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Pi : A Source Book - J. Lennart Berggren

Hardcover

Published: 17th June 2004
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This book documents the history of pi from the dawn of mathematical time to the present. One of the beauties of the literature on pi is that it allows for the inclusion of very modern, yet accessible, mathematics. The articles on pi collected herein include selections from the mathematical and computational literature over four millennia, a variety of historical studies on the cultural significance of the number, and an assortment of anecdotal, fanciful, and simply amusing pieces.

For this new edition, the authors have updated the original material while adding new material of historical and cultural interest. There is a substantial exposition of the recent history of the computation of digits of pi, a discussion of the normality of the distribution of the digits, new translations of works by Viete and Huygen, as well as Kaplansky's never-before-published "Song of Pi."

From the reviews of earlier editions:

"Few mathematics books serve a wider potential readership than does a source book and this particular one is admirably designed to cater for a broad spectrum of tastes: professional mathematicians with research interest in related subjects, historians of mathematics, teachers at all levels searching out material for individual talks and student projects, and amateurs who will find much to amuse and inform them in this leafy tome. The authors are to be congratulated on their good taste in preparing such a rich and varied banquet with which to celebrate pi."
- Roger Webster for the Bulletin of the LMS

"The judicious representative selection makes this a useful addition to one's library as a reference book, an enjoyable survey of developments and a source of elegant and deep mathematics of different eras."
- Ed Barbeau for MathSciNet

"Full of useful formulas and ideas, it is a vast source of inspiration to any mathematician, A level and upwards-a necessity in any maths library."
- New Scientist

From the reviews:

"Few mathematics books serve a wider potential readership than does a source book and this particular one is admirably designed to cater for a broad spectrum of tastes: professional mathematicians with research interest in related subjects, historians of mathematics, teachers at all levels searching out material for individual talks and student projects, and amateurs who will find much to amuse and inform them in this leafy tome. The authors are to be congratulated on their good taste in preparing such a rich and varied banquet with which to celebrate pi."

Roger Webster for the Bulletin of the LMS

"The judicious representative selection makes this a useful addition to one's library as a reference book, an enjoyable survey of developments and a source of elegant and deep mathematics of different eras."

Ed Barbeau for MathSciNet

"Full of useful formulas and ideas, it is a vast source of inspiration to any mathematician, A level and upwards-a necessity in any maths library."

New Scientist

"Should be on every mathematician's coffee-table! ... The seventy articles comprising the source-book proper range from historical articles and classic by such players as Wallis, Huyghens, Newton, and Euler, to the articles on irrationality and transcendence ... . Pi: A source Book is truly an amazing book, irresistible in its own way, and filled with gems. And once it's on your coffee-table, feel free to do more than just browse: it's pretty well-suited for more in-depth study ... . Obviously the book is highly recommended." (Michael Berg, MathDL, January, 2001)

From the reviews of the third edition:

"This is the third edition of the by now classical Pi: a source book. ... contains some notes on the computation of individual (binary) digits of p, some considerations on the normality of the decimal expansion of p, and two more sections on the history of p. This book is still a classic work of reference for anyone with an interest in fascinating Pi." (F. Beukers, Mathematical Reviews, 2005h)

"The book documents the history of pi from the dawn of mathematical time to the present. One of the beauties of the literature on pi is that it allows for the inclusion of very modern, yet accessible, mathematics. The articles on pi collected herein include selections from the mathematical and computational literature over four millennia, a variety of historical studies on the cultural significance of the number, and an assortment of anecdotal, fanciful, and simply amusing pieces." (Zentralblatt fur Didaktik der Mathematik, November, 2004)

"This is the third edition of a comprehensive selection of about 70 articles on the number pi and related constants. ... This edition contains a new supplement on the recent history of the computation of digits of pi ... . Furthermore, new translations of articles by Viete and Huygens have been added. Altogether, this volume provides a fascinating overview of many hundred years of research and will delight and enlighten amateur lovers of pi and professional mathematicians alike." (Ch. Baxa, Monatshefte fur Mathematik, Vol. 148 (1), 2006)

"This is a fascinating reference book, which consists almost entirely of facsimiles of 70 articles about pi, followed by appendices which look at its early history as well as much computational information. ... it should be something to appear in any mathematical library, as it does contain so much material. ... There is something for all levels of readers in this book, from the 14-year old who may wish to know a little of the history, to the professional mathematician seeking information ... ." (Anthony C. Robin, The Mathematical Gazette, Vol. 90 (518), 2006)

