Pi: a source book
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2004
|
| Edition: | 3. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 797 Seiten graphische Darstellungen |
| ISBN: | 0387205713 |
Staff View
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Record in the Search Index
| _version_ | 1804132890140737536 |
|---|---|
| adam_text | Contents
Preface to the Third Edition v
Preface to the Second Edition vi
Preface vii
Acknowledgments x
Introduction xvii
1. The Rhind Mathematical Papyrus Problem 50 (~1650 B.C.) 1
A problem dealing with the area of a round field of given diameter.
2. Engels. Quadrature of the Circle in Ancient Egypt (1977) 3
A conjectural explanation of how the mathematicians of ancient Egypt approximated
the area of a circle.
3. Archimedes. Measurement of a Circle (~250 B.C.) 7
The seminal work in which Archimedes presents the first true algorithm for n.
4. Phillips. Archimedes the Numerical Analyst (1981) 15
A summary of Archimedes work on the computation of it using modern notation.
5. Lam and Ang. Circle Measurements in Ancient China (1986) 20
This paper discusses and contains a translation of Liu Hui s (3rd century) method for
evaluating n and also examines values for n given by Zu Chongzhi (429 500)
6. The Banu Musa: The Measurement of Plane and Solid Figures (~850) 36
This extract gives an explicit statement and proof that the ratio of the circumference to
the diameter is constant.
7. Madhava. The Power Series for Arctan and Pi (~1400) 45
These theorems by a fifteenth century Indian mathematician give Gregory s series for
arctan with remainder terms and Leibniz s series for rt.
8. Hope Jones. Ludolph (or Ludolff or Lucius) van Ceulen (1938) 51
Correspondence about van Ceulen s tombstone in reference to it containing some digits
ofn.
xii Contents
9. Viete. Variorum de Rebus Mathematicis Reponsorum Liber VII (1593) 53
Two excerpts. One containing the first infinite expression ofn, obtained by relating the
area of a regular 2n gon to that of a regular n gon.
10. Wallis. Computation of k by Successive Interpolations (1655) 68
How Wallis derived the infinite product for n that bears his name.
11. Wallis. Arithmetica Infinitorum (1655)
An excerpt including Prop. 189, 191 and an alternate form of the result that gives Wm.
Brounker s continued fraction expression for 4/pi.
12. Huygens. De Circuli Magnitudine Inventa (1654) 81
Huygens s demonstration of how to triple the number of correct decimals over those in
Archimedes estimate ofn.
13. Gregory. Correspondence with John Collins (1671) 87
A letter to Collins in which he gives his series for arctangent, carried to the ninth
power.
14. Roy. The Discovery of the Series Formula for tt by Leibniz, Gregory,
and Nilakantha (1990) 92
A discussion of the discovery of the series n/4 = 1 — 1/3 + 1/5 ...
15. Jones. The First Use of tt for the Circle Ratio (1706) 108
An excerpt from Jones book, the Synopsis Palmariorum Matheseos: or, a New
Introduction to the Mathematics, London, 1706.
16. Newton. Of the Method of Fluxions and Infinite Series (1737) 110
An excerpt giving Newton s calculation of it to 16 decimal places.
17. Euler. Chapter 10 of Introduction to Analysis of the Infinite (On the Use
of the Discovered Fractions to Sum Infinite Series) (1748) 112
This includes many ofEuler s infinite series for n and powers ofn.
18. Lambert. Memoire Sur Quelques Proprietes Remarquables Des
Quantites Transcendentes Circulaires et Logarithmiques (1761) 129
An excerpt from Lambert s original proof of the irrationality ofn.
19. Lambert. Irrationality of 7T (1969) 141
A translation and Struik s discussion of Lambert s proof of the irrationality ofn.
20. Shanks. Contributions to Mathematics Comprising Chiefly of the
Rectification of the Circle to 607 Places of Decimals (1853) 147
Pages from Shanks s report of his monumental hand calculation ofn.
21. Hermite. Sur La Fonction Exponentielle (1873) 162
The first proof of the transcendence of e.
22. Lindemann. Ueber die Zahl 7T (1882) 194
The first proof of the transcendence ofn.
23. Weierstrass. Zu Lindemann s Abhandlung Uber die Ludolphsche
Zahl (1885) 207
Weierstrass proof of the transcendence ofn.
