The quest for PI

  title={The quest for PI},
  author={David H. Bailey and Simon Plouffe and Peter B. Borwein and Jonathan Michael Borwein},
  journal={The Mathematical Intelligencer},
This article gives a brief history of the analysis and computation of the mathematical constant $\pi = 3.14159 \ldots$, including a number of the formulas that have been used to compute $\pi$ through the ages. Recent developments in this area are then discussed in some detail, including the recent computation of $\pi$ to over six billion decimal digits using high-order convergent algorithms, and a newly discovered scheme that permits arbitrary individual hexadecimal digits of $\pi$ to be… 
A catalogue of mathematical formulas involving π, with analysis
This paper presents a catalogue of mathematical formulas and iterative algorithms for evaluating the mathematical constant π, ranging from Archimedes’ 2200-year-old iteration to some formulas that
Some Background on Kanada's Recent Pi Calculation
in a similar way. This formula was used in numerous computations of π, culminating with Shanks’ computation of π to 707 decimal digits accuracy in 1873 (although it was later found that this result
Finding new mathematical identities via numerical computations
  • D. Bailey
  • Mathematics, Computer Science
  • 1998
A remarkable new formula forπ permits one to directly compute the n-th hexadecimal digit of π, without computing the first n - 1 digits, and without the need of multiple-precision arithmetic software.
Birth, growth and computation of pi to ten trillion digits
The universal real constant pi, the ratio of the circumference of any circle and its diameter, has no exact numerical representation in a finite number of digits in any number/radix system. It has
An interesting (and unexpected) application of the Fast Fourier transform in number theory is described, which enhanced their efficiency and reduced computer time.
We apply a newly-developed computational method, Geometric Random Inner Products (GRIP), to quantify the randomness of number sequences obtained from the decimal digits of π. Several members from the
3-5 プロジェクト・エンジニアリングにおける情報セキュリティ・モデル
The goal of this paper is to present formulas for Apéry Constant, Archimede’s Constant, Logarithm Constant , Logarithsm Constant, Catalan's Constant, and other constants useful for high precision calculations frequently appearing in number theory.
New formulas for approximation of π and other transcendental numbers
Many new formulas and arbitrary high-order methods for the approximation of roots of certain analytic functions are derived, including a nontrivial determinantal generalization of Taylor's theorem.
Computing Bits of Algebraic Numbers
It is shown that computing a bit of a fixed real algebraic number is in C= NC1 $\subseteq \mbox{{\sf L}}$ when the bit position has a verbose (unary) representation and in the counting hierarchy when it has a succinct (binary) representation.


On the rapid computation of various polylogarithmic constants
These algorithms can be easily implemented, require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired make it feasible to compute the billionth binary digit of log(2) or π on a modest work station in a few hours run time.
It is remarkable that the algorithm illustrated in Table 1, which uses no floating-point arithmetic, produces the digits of π. The algorithm starts with some 2s, in columns headed by the fractions
Ramanujan and Pi
Pi, the ratio of any circle’s circumference to its diameter, was computed in 1987 to an unprecedented level of accuracy: more than 100 million decimal places. Last year also marked the centenary of
Computation of π Using Arithmetic-Geometric Mean
A new formula for π is derived. It is a direct consequence of Gauss’ arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendre’s relation for elliptic
Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi
The year 1987 was the centenary of Ramanujan’s birth. He died in 1920 Had he not died so young, his presence in modern mathematics might be more immediately felt. Had he lived to have access to
Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity
Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration. Theta Functions and the Arithmetic-Geometric Mean Iteration. Jacobi's Triple-Product and Some Number Theoretic Applications.
The Computation of π to 29,360,000 Decimal Digits Using Borweins’ Quartically Convergent Algorithm
Paper 7: David H. Bailey, “The computation of pi to 29,360,000 decimal digits using Borweins’ quartically convergent algorithm,” Mathematics of Computation, vol. 50 (1988), p. 283–296. Reprinted by
A History of Pi
The history of pi, says the author, is a mirror of the history of man, and Petr Beckmann holds up this mirror, giving the background of the times when pi made progress -- and also when it did not, because science was being stifled by militarism or religious fanaticism.
The Works of Archimedes
THIS is a companion volume to Dr. T. L. Heath's valuable edition of the “Treatise on Conic Sections” by Apollonius of Perga, and the same patience, learning and skill which have turned the latter
Calculation of π to 100,000 Decimals
The following comparison of the previous calculations of π performed on electronic computers shows the rapid increase in computational speeds which has taken place.