Tham khảo Định lý Pythagoras

  1. 1 2 Judith D. Sally; Paul Sally (2007). “Chapter 3: Pythagorean triples”. Roots to research: a vertical development of mathematical problems. American Mathematical Society Bookstore. tr. 63. ISBN 0-8218-4403-2. 
  2. 1 2 Posamentier, Alfred. The Pythagorean Theorem: The Story of Its Power and Beauty, p. 23 (Prometheus Books 2010).
  3. ↑ O'Connor, J J; Robertson, E F (tháng 12 năm 2000). “Pythagoras's theorem in Babylonian mathematics”. School of Mathematics and Statistics. University of St. Andrews, Scotland. Truy cập ngày 25 tháng 1 năm 2017. In this article we examine four Babylonian tablets which all have some connection with Pythagoras's theorem. Certainly the Babylonians were familiar with Pythagoras's theorem. 
  4. ↑ George Johnston Allman (1889). Greek Geometry from Thales to Euclid . Hodges, Figgis, & Co. tr. 26. ISBN 1-4326-0662-X. The discovery of the law of three squares, commonly called the "theorem of Pythagoras" is attributed to him by – amongst others – Vitruvius, Diogenes Laertius, Proclus, and Plutarch ... 
  5. 1 2 (Heath 1921, Vol I, p. 144)
  6. ↑ According to Heath 1921, Vol I, p. 147, Vitruvius says that Pythagoras first discovered the triangle (3,4,5); the fact that the latter is right-angled led to the theorem.
  7. 1 2 Otto Neugebauer (1969). The exact sciences in antiquity (ấn bản 2). Courier Dover Publications. tr. 36. ISBN 0-486-22332-9. . For a different view, see Dick Teresi (2003). Lost Discoveries: The Ancient Roots of Modern Science. Simon and Schuster. tr. 52. ISBN 0-7432-4379-X. , where the speculation is made that the first column of tablet 322 in the Plimpton collection supports a Babylonian knowledge of some elements of trigonometry. That notion is pretty much laid to rest, however, by Eleanor Robson (2002). “Words and Pictures: New Light on Plimpton 322”. The American Mathematical Monthly (Mathematical Association of America) 109 (2): 105–120. JSTOR 2695324. doi:10.2307/2695324.  (pdf file). The generally accepted view today is that the Babylonians had no awareness of trigonometric functions. See also Abdulrahman A. Abdulaziz (2010). "The Plimpton 322 Tablet and the Babylonian Method of Generating Pythagorean Triples". arΧiv:1004.0025 [math.HO].  §2, p. 7.
  8. ↑ Mario Livio (2003). The golden ratio: the story of phi, the world's most astonishing number. Random House, Inc. tr. 25. ISBN 0-7679-0816-3. 
  9. ↑ Benson, Donald. The Moment of Proof: Mathematical Epiphanies, pp. 172–173 (Oxford University Press, 1999).
  10. ↑ Maor, Eli. The Pythagorean Theorem: A 4,000-year History, p. 61 (Princeton University Press, 2007).
  11. ↑ (Loomis 1968)
  12. ↑ (Maor 2007, tr. 39)
  13. 1 2 Stephen W. Hawking (2005). God created the integers: the mathematical breakthroughs that changed history. Philadelphia: Running Press Book Publishers. tr. 12. ISBN 0-7624-1922-9. This proof first appeared after a computer program was set to check Euclidean proofs.
  14. ↑ See for example Pythagorean theorem by shear mapping, Saint Louis University website Java applet
  15. ↑ Jan Gullberg (1997). Mathematics: from the birth of numbers. W. W. Norton & Company. tr. 435. ISBN 0-393-04002-X. 
  16. ↑ Elements 1.47 by Euclid. Retrieved 19 December 2006.
  17. ↑ Euclid's Elements, Book I, Proposition 47: web page version using Java applets from Euclid's Elements by Prof. David E. Joyce, Clark University
  18. ↑ The proof by Pythagoras probably was not a general one, as the theory of proportions was developed only two centuries after Pythagoras; see (Maor 2007, tr. 25)
  19. ↑ Alexander Bogomolny. “Pythagorean theorem, proof number 10”. Cut the Knot. Truy cập ngày 27 tháng 2 năm 2010. 
  20. ↑ (Loomis 1968, tr. 113, Geometric proof 22 and Figure 123)
  21. ↑ Schroeder, Manfred Robert (2012). Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. Courier Corporation. tr. 3–4. ISBN 0486134784. 