24. Hilbert. Ueber die Transzendenz der Zahlen e und tt (1893) 226
Hilbert s short and elegant simplification of the transcendence proofs for e and n.
25. Goodwin. Quadrature of the Circle (1894) 230
The dubious origin of the attempted legislation of the value ofn in Indiana.
Contents xiii
26. Edington. House Bill No. 246, Indiana State Legislature, 1897 (1935) 231
A summary of the action taken by the Indiana State Legislature to fix the value of it
(including a copy of the actual bill that was proposed).
27. Singmaster. The Legal Values of Pi (1985) 236
A history of the attempt by Indiana to legislate the value of re.
28. Ramanujan. Squaring the Circle (1913) 240
A geometric approximation to n.
29. Ramanujan. Modular Equations and Approximations to n (1914) 241
Ramanujan s seminal paper on pi that includes a number of striking series and
algebraic approximations.
30. Watson. The Marquis and the Land Agent: A Tale of the Eighteenth
Century (1933)
A Presidential address to the Mathematical Association in which the author gives an
account of some of the elementary work on arcs and ellipses and other curves which
led up to the idea of inverting an elliptic integral, and so laying the foundations of
elliptic functions and doubly periodic functions generally.
31. Ballantine. The Best (?) Formula for Computing it to a Thousand
Places (1939) 271
An early attempt to orchestrate the calculation ofn more cleverly.
32. Birch. An Algorithm for Construction of Arctangent Relations (1946) 274
The object of this note is to express n/4 as a sum ofarctan relations in powers of 10.
33. Niven. A Simple Proof that n is Irrational (1947) 276
A very concise proof of the irrationality ofn.
34. Reitwiesner. An ENIAC Determination of n and e to 2000 Decimal
Places (1950) 277
One of the first computer based computations.
35. Schepler. The Chronology of Pi (1950) 282
A fairly reliable outline of the history ofn from 3000 B.C. to 1949.
36. Mahler. On the Approximation of ?r (1953) 306
The aim of this paper is to determine an explicit lower bound free of unknown
constants for the distance of n from a given rational or algebraic number.
37. Wrench, Jr. The Evolution of Extended Decimal Approximations to tt
(1960) 319
A history of the calculation of the digits ofn to 1960.
38. Shanks and Wrench, Jr. Calculation of it to 100,000 Decimals (1962) 326
A landmark computation ofn to more than 100,000 places.
39. Sweeny. On the Computation of Euler s Constant (1963) 350
The computation of Euler s constant to 3566 decimal places.
40. Baker. Approximations to the Logarithms of Certain Rational
Numbers (1964) 359
The main purpose of this deep and fundamental paper is to deduce results concerning
the accuracy with which the natural logarithms of certain rational numbers may be
approximated by rational numbers, or, more generally, by algebraic numbers of
bounded degree.
xiv Contents
41. Adams. Asymptotic Diophantine Approximations to e (1966) 368
An asymptotic estimate for the rational approximation to e which disproves the
conjecture that e behaves like almost all numbers in this respect.
42. Mahler. Applications of Some Formulae by Hermite to the
Approximations of Exponentials of Logarithms (1967) 372
An important extension ofHilbert s approach to the study of transcendence.
43. Eves. In Mathematical Circles; A Selection of Mathematical Stories
and Anecdotes (excerpt) (1969) 400
A collection of mathematical stories and anecdotes about tt.
44. Eves. Mathematical Circles Revisited; A Second Collection of
Mathematical Stories and Anecdotes (excerpt) (1971) 402
A further collection of mathematical stories and anecdotes about n.
45. Todd. The Lemniscate Constants (1975) 412
A unifying account of some of the methods used for computing the lemniscate constants.
46. Salamin. Computation of tt Using Arithmetic Geometric Mean (1976) 418
The first quadratically converging algorithm for tt based on Gauss s AGM and on
Legendre s relation for elliptic integrals.
47. Brent. Fast Multiple Precision Evaluation of Elementary Functions
(1976) 424
This paper contains the Gauss Legendre method and some different algorithms for
log and exp (using Landen transformations).
48. Beukers. A Note on the Irrationality of C(2) and C(3) (1979) 434
A short and elegant recasting of Apery s proof of the irrationality o/£(3) (and f (2)j.
49. van der Poorten. A Proof that Euler Missed... Apery s Proof of the
Irrationality of C(3) (1979) 439
An illuminating account of Apery s astonishing proof of the irrationality o/£(3).
