An imaginary tale: The story of √-1

An Imaginary Tale: The Story of √-1

Volumn , Issue , 2010, Pages

  Nahin, Paul J ^a  

^a NONE

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References (86)

1. H. M. Edwards, Riemann's Zeta Function, Academic Press 1974, p. 166.
2. For more on this, see Aleksandar Ivić, The Riemann Zeta-Function, Wiley- Interscience 1985, particularly p. 197.
3. Richard J. Gillings, Mathematics in the Time of the Pharaohs, MIT Press 1972, pp. 246-47.
4. George Sarton, A History of Science (volume 1), Harvard University Press 1952, p. 39.
5. For interesting speculations on how the Egyptians might have reasoned, see Gillings (note 1), pp. 187-93, and B. L. van der Waerden, Science Awakening, P. Noordhoff 1954, pp. 34-35. This last book has a photograph of the frustum calculation portion of the MMP.
6. W. W. Beman, "A Chapter in the History of Mathematics," Proceedings of the American Association for the Advancement of Science 46 (1897):33-50.
7. The Ganita-Sara-Sangraha of Mahaviracarya (with English translation and notes by M. Rangacarya), The Government Press, Madras 1912, p. 7 (of the translation).
8. If you're wondering why I am ignoring the negative root, that's good. You should be wondering. The negative root is perfectly valid, but if you use it from this point on in the analysis you'll find that you'll get exactly the same answer as I will get with the positive root. Try it and see.
9. An informative and entertaining biography of Cardan's amazing life is the older but still recommended book by Oystein Ore, Cardano, the Gambling Scholar, Princeton 1953. Any man who is modern enough in intellect to solve the cubic, and still medieval enough to cast the horoscope for Christ, for which he was imprisoned in 1570 on the charge of heresy, is worth reading about.
10. A modern student of mathematics, science, and engineering would find this claim obvious. That is, given any complex number, its conjugate is the number with all occurrences of √-1 = i changed to -√-1 = -i.
11. E. T. Bell, The Development of Mathematics (2nd edition), McGraw-Hill 1945, p. 149.
12. R. B. McClenon, "A Contribution of Leibniz to the History of Complex Numbers," American Mathematical Monthly 30 (November 1930):369-74. Huygens was just as puzzled as was Leibniz, as is shown by his reply to Leibniz: "One would never have believed that √1+√-3 + √1-√-3 = 6 and there is something hidden in this which is incomprehensible to us."
13. Gilbert Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press 1986, p. 330.
14. La Geometrie was not a book in its own right, but rather it was "just" the third illustrative appendix to Descartes' Discourse on the Method of Reasoning and Seeking Truth in Science. It was called "the greatest single step ever made in the progress of the exact sciences" by John Stuart Mill, and it is a masterpiece. It is well worth the time and effort, especially by high school geometry teachers, to read the excellent English translation by David E. Smith and Marcia L. Latham, The Geometry of René Descartes, Dover 1954.
15. Descartes himself offers no proof but Smith and Latham (note 1) sketch a possible approach. Their suggestion is, I think, more involved than necessary, and there is a much easier solution. Hint: look at the triangle NQR in figure 2.4 and use the Pythagorean theorem twice to find QR. Then MQ = 1/2a - 1/2QR and MR = MQ + QR.
16. When the expression for T is real, then the smaller of the two possible values, the one given by using the minus sign, represents the time at which the man first catches the bus. But if the man does not hop onto the bus, but rather keeps on running, he will move out in front of the bus. You can easily verify that the man is indeed running faster than the bus is moving-after all, that's why he caught up to it. But the bus is accelerating and so eventually, at the second larger value of T, the bus catches up to the running man and again, for the second time, the condition xm = xb is satisfied.
17. For much more on this particular problem see Nathan Altshiller Court, "Imaginary Elements in Pure Geometry-What They Are and What They Are Not," Scripta Mathematica 17 (1951):55-64 and 190-201.
18. For more on the topic, in general, see J. L. S. Hatton, The Theory of the Imaginary in Geometry, Together with the Trigonometry of the Imaginary, Cambridge University Press 1920.
19. Florian Cajori, "Historical Note on the Graphic Representation of Imaginaries Before the Time of Wessel," American Mathematical Monthly 19 (October-November 1912):167-71.