  22. ↑ Alexander Bogomolny. “Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #3”. Cut the Knot. Truy cập ngày 4 tháng 11 năm 2010. 
  23. ↑ Alexander Bogomolny. “Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #4”. Cut the Knot. Truy cập ngày 4 tháng 11 năm 2010. 
  24. ↑ Published in a weekly mathematics column: James A Garfield (1876). “Pons Asinorum”. The New England Journal of Education 3 (14): 161.  as noted in William Dunham (1997). The mathematical universe: An alphabetical journey through the great proofs, problems, and personalities. Wiley. tr. 96. ISBN 0-471-17661-3.  and in A calendar of mathematical dates: April 1, 1876 by V. Frederick Rickey
  25. ↑ Prof. David Lantz' animation from his web site of animated proofs
  26. ↑ Mike Staring (1996). “The Pythagorean proposition: A proof by means of calculus”. Mathematics Magazine (Mathematical Association of America) 69 (1): 45–46. JSTOR 2691395. doi:10.2307/2691395. 
  27. ↑ Bogomolny, Alexander. “Pythagorean Theorem”. Interactive Mathematics Miscellany and Puzzles. Alexander Bogomolny. Truy cập ngày 9 tháng 5 năm 2010. 
  28. ↑ Bruce C. Berndt (1988). “Ramanujan – 100 years old (fashioned) or 100 years new (fangled)?”. The Mathematical Intelligencer 10 (3): 24. doi:10.1007/BF03026638. 
  29. ↑ Judith D. Sally; Paul J. Sally Jr. (21 tháng 12 năm 2007). “Theorem 2.4 (Converse of the Pythagorean theorem).”. Roots to Research. American Mathematical Society. tr. 54–55. ISBN 0-8218-4403-2. 
  30. ↑ Euclid's Elements, Book I, Proposition 48 From D.E. Joyce's web page at Clark University
  31. ↑ Casey, Stephen, "The converse of the theorem of Pythagoras", Mathematical Gazette 92, July 2008, 309–313.
  32. ↑ Mitchell, Douglas W., "Feedback on 92.47", Mathematical Gazette 93, March 2009, 156.
  33. ↑ Ernest Julius Wilczynski; Herbert Ellsworth Slaught (1914). “Theorem 1 and Theorem 2”. Plane trigonometry and applications. Allyn and Bacon. tr. 85. 
  34. ↑ Dijkstra, Edsger W. (7 tháng 9 năm 1986). “On the theorem of Pythagoras”. EWD975. E. W. Dijkstra Archive. 
  35. ↑ Law, Henry (1853). “Corollary 5 of Proposition XLVII (Pythagoras's Theorem)”. The Elements of Euclid: with many additional propositions, and explanatory notes, to which is prefixed an introductory essay on logic. John Weale. tr. 49. 
  36. ↑ Shaughan Lavine (1994). Understanding the infinite. Harvard University Press. tr. 13. ISBN 0-674-92096-1. 
  37. ↑ (Heath 1921, Vol I, pp. 65); Hippasus was on a voyage at the time, and his fellows cast him overboard. See James R. Choike (1980). “The pentagram and the discovery of an irrational number”. The College Mathematics Journal 11: 312–316. 
  38. 1 2 3 A careful discussion of Hippasus's contributions is found inKurt Von Fritz (tháng 4 năm 1945). “The Discovery of Incommensurability by Hippasus of Metapontum”. Annals of Mathematics. Second Series (Annals of Mathematics) 46 (2): 242–264. JSTOR 1969021. doi:10.2307/1969021. 
  39. ↑ Jon Orwant; Jarkko Hietaniemi; John Macdonald (1999). “Euclidean distance”. Mastering algorithms with Perl. O'Reilly Media, Inc. tr. 426. ISBN 1-56592-398-7. 
  40. ↑ Wentworth, George (2009). Plane Trigonometry and Tables. BiblioBazaar, LLC. tr. 116. ISBN 1-103-07998-0. , Exercises, page 116
  41. ↑ Lawrence S. Leff (2005). PreCalculus the Easy Way (ấn bản 7). Barron's Educational Series. tr. 296. ISBN 0-7641-2892-2. 
  42. ↑ WS Massey (tháng 12 năm 1983). “Cross products of vectors in higher-dimensional Euclidean spaces”. The American Mathematical Monthly (Mathematical Association of America) 90 (10): 697–701. JSTOR 2323537. doi:10.2307/2323537. 
