Pi: The Next Generation

1. Front Matter

   Pages i-xiv

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2. Computation of π using arithmetic-geometric mean (1976)

   • Eugene Salamin

   Pages 1-8

3. Fast multiple-precision evaluation of elementary functions (1976)

   • Richard P. Brent

   Pages 9-20

4. The arithmetic-geometric mean of Gauss (1984)

   • David A. Cox

   Pages 21-78

5. The arithmetic-geometric mean and fast computation of elementary functions (1984)

   • J. M. Borwein, P. B. Borwein

   Pages 79-96

6. A simplified version of the fast algorithms of Brent and Salamin (1985)

   • D. J. Newman

   Pages 97-102

7. Is pi normal? (1985)

   • S. Wagon

   Pages 103-107

8. The computation of π to 29,360,000 decimal digits using Borweins quartically convergent algorithm (1988)

   • David H. Bailey

   Pages 109-124

9. Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary (1988)

   • Gert Almkvist, Bruce Berndt

   Pages 125-150

10. Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of pi calculation (1988)

    • Yasumasa Kanada

    Pages 151-164

11. Ramanujan and pi (1988)

    • Jonathan M. Borwein, Peter B. Borwein

    Pages 165-174

12. Ramanujan, modular equations, and approximations to pi or how to compute one billion digits of pi (1989)

    • Jonathan M. Borwein, Peter B. Borwein, David H. Bailey

    Pages 175-195

13. Pi, Euler numbers, and asymptotic expansions (1989)

    • Jonathan M. Borwein, Peter B. Borwein, Karl Dilcher

    Pages 197-205

14. A spigot algorithm for the digits of π (1995)

    • Stanley Rabinowitz, Stan Wagon

    Pages 207-217

15. On the rapid computation of various polylogarithmic constants (1997)

    • David H. Bailey, Peter B. Borwein, Simon Plouffe

    Pages 219-231

16. Similarities in irrationality proofs for π, ln 2, ζ(2), and ζ(3) (2001)

    • Dirk Huylebrouck

    Pages 233-244

17. Unbounded spigot algorithms for the digits of pi (2006)

    • Jeremy Gibbons

    Pages 245-257

18. Mathematics by experiment: Plausible reasoning in the 21st Century (2008)

    • David H. Bailey, Jonathan M. Borwein

    Pages 259-291

19. Approximations to π derived from integrals with nonnegative integrands (2009)

    • Stephen K. Lucas

    Pages 293-301

20. Ramanujan’s series for 1/π: A survey (2009)

    • Nayandeep Deka Baruah, Bruce C. Berndt, Heng Huat Chan

    Pages 303-325

This book contains a compendium of 25 papers published since the 1970s dealing with pi and associated topics of mathematics and computer science.  The collection begins with a Foreword by Bruce Berndt. Each contribution is preceded by a brief summary of its content as well as a short key word list indicating how the content relates to others in the collection. The volume includes articles on actual computations of pi, articles on mathematical questions related to pi (e.g., “Is pi normal?”), articles presenting new and often amazing techniques for computing digits of pi (e.g., the “BBP” algorithm for pi, which permits one to compute an arbitrary binary digit of pi without needing to compute any of the digits that came before), papers presenting important fundamental mathematical results relating to pi, and papers presenting new, high-tech techniques for analyzing pi (i.e., new graphical techniques that permit one to visually see if pi and other numbers are “normal”).  

This volume is a companion to Pi: A Source Book whose third edition released in 2004.  The present collection begins with 2 papers from 1976, published by Eugene Salamin and Richard Brent, which describe “quadratically convergent” algorithms for pi and other basic mathematical functions, derived from some mathematical work of Gauss. Bailey and Borwein hold that these two papers constitute the beginning of the modern era of computational mathematics.  This time period (1970s) also corresponds with the introduction of high-performance computer systems (supercomputers), which since that time have increased relentlessly in power, by approximately a factor of 100,000,000, advancing roughly at the same rate as Moore’s Law of semiconductor technology.  This book may be of interest to a wide range of mathematical readers; some articles cover more advanced research questions suitable for active researchers in the field, but several are highly accessible to undergraduate mathematics students.

• Catalan's constant
• Euler numbers
• approximations to pi
• arithmetic-geometric mean Gauss
• asymptotic expansions
• computation pi
• ladies diary
• pi normal
• proofs pi
• spigot algorithms

“Pi: The Next Generation is compiled as a sourcebook on the recent history of π from 1975 on, and on computational issues. … Reading the papers in this book I found many aspects on the mathematics and history of π which I did not know before and I enjoyed reading it very much. As the older book on π this one will also soon become a standard reference tool for working mathematicians and historians of mathematics alike.” (Thomas Sonar, London Mathematical Society Newsletter, newsletter.lms.ac.uk, November, 2017)

“Each reprinted paper is accompanied by a brief introduction explaining its significance. The papers range from historical surveys to popular expositions to research articles. Although I knew most of the papers already, I still found it delightful to browse at random. It would make a good selection for a high school or college library.” (Jeffrey O. Shallit, Mathematical Reviews, May, 2017)

• Berkeley, USA

  David H. Bailey

• Centre for Computer Assisted Research Ma, University of Newcastle, Newcastle, Australia

  Jonathan M. Borwein

David H. Bailey currently has two affiliations for his professional research work. Dr. Bailey is Senior Scientist, Computational Research Department, Lawrence Berkeley National Laboratory from which he officially retired in June 2013 but continues as an active researcher. Since February 2013, Bailey is also a Research Fellow, Department of Computer Science, University of California, Davis.


Jonathan M. Borwein is currently Laureate Professor in the School of Mathematical and Physical Sciences at the University of Newcastle (NSW) with adjunct appointments at Dalhousie and at Simon Fraser. He received his Doctorate from Oxford in 1974, and has published extensively in optimization, analysis, and computational mathematics, and has received various prizes both for research and for exposition. He directs the University of Newcastle’s Priority Research Centre in Computer Assisted Research Mathematics and its Applications (CARMA).

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