The Rhind mathematical papyrus-problem 50 (ca. 1650 B.C.)p. 1
Engels : Quadrature of the circle in ancient Egypt (1977)p. 3
Archimedes : Measurement of a circle (ca. 250 B.C.)p. 7
Phillips : Archimedes the numerical analyst (1981)p. 15
Lam and Ang : Circle measurements in ancient China (1986)p. 20
The Banu Musa : the measurement of plane and solid figures (ca. 850)p. 36
Madhava : The power series for Arctan and Pi (ca. 1400)p. 45
Hope-Jones : Ludolph (or Ludolff or Lucius) van Ceulen (1938)p. 51
Viete : Variorum de Rebus mathematicis reponsorum liber VII (1593)p. 53
Wallis : Computation of [pi] by successive interpolations (1655)p. 68
Wallis : Arithmetica infinitorum (1655)p. 78
Huygens : De Circuli magnitudine inventa (1654)p. 81
Gregory : correspondence with John Collins (1671)p. 87
Roy : The discovery of the series formula for [pi] by Leibniz, Gregory, and Nilakantha (1990)p. 92
Jones : The first use of [pi] for the circle ratio (1706)p. 108
Newton : Of the method of fluxions and infinite series (1737)p. 110
Euler : chapter 10 of Introduction to analysis of the infinite (on the use of the discovered fractions to sum infinite series) (1748)p. 112
Lambert : Memoire Sur Quelques Proprietes Remarquables Des Quantites Transcendentes Circulaires et Logarithmiques (1761)p. 129
Lambert : Irrationality of [pi] (1969)p. 141
Shanks : Contributions to mathematics comprising chiefly of the rectification of the circle to 607 places of decimals (1853)p. 147
Hermite : Sur La Fonction Exponentielle (1873)p. 162
Lindemann : Ueber die Zahl [pi] (1882)p. 194
Weiserstrass : Zu Lindemann's Abhandlung "Uber die Ludolphsche Zahl" (1885)p. 207
Hilbert : Ueber die Transzendenz der Zahlen e und [pi] (1893)p. 226
Goodwin : Quadrature of the circle (1894)p. 230
Edington : House bill no. 246, Indiana State Legislature, 1897 (1935)p. 231
Singmaster : The legal values of Pi (1985)p. 236
Ramanujan : Squaring the circle (1913)p. 240
Ramanujan : Modular equations and approximations to [pi] (1914)p. 241
Watson. The Aarquis and the land agent : a tale of the eighteenth century (1933)
Ballantine. The best (?) formula for computing [pi] to a thousand places (1939)p. 271
Birch. An algorithm for construction of arctangent relations (1946)p. 274
Niven. A simple proof that [pi] is irrational (1947)p. 276
Reitwiesner. An ENLAC determination of [pi] and e to 2000 decimal places (1950)p. 277
Schepler. The chronology of Pi (1950)p. 282
Mahler. On the approximation of [pi] (1953)p. 306
Wrench, Jr. the evolution of extended decimal approximations to [pi] (1960)p. 319
Shanks and Wrench, Jr. calculation of [pi] to 100,000 decimals (1962)p. 326
Sweeny. On the computation of Euler's constant (1963)p. 350
Baker. Approximations to the logarithms of certain rational numbers (1964)p. 359
Adams. Asymptotic diophantine approximations to e (1966)p. 368
Mahler. Applications of some formulae by hermite to the approximations of exponentials of logarithms (1967)p. 372
Eves. In mathematical circles; a selection of mathematical stories and anecdotes (excerpt) (1969)p. 400
Eves. Mathematical circles revisited; a second collection of mathematical stories and anecdotes (excerpt) (1971)p. 402
Todd. The lemniscate constants (1975)p. 412
Salamin. Computation of [pi] using arithmetic-geometric mean (1976)p. 418
Brent. Fast multiple-precision evaluation of elementary functions (1976)p. 424
Beukers. A note on the irrationality of [Zeta](2) and [Zeta](3) (1979)p. 434
Van der Poorten. A proof that Euler missed ... Apery's proof of the irrationality of [Zeta](3) (1979)p. 439
Brent and McMillan. Some new algorithms for high-precision computation of Euler's constant (1980)p. 448
Apostol. A proof that Euler missed : evaluating [zeta](2) the easy way (1983)p. 456
O'Shaughnessy. Putting God back in math (1983)p. 458
Stern. A remarkable approximation to [pi] (1985)p. 460
Newman and Shanks. On a sequence arising in series for [pi] (1984)p. 462
Cox. The arithmetic-geometric mean of gauss (1984)p. 481
Borwein and Borwein. The arithmetic-geometric mean and fast computation of elementary functions (1984)p. 537
Newman. A simplified version of the fast algorithms of Brent and Salamin (1984)p. 553
Wagon. Is Pi normal? (1985)p. 557
Keith. Circle digits : a self-referential story (1986)p. 560
Bailey. The computation of [pi] to 29,360,000 decimal digits using Borwein's quartically convergent algorithm (1988)p. 562
Kanada. Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of [pi] calculation (1988)p. 576
Borwein and Borwein. Ramanujan and Pi (1988)p. 588
Chudnovsky and Chudnovsky. Approximations and complex multiplication according to Ramanujan (1988)p. 596
Borwein, Borwein and Bailey. Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi (1989)p. 623
Borwein, Borwein and Dilcher. Pi, Euler numbers, and asymptotic expansions (1989)p. 642
Beukers, Bezivin, and Robba. An alternative proof of the Lindemann-Weierstrass theorem (1990)p. 649
Webster. The tale of Pi (1991)p. 654
Eco. An except from Foucault's pendulum (1993)p. 658
Keith. Pi mnemonics and the art of constrained writing (1996)p. 659
Bailey, Borwein, and Plouffe. On the rapid computation of various polylogarithmic constraints (1997)p. 663
On the early history of Pip. 677
A computational chronology of Pip. 683
Selected formulae for Pip. 686
Translations of Viete and Huygensp. 690
A pamphlet on Pip. 721
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387205717
ISBN-10: 0387205713
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 797
Published: 17th June 2004
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 25.4 x 17.8  x 4.2
Weight (kg): 3.88
Edition Number: 3
Edition Type: Revised