Pi: A Source Book [3ed.] 1441919155, 9781441919151, 9780387205717, 0387205713

This book documents the history of pi from the dawn of mathematical time to the present. One of the beauties of the lite

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English Pages 797 [819] Year 2004

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Pi: A Source Book [3ed.]
1441919155, 9781441919151, 9780387205717, 0387205713

Author / Uploaded
Berggren
Lennart; Borwein
Peter B.; Borwein
Jonathan M

Table of contents :
Front Matter....Pages i-xix
Extract from the Rhind Papyrus....Pages 1-2
Quadrature of the Circle in Ancient Egypt....Pages 3-6
Measurement of a Circle....Pages 7-14
Archimedes the Numerical Analyst....Pages 15-19
Circle Measurements in Ancient China....Pages 20-35
[V.] The Ratio of the Diameter of any Circle to its Circumference is One [That is, is The Same for All Circles].....Pages 36-44
Appendix....Pages 45-50
Correspondence. Ludolph (or Ludolff or Lucius) van Ceulen....Pages 51-52
Capvt XVIII....Pages 53-67
Wallis. Computation of π by Successive Interpolations....Pages 68-77
Wallis. Arithmetica Infinitorum....Pages 78-80
Huygens. De Circuli Magnitudine Inventa (1654)....Pages 81-86
Gregory. Correspondence with John Collins (1671)....Pages 87-91
The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha....Pages 92-107
The First Use of π for the Circle Ratio....Pages 108-109
Newton. Of the Method of Fluxions and Infinite Series (1737)....Pages 110-111
On the Use of the Discovered Factors to Sum Infinite Series....Pages 112-128
Mémoire sur Quelques Propriétés Remarquables des Quantités Transcendentes Circulaires et Logarithmiques....Pages 129-140
Lambert. Irrationality of π....Pages 141-146
Contributions to Mathematics Comprising Chiefly the Rectification of the Circle to 607 Places of Decimals....Pages 147-161
Sur La Fonction Exponentielle....Pages 162-193
Ueber die Zahl π....Pages 194-206
Zu Lindemann ’s Abhandlung: „Über die Ludolph ’sche Zahl”....Pages 207-225
Ueber die Transcendenz der Zahlen e und π....Pages 226-229
Quadrature of the Circle....Pages 230-230
House Bill No. 246, Indiana State Legislature, 1897....Pages 231-235
The Legal Values of Pi....Pages 236-239
Squaring the Circle....Pages 240-240
Modular Equations and Approximations to π....Pages 241-257
The Marquis and the Land-Agent; A Tale of the Eighteenth Century....Pages 258-270
The Best (?) Formula for Computing π to a Thousand Places....Pages 271-273
An Algorithm for the Construction of Arctangent Relations....Pages 274-275
A Simple Proof that π is Irrational....Pages 276-276
An ENIAC Determination of π and e to more than 2000 Decimal Places....Pages 277-281
The Chronology of P I....Pages 282-305
On the Approximation of π....Pages 306-318
The evolution of extended decimal approximations to π....Pages 319-325
Calculation of π to 100,000 Decimals....Pages 326-349
On the Computation of Euler’s Constant....Pages 350-358
Approximations to the logarithms of certain rational numbers....Pages 359-367
Asymptotic Diophantine Approximations to E....Pages 368-371
Applications of Some Formulae by Hermite to the Approximation of Exponentials and Logarithms....Pages 372-399
In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes....Pages 400-401
In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes....Pages 402-411
The Lemniscate Constants....Pages 412-417
Computation of π Using Arithmetic-Geometric Mean....Pages 418-423
Fast Multiple-Precision Evaluation of Elementary Functions....Pages 424-433
A Note on the Irrationality of ζ(2) and ζ(3)....Pages 434-438
A Proof that Euler Missed .......Pages 439-447
Some New Algorithms for High-Precision Computation of Euler’s Constant....Pages 448-455
A Proof that Euler Missed: Evaluating ζ(2) the Easy Way....Pages 456-457
Putting God Back In Math....Pages 458-459
69.30 A remarkable approximation to π ....Pages 460-461
On a Sequence Arising in Series for π ....Pages 462-480
The Arithmetic-Geometric Mean of Gauss....Pages 481-536
The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions....Pages 537-552
A Simplified Version of the Fast Algorithms of Brent and Salamin....Pages 553-556
Is π Normal?....Pages 557-559
Circle Digits A Self-Referential Story....Pages 560-561
The Computation of π to 29,360,000 Decimal Digits Using Borweins’ Quartically Convergent Algorithm....Pages 562-575
Vectorization of Multiple-Precision Arithmetic Program and 201,326,000 Decimal Digits of π Calculation....Pages 576-587
Ramanujan and Pi....Pages 588-595
Approximations and complex multiplication according to Ramanujan....Pages 596-622
Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi....Pages 623-641
Pi, Euler Numbers, and Asymptotic Expansions....Pages 642-648
An Alternative Proof of the Lindemann-Weierstrass Theorem....Pages 649-653
The Tail of π....Pages 654-657
The Deconstruction of Pi....Pages 658-658
Pi Mnemonics and the Art of Constrained Writing....Pages 659-662
On the Rapid Computation of Various Polylogarithmic Constants....Pages 663-676
Back Matter....Pages 677-797