20. Julian Lowell Coolidge, The Geometry of the Complex Domain, Oxford 1924, p. 14.
21. It is traditional to call Wessel a Norwegian, but in fact when he was born Norway was actually part of Denmark, and he spent most of his life in Denmark, dying in Copenhagen.
22. Wessel's paper was unearthed in 1895 by an antiquarian, and its significance recognized by the Danish mathematician Sophus Christian Juel (1855-1935).
23. For more on this discovery, see Viggo Brun, "Caspar Wessel et l'introduction géométrique des numbres complexes," Revue d'Histoire des Sciences et de Leurs Applications 12 (1959):19-24.
24. E. T. Bell, Men of Mathematics. Simon and Schuster 1986, p. 234.
25. o, and try again."
26. James Gleick, Genius, Pantheon 1992, p. 35.
27. See my book The Science of Radio, AIP Press 1995, pp. 173-75, for how such expressions occur in the theory of single-sideband radio.
28. The exchanges that appeared in the Annales were reproduced by Hoüel's book, and much of it was again reproduced in the English translation of Hoüel's reprint that was prepared by Dartmouth mathematics professor A. S. Hardy, Imaginary Quantities, D. Van Nostrand 1881.
29. Julian Lowell Coolidge, The Geometry of the Complex Domain, Oxford 1924, p. 24.
30. G. Windred, "History of the Theory of Imaginary and Complex Quantities," The Mathematical Gazette 14 (1929):533-41.
31. The mathematician was Sylvestre Francois Lacroix (1765-1843), who was famous for his enormously influential textbooks.
32. It is briefly discussed by Coolidge and Windred (notes 7 and 8), with the general assessment being that Mourey's book (La Vraie Théorie des Quantités Négatives et des Prétendues Imaginaires) was a work of quality.
33. See Bell's biographical essay on Hamilton, "An Irish Tragedy," Men of Mathematics (note 3), pp. 340-61.
34. John O'Neill, "Formalism, Hamilton and Complex Numbers," Studies in History and Philosophy of Science 17 (September 1986):351-72.
35. Quoted from Umberto Bottazzini, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag 1986, p. 96.
36. John Stillwell, Mathematics and Its History, Springer-Verlag 1991, p. 188.
37. A. R. Forsyth, "Old Tripos Days at Cambridge," The Mathematical Gazette 19 (1935):162-79.
38. The numerical examples used in this section are taken from Juan E. Sornito, "Vector Representation of Multiplication and Division of Complex Numbers," Mathematics Teacher 47 (May 1954):320-22, 382.
39. See, for example, E. F. Krause, Taxicab Geometry, Dover 1986, where the so-called city-block distance function ds = dx + dy is explored in great detail. The geometric interpretation of special relativity is actually due to the German mathematician Hermann Minkowski (1864-1909), who was at one time Einstein's teacher. The theory, however, is all Einstein's.
40. See, for example, my book Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction, AIP Press 1993, pp. 287-303.
41. This is actually only true for what is called the flat spacetime of special relativity. Curved spacetimes, such as occur in the general theory of relativity (Einstein's theory of gravity), have more complicated metrics. You can find more on this in my Time Machines (note 3), p. 314.
42. For why Einstein made this assumption see my Time Machines (note 3), pp. 291 and 302-3.
43. In 1889 the Russian mathematician Sophie Kowalevski (1850-91) used complex time to study the mechanics of a rotating mass. Her work is described in Michèle Audin, Spinning Tops, Cambridge University Press 1996.
44. Paul R. Heyl, "The Skeptical Physicist," Scientific Monthly 46 (March 1938):225-29.
45. See also my Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction, AIP Press 1993, pp. 341-52.
46. Edward Kasner, "The Ratio of the Arc to the Chord of an Analytic Curve Need Not Be Unity," Bulletin of the American Mathematical Society 20 (July 1914):524-31.
47. A brief but elegant, geometry-only derivation of the inverse square law is in George Gamow's Gravity, Doubleday 1962.
48. See also S. K. Stein, "Exactly How Did Newton Deal with His Planets?" Mathematical Intelligencer 18 (Spring 1996):6-11.
49. The presentation in this section was inspired by Donald G. Saari, "A Visit to the Newtonian N-Body Problem via Elementary Complex Variables," American Mathematical Monthly 97 (February 1990):105-19.
