圓周率
參考資料 > 引用

↑  Sloane, N.J.A. (编). . The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80  Arndt & Haenel 2006
1 2  Kennedy, E. S., , Journal for the History of Astronomy: 65, Bibcode:1978JHA.....9...65K, doi:10.1177/002182867800900106 。克劳狄乌斯·托勒密使用了60进制下的三位小数去近似，随后卡西将其扩展到了九位小数。参见Aaboe, Asger, , New Mathematical Library 13, 纽约: Random House: 125, 1964  [2017-12-08], （原始内容存档于2017-02-01）
1 2  Sloane, N.J.A. (编). . The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
↑  George E. Andrews, Richard Askey, Ranjan Roy. . Cambridge University Press. 1999: 58. ISBN 0-521-78988-5.
↑  Gupta, R. C. . Ganita Bharati. 1992, 14 (1-4): 68–71.
↑  刘霞. . 科技日报微信公众号.   [2024-03-18]. （原始内容存档于2024-03-18）.
↑  David Bailey; Jonathan Borwein; Peter Borwein; Simon Plouffe, , The Mathematical Intelligencer, 1997, 19 (1): 50–56, doi:10.1007/bf03024340
1 2  Bronshteĭn & Semendiaev 1971，第200, 209頁
↑  . 中央通訊社. 2021-03-14  [2021-04-11]. （原始内容存档于2021-06-28）.
↑  . 衛報. 2015-03-13  [2021-04-12]. （原始内容存档于2015-07-22） （英语）.
↑  Boeing, Niels.   [The World is Pi]. Zeit Online. 2016-03-14  [2016-09-13]. （原始内容存档于2016-03-17） （德语）. 中文：“鲁道夫数”，或称“圆周率”，同样獲得一个如今广为人知的符号——威廉·琼斯1706年提出使用字母π，因为这是希腊语“周长”（περίμετρος）的开头字母。莱昂哈德·欧拉在其数学著作中確立了π的使用。
↑  . Dictionary.com.   [2017-07-15]. （原始内容存档于2017-07-18）.
↑  Tom Apostol,  2nd, Wiley, 1967 。 Page 102: "From a logical point of view, this is unsatisfactory at the present stage because we have not yet discussed the concept of arc length."  Arc length is introduced on page 529.
1 2 3  Remmert, Reinhold, , , Springer: 129, 1991
↑  Remmert (1991)。魏尔斯特拉斯使用的积分实为 \pi =\int _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}.
↑  Richard Baltzer, , Hirzel: 195, 1870  [2017-12-08], （原始内容存档于2017-01-29）
↑  Edmund Landau, , Noordoff: 193, 1934
1 2  Rudin, Walter. . McGraw-Hill. 1976. ISBN 0-07-054235-X. , p 183.
↑  Rudin, Walter. . McGraw-Hill. 1986. , p 2.
↑  Lars Ahlfors, , McGraw-Hill: 46, 1966
↑  Nicolas Bourbaki, , Springer, 1981 , §VIII.2
1 2  Nicolas Bourbaki, , Springer, 1979 , §II.3.
↑  Salikhov, V. . Russian Mathematical Survey. 2008, 53 (3): 570–572. Bibcode:2008RuMaS..63..570S. ISSN 0036-0279. doi:10.1070/RM2008v063n03ABEH004543.
1 2  Preuss, Paul. . Lawrence Berkeley National Laboratory. 2001-07-23  [2007-11-10]. （原始内容存档于2007-10-20）.
↑  Mayer, Steve. .   [2007-11-04]. （原始内容存档于2000-09-29）.
↑  Posamentier & Lehmann 2004，第25頁
↑  Eymard & Lafon 1999，第129頁
↑  Beckmann 1989，第37頁
Schlager, Neil; Lauer, Josh. . Gale Group. 2001. ISBN 0-7876-3933-8. ，第185页
1 2  Eymard & Lafon 1999，第78頁
↑  Sloane, N.J.A. (编). . The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.  Retrieved 12 April 2012.
↑  Lange, L. J. . The American Mathematical Monthly. 1999-05, 106 (5): 456–458. JSTOR 2589152. doi:10.2307/2589152.