50. Brent and McMillan. Some New Algorithms for High Precision
Computation of Euler s Constant (1980) 448
Several new algorithms for high precision calculation of Euler s constant, including
one which was used to compute 30,100 decimal places.
51. Apostol. A Proof that Euler Missed: Evaluating £(2) the Easy Way
(1983) 456
This note shows that one of the double integrals considered by Beukers ([48] in the
table of contents) can be used to establish directly that f (2) = pi2/6.
52. O Shaughnessy. Putting God Back in Math (1983) 458
An article about the Institute of Pi Research, an organization that pokes fun at
creationists by pointing out that even the Bible makes mistakes.
53. Stern. A Remarkable Approximation to ix (1985) 460
Justification of the value ofn in the Bible through numerological interpretations.
54. Newman and Shanks. On a Sequence Arising in Series for tt (1984) 462
More connections between tt and modular equations.
55. Cox. The Arithmetic Geometric Mean of Gauss (1984) 481
An extensive study of the complex analytic properties of the AGM.
Contents xv
56. Borwein and Borwein. The Arithmetic Geometric Mean and Fast
Computation of Elementary Functions (1984) 537
The relationship between the AGM iteration and fast computation of elementary
functions (one of the by products is an algorithm for n).
57. Newman. A Simplified Version of the Fast Algorithms of Brent and
Salamin (1984) 553
Elementary algorithms for evaluating e* and it using the Gauss AGM without explicit
elliptic function theory.
58. Wagon. Is Pi Normal? (1985) 557
A discussion of the conjecture that re has randomly distributed digits.
59. Keith. Circle Digits: A Self Referential Story (1986) 560
A mnemonic for the first 402 decimal places of it.
60. Bailey. The Computation of n to 29,360,000 Decimal Digits Using
Borwein s Quartically Convergent Algorithm (1988) 562
The algorithms used, both for n and for performing the required multiple precision
arithmetic.
61. Kanada. Vectorization of Multiple Precision Arithmetic Program and
201,326,000 Decimal Digits of n Calculation (1988) 576
Details of the computation and statistical tests of the first 200 million digits ofn.
62. Borwein and Borwein. Ramanujan and Pi (1988) 588
This article documents Ramanujan s life, his ingenious approach to calculating n, and
how his approach is now incorporated into modern computer algorithms.
63. Chudnovsky and Chudnovsky. Approximations and Complex
Multiplication According to Ramanujan (1988) 596
This excerpt describes Ramanujan s original quadratic period—quasiperiod relations
for elliptic curves with complex multiplication and their applications to representations
of fractions ofn and other logarithms in terms of rapidly convergent nearly integral
(hypergeometric) series.
64. Borwein, Borwein and Bailey. Ramanujan, Modular Equations, and
Approximations to Pi or How to Compute One Billion Digits of Pi
(1989) 623
An exposition of the computation ofn using mathematics rooted in Ramanujan s work.
65. Borwein, Borwein and Dilcher. Pi, Euler Numbers, and Asymptotic
Expansions (1989) 642
An explanation as to why the slowly convergent Gregory series for n, truncated at
500,000 terms, gives n to 40places with only the 6th, 17th, 18th, and 29th places being
incorrect.
66. Beukers, Bezivin, and Robba. An Alternative Proof of the
Lindemann Weierstrass Theorem (1990) 649
The Lindemann Weierstrass theorem as a by product of a criterion for rationality of
solutions of differential equations.
67. Webster. The Tale of Pi (1991) 654
Various anecdotes about n from the 14th annual IMO Lecture to the Royal Society.
xvi Contents
68. Eco. An excerpt from Foucault s Pendulum (1993) 658
The unnumbered perfection of the circle itself.
69. Keith. Pi Mnemonics and the Art of Constrained Writing (1996) 659
A mnemonic for n based on Edgar Allen Poe s poem The Raven.
70. Bailey, Borwein, and Plouffe. On the Rapid Computation of Various
Polylogarithmic Constants (1997) 663
A fast method for computing individual digits ofn in base 2.