  43. ↑ Pertti Lounesto (2001). “§7.4 Cross product of two vectors”. Clifford algebras and spinors (ấn bản 2). Cambridge University Press. tr. 96. ISBN 0-521-00551-5. 
  44. ↑ Francis Begnaud Hildebrand (1992). Methods of applied mathematics (ấn bản 2). Courier Dover Publications. tr. 24. ISBN 0-486-67002-3. 
  45. ↑ Heath, T. L., A History of Greek Mathematics, Oxford University Press, 1921; reprinted by Dover, 1981.
  46. ↑ Euclid's Elements: Book VI, Proposition VI 31: "In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle."
  47. 1 2 Putz, John F. and Sipka, Timothy A. "On generalizing the Pythagorean theorem", The College Mathematics Journal 34 (4), September 2003, pp. 291–295.
  48. ↑ Lawrence S. Leff (1 tháng 5 năm 2005). cited work. Barron's Educational Series. tr. 326. ISBN 0-7641-2892-2. 
  49. ↑ Howard Whitley Eves (1983). “§4.8:...generalization of Pythagorean theorem”. Great moments in mathematics (before 1650). Mathematical Association of America. tr. 41. ISBN 0-88385-310-8. 
  50. ↑ Aydin Sayili (tháng 3 năm 1960). “Thâbit ibn Qurra's Generalization of the Pythagorean Theorem”. Isis 51 (1): 35–37. JSTOR 227603. doi:10.1086/348837. 
  51. ↑ Judith D. Sally; Paul Sally (21 tháng 12 năm 2007). “Exercise 2.10 (ii)”. Roots to Research: A Vertical Development of Mathematical Problems. tr. 62. ISBN 0-8218-4403-2. 
  52. ↑ For the details of such a construction, see George Jennings (1997). “Figure 1.32: The generalized Pythagorean theorem”. Modern geometry with applications: with 150 figures (ấn bản 3). Springer. tr. 23. ISBN 0-387-94222-X. 
  53. ↑ Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN 9780883853481, pp. 77–78 (excerpt, tr. 77, tại Google Books)
  54. ↑ Rajendra Bhatia (1997). Matrix analysis. Springer. tr. 21. ISBN 0-387-94846-5. 
  55. ↑ For an extended discussion of this generalization, see, for example, Willie W. Wong 2002, A generalized n-dimensional Pythagorean theorem.
  56. ↑ Ferdinand van der Heijden; Dick de Ridder (2004). Classification, parameter estimation, and state estimation. Wiley. tr. 357. ISBN 0-470-09013-8. 
  57. ↑ Qun Lin; Jiafu Lin (2006). Finite element methods: accuracy and improvement. Elsevier. tr. 23. ISBN 7-03-016656-6. 
  58. ↑ Howard Anton; Chris Rorres (2010). Elementary Linear Algebra: Applications Version (ấn bản 10). Wiley. tr. 336. ISBN 0-470-43205-5. 
  59. 1 2 3 Karen Saxe (2002). “Theorem 1.2”. Beginning functional analysis. Springer. tr. 7. ISBN 0-387-95224-1. 
  60. ↑ Douglas, Ronald G. (1998). Banach Algebra Techniques in Operator Theory, 2nd edition. New York, New York: Springer-Verlag New York, Inc. tr. 60–61. ISBN 978-0-387-98377-6. 
  61. ↑ Stephen W. Hawking (2005). cited work. tr. 4. ISBN 0-7624-1922-9. 
  62. ↑ Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics (ấn bản 2). tr. 2147. ISBN 1-58488-347-2. The parallel postulate is equivalent to the Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean theorem. 
  63. ↑ Alexander R. Pruss (2006). The principle of sufficient reason: a reassessment. Cambridge University Press. tr. 11. ISBN 0-521-85959-X. We could include...the parallel postulate and derive the Pythagorean theorem. Or we could instead make the Pythagorean theorem among the other axioms and derive the parallel postulate. 
  64. ↑ Victor Pambuccian (tháng 12 năm 2010). “Maria Teresa Calapso's Hyperbolic Pythagorean Theorem”. The Mathematical Intelligencer 32 (4): 2. doi:10.1007/s00283-010-9169-0. 
  65. ↑ Barrett O'Neill (2006). “Exercise 4”. Elementary differential geometry (ấn bản 2). Academic Press. tr. 441. ISBN 0-12-088735-5. 
  66. ↑ Saul Stahl (1993). “Theorem 8.3”. The Poincaré half-plane: a gateway to modern geometry. Jones & Bartlett Learning. tr. 122. ISBN 0-86720-298-X. 