↑  Ayers 1964，第100頁
1 2  Bronshteĭn & Semendiaev 1971，第592頁
↑  Maor, Eli, E: The Story of a Number, Princeton University Press, 2009, p 160, ISBN 978-0-691-14134-3 ("five most important" constants)
↑  Weisstein, Eric W. (编). . at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.
↑  大卫·希尔伯特; 理查·科朗特, , Wiley: 286&endash;290, 1966
↑  H Dym; H P McKean, , Academic Press: 47, 1972
↑  Isaac Chavel, , Cambridge University Press, 2001
↑  Capogna, L.; Danielli, D.,; Pauls, S.D.; Tyson, J., , Progress in Mathematics, Birkhäuser, 2007 , Chapter 7.
↑  Talenti, Giorgio, , Annali di Matematica Pura ed Applicata: 353–372,   [2016-09-13], ISSN 1618-1891, doi:10.1007/BF02418013, （原始内容存档于2017-01-29）
↑  L. Esposito; C. Nitsch; C. Trombetti, , arXiv:1110.2960
↑  M Del Pino; J Dolbeault, , Journal de Mathématiques Pures et Appliquées, 2002, 81 (9): 847–875, doi:10.1016/s0021-7824(02)01266-7
↑  Gerald Folland, , Princeton University Press: 5, 1989
↑  Howe 1980
↑  Bronshteĭn & Semendiaev 1971，第106–107, 744, 748頁
↑  Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, 1968, pp 174–190.
↑  H Dym; H P McKean, , Academic Press, 1972 ; Section 2.7
↑  Elias Stein; Guido Weiss, , Princeton University Press: 6, 1971 ; Theorem 1.13.
↑  Petrie, W.M.F. Wisdom of the Egyptians (1940)
↑  Verner, Miroslav. The Pyramids: The Mystery, Culture, and Science of Egypt's Great Monuments. Grove Press. 2001 (1997). ISBN 0-8021-3935-3
↑  Rossi, Corinna Architecture and Mathematics in Ancient Egypt, Cambridge University Press. 2007. ISBN 978-0-521-69053-9.
↑  Legon, J. A. R. On Pyramid Dimensions and Proportions (1991) Discussions in Egyptology (20) 第25-34页  （页面存档备份，存于）
↑  "We can conclude that although the ancient Egyptians could not precisely define the value of π, in practice they used it". Verner, M. . 2003. ，第70页。
Petrie. . 1940. ，第30页。
参见Legon, J. A. R. . Discussions in Egyptology. 1991, 20: 第25–34页  [2016-09-13]. （原始内容存档于2011-07-18）. .
参见Petrie, W. M. F. . Nature. 1925, 116 (2930期): 第942页. Bibcode:1925Natur.116..942P. doi:10.1038/116942a0.
↑  Egyptologist: Rossi, Corinna, Architecture and Mathematics in Ancient Egypt, Cambridge University Press, 2004, 第60-70页，200, ISBN 9780521829540.

对此的怀疑：Shermer, Michael（英语：）, The Skeptic Encyclopedia of Pseudoscience, ABC-CLIO, 2002, 第407-408页, ISBN 9781576076538.

参见Fagan, Garrett G., Archaeological Fantasies: How Pseudoarchaeology Misrepresents The Past and Misleads the Public, Routledge, 2006, ISBN 9780415305938.

若需要一个没有π参与的解释的列表，请见Roger Herz-Fischler. . Wilfrid Laurier University Press. 2000: 第67–77页，第165–166页  [2013-06-05]. ISBN 9780889203242. （原始内容存档于2016-11-29）.
↑  Chaitanya, Krishna. A profile of Indian culture. （页面存档备份，存于） Indian Book Company (1975). 第133页。
↑  . Mathworks.com.   [2013-03-12]. （原始内容存档于2013-02-25）.
↑  Boyer & Merzbach 1991，第168頁
↑  Arndt & Haenel 2006，第15–16, 175, 184–186, 205頁Grienberger在1630年已計算到39位小數，Sharp在1699年計算到71位小數
↑  《周髀算經》注中， 趙爽指出「圓徑一而周三，方徑一而匝四」。
1 2  Boyer & Merzbach 1991，第202頁
↑  . 臺灣商務印書館. 2006: 286. ISBN 978-957-05-2072-9.
↑  Azarian, Mohammad K. . Missouri Journal of Mathematical Sciences. 2010, 22 (2): 64–85  [2016-09-13]. （原始内容存档于2015-01-14）.