Appendix I—On the Early History of Pi 677
Appendix II—A Computational Chronology of Pi 683
Appendix III—Selected Formulae for Pi 686
Appendix IV—Translations of Viete and Huygens 690
Bibliography 710
Credits 717
A Pamphlet on Pi 721
Contents 723
1. Pi and Its Friends 725
2. Normality of Numbers 741
3. Historia Cyclometrica 753
4. Demotica Cyclometrica 771
References 779
Index 783
|
| any_adam_object | 1 |
| author | Berggren, J. L. 1941- Borwein, Jonathan M. 1951-2016 Borwein, Peter B. 1953-2020 |
| author_GND | (DE-588)122813553 (DE-588)115617884 (DE-588)1018056440 |
| author_facet | Berggren, J. L. 1941- Borwein, Jonathan M. 1951-2016 Borwein, Peter B. 1953-2020 |
| author_role | aut aut aut |
| author_sort | Berggren, J. L. 1941- |
| author_variant | j l b jl jlb j m b jm jmb p b b pb pbb |
| building | Verbundindex |
| bvnumber | BV019424048 |
| callnumber-first | Q - Science |
| callnumber-label | QA484 |
| callnumber-raw | QA484 |
| callnumber-search | QA484 |
| callnumber-sort | QA 3484 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SG 590 SK 180 |
| ctrlnum | (OCoLC)53814116 (DE-599)BVBBV019424048 |
| dewey-full | 516.22 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.22 |
| dewey-search | 516.22 |
| dewey-sort | 3516.22 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| era | Geschichte gnd Geschichte 1650 v. Chr.-1996 gnd |
| era_facet | Geschichte Geschichte 1650 v. Chr.-1996 |
| format | Book |
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| genre | (DE-588)4135952-5 Quelle gnd-content |
| genre_facet | Quelle |
| id | DE-604.BV019424048 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T20:00:00Z |
| institution | BVB |
| isbn | 0387205713 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-012885638 |
| oclc_num | 53814116 |
| open_access_boolean | |
| owner | DE-12 DE-29 DE-20 DE-11 DE-19 DE-BY-UBM DE-706 |
| owner_facet | DE-12 DE-29 DE-20 DE-11 DE-19 DE-BY-UBM DE-706 |
| physical | XIX, 797 Seiten graphische Darstellungen |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Springer |
| record_format | marc |
| spelling | Berggren, J. L. 1941- (DE-588)122813553 aut Pi a source book Len Berggren ; Jonathan M. Borwein ; Peter Borwein 3. ed. New York [u.a.] Springer 2004 XIX, 797 Seiten graphische Darstellungen txt rdacontent n rdamedia nc rdacarrier Geschichte gnd rswk-swf Geschichte 1650 v. Chr.-1996 gnd rswk-swf Pi (Le nombre) Pi Geschichte (DE-588)4020517-4 gnd rswk-swf Pi Zahl (DE-588)4174646-6 gnd rswk-swf (DE-588)4135952-5 Quelle gnd-content Pi Zahl (DE-588)4174646-6 s Geschichte z DE-604 Geschichte 1650 v. Chr.-1996 z 1\p DE-604 Geschichte (DE-588)4020517-4 s 2\p DE-604 Borwein, Jonathan M. 1951-2016 (DE-588)115617884 aut Borwein, Peter B. 1953-2020 (DE-588)1018056440 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012885638&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Berggren, J. L. 1941- Borwein, Jonathan M. 1951-2016 Borwein, Peter B. 1953-2020 Pi a source book Pi (Le nombre) Pi Geschichte (DE-588)4020517-4 gnd Pi Zahl (DE-588)4174646-6 gnd |
| subject_GND | (DE-588)4020517-4 (DE-588)4174646-6 (DE-588)4135952-5 |
| title | Pi a source book |
| title_auth | Pi a source book |
| title_exact_search | Pi a source book |
| title_full | Pi a source book Len Berggren ; Jonathan M. Borwein ; Peter Borwein |
| title_fullStr | Pi a source book Len Berggren ; Jonathan M. Borwein ; Peter Borwein |
| title_full_unstemmed | Pi a source book Len Berggren ; Jonathan M. Borwein ; Peter Borwein |
| title_short | Pi |
| title_sort | pi a source book |
| title_sub | a source book |
| topic | Pi (Le nombre) Pi Geschichte (DE-588)4020517-4 gnd Pi Zahl (DE-588)4174646-6 gnd |
| topic_facet | Pi (Le nombre) Pi Geschichte Pi Zahl Quelle |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012885638&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT berggrenjl piasourcebook AT borweinjonathanm piasourcebook AT borweinpeterb piasourcebook |