50. William J. Kaufmann III, Universe (2nd edition), W. H. Freeman 1988, p. 56.
51. See my book The Science of Radio, AIP Press 1995, for the details of how that came to pass.
52. How he did this, and much more, can be found in Raymond Ayoub, "Euler and the Zeta Function," American Mathematical Monthly 81 (December 1974):1067-86.
53. See also Ronald Calinger, "Leonhard Euler: The First St. Petersburg Years (1727- 1741)," Historia Mathematica 23 (May 1996): 121-66.
54. -(π/2) = ii," American Mathematical Monthly 28 (March 1921):116-21.
55. For a full English translation of "Logometria" see Ronald Gowing, Roger Cotes-Natural Philosopher, Cambridge University Press 1983.
56. R. C. Archibald, "Euler Integrals and Euler's Spiral-Sometimes Called Fresnel Integrals and the Clothoide or Cornu's Spiral," American Mathematical Monthly 25 (June 1918):276-82.
57. Historical comments about, and citations to, the work of Cauchy and Poisson related to these integrals can be found in Horace Lamb, "On Deep-Water Waves," Proceedings of the London Mathematical Society 2 (November 10, 1904):371-400.
58. In two letters, dated October 13, 1729 and January 8, 1730, to one of his frequent correspondents, the German Christian Goldbach in Moscow. The name and symbol of gamma is, however, due to Legendre, who introduced the modern terminology in 1808. For a tutorial on this part of Euler's work, see Philip J. Davis, "Leonhard Euler's Integral: A Historical Profile of the Gamma Function," American Mathematical Monthly 66 (December 1959):849-69.
59. Much of my commentary on Cauchy's work is based on H. J. Ettlinger, "Cau-chy's Paper of 1814 on Definite Integrals," Annals of Mathematics 23 (1921-22):255-70
60. Philip E. B. Jourdain, "The Theory of Functions with Cauchy and Gauss," Bibliotheca Mathematica 6 (1905): 190-207.
61. Neither Ettlinger nor I follow Cauchy's original presentation or notation slavishly (Jourdain does, largely), although the ideas he developed will be pointed out. Very interesting reading, too, is the second half of volume 2 of the series Mathematics of the l9th Century, on the history of analytic function theory. Edited by A. N. Kolmogorov and A. P. Yushkevich, it was translated from the Russian by Roger Cooke and published by Birkhauser Verlag in 1996. I also found helpful the extensive historical commentary in Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press 1972, pp. 626-70.
62. Some of Kline's comments concerning the original debates over the 1814 memoir were clarified for me in I. Grattan-Guinness, The Development of the Foundations of Mathematical Analysis from Euler to Riemann, MIT Press 1979, pp. 24-45.
63. Finally, see Frank Smithies, Cauchy and the Creation of Complex Function Theory, Cambridge University Press 1997.
64. Cauchy didn't use this term, but the concept of analyticity is contained in his memoir. And, as Ettlinger (note 1) observes, "although geometrical representation is now an essential feature of every presentation of the theory of functions, Cauchy used neither figures nor geometrical language." Following Ettlinger and all modern texts I use both aids in this chapter a lot.
65. See volume 4 of Stokes' Mathematical and Physical Papers, Cambridge University Press 1904, pp. 77-109.
66. Jesper Ltitzen, Joseph Liouville 1809-1882: Master of Pure and Applied Mathematics, Springer-Verlag 1990, p. 586.
67. See also the chapter on integral theorems by J. J. Cross in Wranglers and Physicists: Studies on Cambridge Mathematical Physics in the Nineteenth Century (P. M. Harman, editor), Manchester University Press 1985.
68. I. Grattan-Guinness, "Why Did George Green Write His Essay of 1828 on Electricity and Magnetism?" American Mathematical Monthly 102 (May 1995):387-96.
69. The Polish-born mathematician Mark Kac tells the following wonderfully funny story in his autobiography Enigmas of Chance, Harper & Row 1985, p. 126. Once, when serving on a doctoral examination committee at Cornell, Kac asked the candidate some mathematical questions. As Kac tells it, "He was not terribly good-in mathematics at least. After he had failed to answer a couple of questions, I asked him a really simple one, which was to describe the behavior of the function 1/z in the complex plane. 'The function is analytic, sir, in the whole plane except at z = 0, where it has a singularity,' he answered and it was perfectly correct. 'What is the singularity called?' I continued. The student stopped in his tracks. 'Look at me,' I said. 'What am I?' His face lit up. 'A simple Pole, sir,' which is in fact the correct answer." What a nice person Kac must have been!