↑  O'Connor, John J.; Robertson, Edmund F. . MacTutor History of Mathematics archive. 1999  [2012-08-11]. （原始内容存档于2011-04-12）.
1 2 3  Arndt & Haenel 2006，第182頁
↑  Grienbergerus, Christophorus.  (PDF). 1630  [2016-09-13]. （原始内容 (PDF)存档于2014-02-01） （拉丁语）. 。其計算結果是3.14159 26535 89793 23846 26433 83279 50288 4196 < π < 3.14159 26535 89793 23846 26433 83279 50288 4199
↑  Roy 1990，第101–102頁
1 2 3  Roy 1990，第101–102頁
↑  Joseph 1991，第264頁
1 2  Arndt & Haenel 2006，第188頁。牛顿由Arndt引用。
↑  A060294
1 2  Eymard & Lafon 1999，第53–54頁
1 2  Borwein, J. M.; Borwein, P. B. . Scientific American. 1988, 256 (2): 112–117. Bibcode:1988SciAm.258b.112B. doi:10.1038/scientificamerican0288-112.
↑  Borwein, J. M.; Borwein, P. B.; Dilcher, K. . American Mathematical Monthly. 1989, 96 (8): 681–687. doi:10.2307/2324715.
↑  Arndt & Haenel 2006，第223頁, (formula 16.10). Note that (n − 1)n(n + 1) = n3 − n.
Wells, David.  revised. Penguin. 1997: 35. ISBN 978-0-140-26149-3.
1 2  Posamentier & Lehmann 2004，第284頁
↑  Lambert, Johann, "Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques", reprinted in Berggren, Borwein & Borwein 1997，第129–140頁
↑  Hardy and Wright 1938 and 2000:177 footnote §11.13-14 references Lindemann's proof as appearing at Math. Ann. 20 (1882), 213-25.
↑  cf Hardy and Wright 1938 and 2000:177 footnote §11.13-14. The proofs that e and π are transcendental can be found on pages 170-176. They cite two sources of the proofs at Landau 1927 or Perron 1910; see the "List of Books" at pages 417-419 for full citations.
↑  Arndt & Haenel 2006，第165頁. A facsimile of Jones' text is in Berggren, Borwein & Borwein 1997，第108–109頁
↑  參考Schepler 1950，第220頁： 威廉·奥特雷德用字母π來表示一個圓的周長
↑  Arndt & Haenel 2006，第197頁。参见 Reitwiesner 1950.
↑  Arndt & Haenel 2006，第第111页（5 倍）；第113–114（4 倍）頁.
具体算法情参见Borwein & Borwein 1987。
1 2 3  Bailey, David H.  (PDF). 2003-05-16  [2012-04-12]. （原始内容存档 (PDF)于2012-04-15）.
↑  Arndt & Haenel 2006，第17頁“39 digits of  \pi  are sufficient to calculate the volume of the universe to the nearest atom.”（中文：39位 \pi 已足以计算宇宙到其最近的原子的体积了。）
鉴于人们需要一些额外的数位来解决捨入誤差的问题，阿恩特称几百位小数足以应付任何科学计算。
↑  Schudel, Matt. . The Washington Post. 2009-03-25: B5.
↑  Connor, Steve. . The Independent (London). 2010-01-08  [2012-04-14]. （原始内容存档于2012-04-02）.
↑  Eymard & Lafon 1999，第254頁
↑  Fabrice Bellard. . www.semanticscholar.org. 2010-02-11  [2017-04-10].
1 2  . www.numberworld.org.   [2012-05-30]. （原始内容存档于2014-01-01）.
↑  PSLQ即Partial Sum of Least Squares，偏最小二乘和。
↑  Plouffe, Simon.  (PDF). 2006-04  [2009-04-10]. （原始内容存档 (PDF)于2012-01-14）.
↑  苏淳,  2nd, 北京: 科学出版社: 41, 2010
↑  苏淳,  2nd, 北京: 科学出版社: 34–35,41, 2010
↑  Posamentier & Lehmann 2004，第105頁
↑  Weisstein, Eric W. (编). . at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc.  [2016-09-18]. （原始内容存档于2016-11-18） （英语）.
