Overview: Pi is one of the few concepts in mathematics whose mention evokes a response of recognition and interest in those not concerned professionally with the subject. Yet, despite this, no source book on Pi has ever been published. Mathematicians and historians of mathematics will find this book indispensable. Teachers from the seventh grade onward will find ample resources for anything from special topic courses to individual talks and special student projects.
Author(s): Lennart Berggren, Jonathan Borwein, Peter Borwein (auth.)
Publisher: Springer New York
Year: 2004
Language: English
Tags: History of Mathematical Sciences; Real Functions; Numerical Analysis; Number Theory
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
