. E., The Art of Computer Programming (1981), Reading, MA: Addison-Wesley, Reading, MA |

[14] | R. Preston, The mountains of pi,The New Yorker, 2 March 1992, 36-67. |

[15] | Rabinowitz, S. D.; Wagon, S., A spigot algorithm for pi, American Mathematical Monthly, 103, 195-203 (1995) · Zbl 0853.11102 |

[16] | Salamin, E., Computation of pi using arithmetic-geometric mean, Mathematics of Computation, 30, 565-570 (1976) · Zbl 0345.10003 |

[17] | Shanks, D.; Wrench, J. W., Calculation of pi to 100,000 decimals, Mathematics of Computation, 16, 76-79 (1962) · Zbl 0104.36002 |

[18] | Wagon, S., Isit normal?, Mathematical Intelligencer, 7, 3, 65-67 (1985) · Zbl 0565.10002 |

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