↑  Grünbaum, B., , Trans. Amer. Math. Soc., 1960, 95: 451–465
↑  Posamentier & Lehmann 2004，第105–108頁
1 2  Gibbons, Jeremy, "Unbounded Spigot Algorithms for the Digits of Pi" （页面存档备份，存于）, 2005. Gibbons produced an improved version of Wagon's algorithm.
↑  Rabinowitz, Stanley; Wagon, Stan. . American Mathematical Monthly. 1995-03, 102 (3): 195–203. doi:10.2307/2975006.  A computer program has been created that implements Wagon's spigot algorithm in only 120 characters of software.
↑  Bailey, David H; Borwein, Peter B; and Plouffe, Simon.  (PDF). Mathematics of Computation. 1997-04, 66 (218): 903–913  [2016-09-13]. doi:10.1090/S0025-5718-97-00856-9. （原始内容存档 (PDF)于2012-07-22）.
↑  Arndt & Haenel 2006，第128頁。普勞夫有找到十進制的位數萃取演算法，但其速度比完整計算之前所有位數要慢。
↑  Bellards formula in: Bellard, Fabrice. .   [2007-10-27]. （原始内容存档于2007-09-12）.
↑  Palmer, Jason. . BBC News. 2010-09-16  [2011-03-26]. （原始内容存档于2011-03-17）.
↑  Weisstein, Eric W. (编). . at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.
1 2  Ayers 1964，第60頁
1 2  Bronshteĭn & Semendiaev 1971，第210–211頁
↑  Michael Spivak,  3, Publish or Perish Press, 1999 ; Chapter 6.
↑  Kobayashi, Shoshichi; Nomizu, Katsumi,  Vol. 2 New, Wiley Interscience: 293, 1996 ; Chapter XII Characteristic classes
↑  H. M. Schey (1996) Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, ISBN 0-393-96997-5.
↑  Yeo, Adrian, The pleasures of pi, e and other interesting numbers, World Scientific Pub., 2006, p 21, ISBN 978-981-270-078-0.

Ehlers, Jürgen, Einstein's Field Equations and Their Physical Implications, Springer, 2000, p 7, ISBN 978-3-540-67073-5.
↑  Lars Ahlfors, , McGraw-Hill: 115, 1966
↑  Weisstein, Eric W. (编). . at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.
↑  Joglekar, S. D., Mathematical Physics, Universities Press, 2005, p 166, ISBN 978-81-7371-422-1.
↑  Bronshteĭn & Semendiaev 1971，第191–192頁
↑  埃米爾·阿廷, , Athena series; selected topics in mathematics 1st, Holt, Rinehart and Winston, 1964
↑  Lawrence Evans, , AMS: 615, 1997 。
↑  Bronshteĭn & Semendiaev 1971，第190頁
↑  Benjamin Nill; Andreas Paffenholz, , Advances in Geometry, 2014, 14 (4): 579–586, ISSN 1615-7168, arXiv:1205.1270 , doi:10.1515/advgeom-2014-0001
↑  此理论由Ernesto Cesàro于1881年证明. For a more rigorous proof than the intuitive and informal one given here, see Hardy, G. H., An Introduction to the Theory of Numbers, Oxford University Press, 2008, ISBN 978-0-19-921986-5, theorem 332.
↑  Ogilvy, C. S.; Anderson, J. T., Excursions in Number Theory, Dover Publications Inc., 1988, pp. 29–35, ISBN 0-486-25778-9.
↑  Vladimir Platonov; Andrei Rapinchuk, , Academic Press: 262&endash;265, 1994
↑  Sondow, J., , Proc. Amer. Math. Soc., 1994, 120: 421–424, doi:10.1090/s0002-9939-1994-1172954-7
↑  T. Friedmann; C.R. Hagen. . Journal of Mathematical Physics. 2015, 56 (11). arXiv:1510.07813 . doi:10.1063/1.4930800.
↑  Tate, John T., , , Thompson, Washington, D.C.: 305–347, 1950, ISBN 978-0-9502734-2-6, MR 0217026
↑  H Dym; H P McKean, , Academic Press, 1972 ; Chapter 4
1 2  Mumford, David, , Boston: Birkhauser: 1–117, 1983, ISBN 3-7643-3109-7
↑  Sidney Port; Charles Stone, , Academic Press: 29, 1978
↑
Titchmarsh, E,  2nd, Oxford University: Clarendon Press, 19481986, ISBN 978-0-8284-0324-5 .