  67. ↑ Jane Gilman (1995). “Hyperbolic triangles”. Two-generator discrete subgroups of PSL(2,R). American Mathematical Society Bookstore. ISBN 0-8218-0361-1. 
  68. ↑ Tai L. Chow (2000). Mathematical methods for physicists: a concise introduction. Cambridge University Press. tr. 52. ISBN 0-521-65544-7. 
  69. ↑ Friberg, Joran (1991). “Methods and Traditions of Babylonian Mathematics” (PDF). Historia Mathematica 8: 277, 306. 
  70. ↑ Lynch, Patrick (12 tháng 4 năm 2016). “A 3,800-year journey from classroom to classroom”. Phys.org. Truy cập ngày 13 tháng 4 năm 2016. 
  71. ↑ Lederman, Leon and Teresi, Dick. The God Particle: If the Universe Is the Answer, What Is the Question?, p. 80 (Houghton Mifflin Harcourt 2006).
  72. ↑ Teresi, Dick. Lost Discoveries: The Ancient Roots of Modern Science – from the Babylonians to the Maya, p. 8 (Simon and Schuster 2010).
  73. ↑ (van_der_Waerden 1983, tr. 5)See also Frank Swetz; T. I. Kao (1977). Was Pythagoras Chinese?: An examination of right triangle theory in ancient China. Penn State Press. tr. 12. ISBN 0-271-01238-2. 
  74. ↑ Kim Plofker (2009). Mathematics in India. Princeton University Press. tr. 17–18, with footnote 13 for Sutra identical to the Pythagorean theorem. ISBN 0-691-12067-6. 
  75. ↑ Carl Benjamin Boyer; Uta C. Merzbach (2011). “China and India”. A history of mathematics, 3rd Edition. Wiley. tr. 229. ISBN 978-0470525487. Quote: [In Sulba-sutras,] we find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. Although Mesopotamian influence in the Sulvasũtras is not unlikely, we know of no conclusive evidence for or against this. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides. Less easily explained is another rule given by Apastamba – one that strongly resembles some of the geometric algebra in Book II of Euclid's Elements. (...) 
  76. ↑ Robert P. Crease (2008). The great equations: breakthroughs in science from Pythagoras to Heisenberg. W W Norton & Co. tr. 25. ISBN 0-393-06204-X. 
  77. ↑ A rather extensive discussion of the origins of the various texts in the Zhou Bi is provided by Christopher Cullen (2007). Astronomy and Mathematics in Ancient China: The 'Zhou Bi Suan Jing'. Cambridge University Press. tr. 139 ff. ISBN 0-521-03537-6. 
  78. ↑ This work is a compilation of 246 problems, some of which survived the book burning of 213 BC, and was put in final form before 100 AD. It was extensively commented upon by Liu Hui in 263 AD. Philip D Straffin, Jr. (2004). “Liu Hui and the first golden age of Chinese mathematics”. Trong Marlow Anderson; Victor J. Katz; Robin J. Wilson. Sherlock Holmes in Babylon: and other tales of mathematical history. Mathematical Association of America. tr. 69 ff. ISBN 0-88385-546-1.  See particularly §3: Nine chapters on the mathematical art, pp. 71 ff.
  79. ↑ Kangshen Shen; John N. Crossley; Anthony Wah-Cheung Lun (1999). The nine chapters on the mathematical art: companion and commentary. Oxford University Press. tr. 488. ISBN 0-19-853936-3. 
  80. ↑ In particular, Li Jimin; see Centaurus, Volume 39. Copenhagen: Munksgaard. 1997. tr. 193, 205. 
  81. ↑ Chen, Cheng-Yih (1996). “§3.3.4 Chén Zǐ's formula and the Chóng-Chã method; Figure 40”. Early Chinese work in natural science: a re-examination of the physics of motion, acoustics, astronomy and scientific thoughts. Hong Kong University Press. tr. 142. ISBN 962-209-385-X. 
  82. ↑ Wen-tsün Wu (2008). “The Gougu theorem”. Selected works of Wen-tsün Wu. World Scientific. tr. 158. ISBN 981-279-107-8. 
  83. ↑ (Euclid 1956, tr. 351) page 351
  84. ↑ An extensive discussion of the historical evidence is provided in (Euclid 1956, tr. 351) page=351
  85. ↑ Asger Aaboe (1997). Episodes from the early history of mathematics. Mathematical Association of America. tr. 51. ISBN 0-88385-613-1. ...it is not until Euclid that we find a logical sequence of general theorems with proper proofs. 

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