↑  Stein, Elias, , Princeton University Press, 1970 ; Chapter II.
1 2  Klebanoff, Aaron.  (PDF). Fractals. 2001, 9 (4): 393–402  [2012-04-14]. doi:10.1142/S0218348X01000828. （原始内容 (PDF)存档于2012-04-06）.
↑  Peitgen, Heinz-Otto, Chaos and fractals: new frontiers of science, Springer, 2004, pp. 801–803, ISBN 978-0-387-20229-7.
↑  Halliday, David; Resnick, Robert; Walker, Jearl, Fundamentals of Physics, 5th Ed., John Wiley & Sons, 1997, p 381, ISBN 0-471-14854-7.
↑  Imamura, James M. . University of Oregon. 2005-08-17  [2007-09-09]. （原始内容存档于2007-10-12）.
↑  C. Itzykson, J-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980.
↑  Low, Peter, Classical Theory of Structures Based on the Differential Equation, CUP Archive, 1971, pp 116–118, ISBN 978-0-521-08089-7.
↑  Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, 1967, p 233, ISBN 0-521-66396-2.
↑  Hans-Henrik Stølum. . Science. 1996-03-22, 271 (5256): 1710–1713. Bibcode:1996Sci...271.1710S. doi:10.1126/science.271.5256.1710.
↑  Posamentier & Lehmann 2004，第140–141頁
↑  . Guinness World Records.   [2016-09-13]. （原始内容存档于2016-02-14）.
↑  Otake, Tomoko. . The Japan Times. 2006-12-17  [2007-10-27]. （原始内容存档于2013-08-18）.
↑  Raz, A.; Packard, M. G. . Neurocase. 2009, 15: 361–372. PMC 4323087 . PMID 19585350. doi:10.1080/13554790902776896.
↑  Keith, Mike. .   [2009-07-29]. （原始内容存档于2009-01-18）.
↑  Keith, Michael. . Vinculum Press. 2010-02-17. ISBN 978-0963009715.
↑  Pickover, Clifford A., , Wiley & Sons: 59, 1995, ISBN 9780471118572
↑  Peterson, Ivars, , MAA spectrum, Mathematical Association of America: 17, 2002  [2017-12-08], ISBN 9780883855379, （原始内容存档于2017-02-01）
↑  BBC, ,   [2016-09-13], （原始内容存档于2014-12-23） , showing a visualization of the historically first exact formula, starting at 35 min and 20 sec into the second part of the documentary.
↑  Posamentier & Lehmann 2004，第118頁
↑  Arndt & Haenel 2006，第14頁这部分情节在改编的电影《超時空接觸》中省略。
↑  Gill, Andy. . The Independent. 2005-11-04  [2016-09-13]. （原始内容存档于2006-10-15）. the almost autistic satisfaction of the obsessive-compulsive mathematician fascinated by 'Pi' (which affords the opportunity to hear Bush slowly sing vast chunks of the number in question, several dozen digits long)
↑  . PI DAY. 2008  [2016-09-13]. （原始内容存档于2013-07-04）.
↑  . MIT.   [2012-04-12]. （原始内容存档于2009-01-19）.
↑  . USAToday.com. 2015-03-14  [2015-03-14]. （原始内容存档于2015-03-15）.
↑  . FinancialPost.com. Reuters. 2011-07-05  [2011-08-16]. （原始内容存档于2011-08-09）.
↑  Eagle, Albert. . Galloway and Porter, Ltd. 1958: ix.
↑  Sequence A019692
↑  Abbott, Stephen. . Math Horizons. 2012-04, 19 (4). ISSN 1072-4117. doi:10.4169/mathhorizons.19.4.34 （英语）.
↑  Palais, Robert. . The Mathematical Intelligencer. 2001-01, 23 (3). ISSN 0343-6993. doi:10.1007/BF03026846 （英语）.
↑  . Light Years - CNN. 2012-06-28  [2012-06-28]. （原始内容存档于2013-01-12）.
↑  . Telegraph India. 2011-06-30  [2016-09-13]. （原始内容存档于2013-07-13）.
↑  Posamentier & Lehmann 2004，第36–37頁
↑  Hallerberg, Arthur E. . Mathematics Magazine. 1977-05, 50 (3). ISSN 0025-570X. doi:10.1080/0025570X.1977.11976632 （英语）.