70. Please don't think I am being disrespectful by calling Cauchy's solution "awkward." Newer, better ways of solving problems are welcomed in mathematics and, indeed, are hoped for. The method I use in the text used to be a common textbook approach, but even it has been improved upon. The answer derived in the text is actually valid even for non-integer values of m and n, but the method I use does impose that restriction. You can find more discussion on this important integral, and its more general evaluation, in Orin J. Farrell and Bertram Ross, "Note on Evaluating Certain Real Integrals by Cauchy's Residue Theorem," American Mathematical Monthly 68 (February 1968):151-52.
71. Such a book is the one by A. David Wunsch, Complex Variables with Applications (2nd edition), Addison-Wesley 1994.
72. Wunsch, a professor of electrical engineering at the Lowell campus of the University of Massachusetts, has done an absolutely first-rate job of presenting the details of complex function theory, including conformal mapping, Laurent series, and system stability. He writes in the language of the engineer, while at the same time giving up none of the mathematical integrity of the subject. For the more mathematically inclined who also have an interest in keeping one foot in physical reality, I highly recommend Tristan Needham, Visual Complex Analysis, Oxford University Press 1997.
73. For a discussion of the details of doing this integral, i.e., of evaluating in the complex plane, see E. C. Titchmarsh (revised by D. R. Heath-Brown), The Theory of the Riemann Zeta-Function, Oxford University Press 1986, pp. 18-20.
74. This story is reprinted in The Early Asimov, Doubleday 1972.
75. Marilyn vos Savant, The World's Most Famous Math Problem (the Proof of Fermat's Last Theorem and Other Mathematical Mysteries), St. Martin's Press 1993, p. 61. This book, which the author happily tells us she wrote in mere weeks, is full of other similarly uninformed statements.
76. A reproduction of this holographic letter is in Scripta Mathematica 1 (1932): 88-90.
77. For more on all this, see John Stillwell, Mathematics and Its History, Springer-Verlag 1989, pp. 195-200.
78. An early, little-known dispute over the number of complex roots to a polynomial equation took place between the Scottish mathematician Colin MacLaurin (1698-1746) and his obscure countryman George Campbell (?-1766). A detailed discussion of this unhappy business is given by Stella Mills, "The Controversy Between Colin MacLaurin and George Campbell Over Complex Roots, 1728-1729," Archive for History of Exact Sciences 28 (1983): 149-64.
79. For even more on the state of knowledge concerning the roots of equations at that time see Robin Rider Hamburg, "The Theory of Equations in the 18th Century: The Work of Joseph Lagrange," Archive for History of Exact Sciences 16 (1976): 17-36.
80. For example, solve ix2 - 2x + 1 = 0 and show that x = - i ± √- 1 + i. Then show that x = - i + √- 1 + i and x = - i - √- 1 + i are not a conjugate pair. In fact, the conjugate of x = -i + √- 1 + i is i + √- 1 - i ≠ -i - √ - 1 + i.
81. E. T. Bell, The Development of Mathematics (2nd edition), McGraw-Hill 1945, p. 176.
82. James Pierpont, "On the Complex Roots of a Transcendental Equation Occurring in the Electron Theory," Annals of Mathematics 30 (1929):81-91.
83. The inspiration for Professor Pierpont was G. A. Shott, "The Theory of the Linear Oscillator and Its Bearing on the Electron Theory," Philosophical Magazine 3 (April 1927):739-52.
84. Newton (1642-1727) derived his fast-convergent series for π during 1665-66, when he fled plague-ridden London to return home to the family farm at Woolsthorpe in Lincolnshire. Thus, his series predates the one due to Leibniz-Gregory (discussed in chapter 6), but it did not appear in print until many years later, in the 1736 posthumous publication of an English translation (The Method of Fluxions and Infinite Series) of his original Latin manuscript. It was during this same period in Woolsthorpe that Newton discovered but didn't publish the power series expansion for ln(1 + x) before Mercator (see the discussion in section 6.3). This is the series used by Schellbach to calculate π from √- 1-again, as discussed in chapter 6-and which Newton used to calculate various "interesting" numbers out to as many as sixty-eight decimal places.
85. H. S. Uhler, "On the Numerical Value of ii," American Mathematical Monthly 28 (March 1921):114-16.
86. Kπ, COSH(Kπ) and SINH(Kπ) to 136 Figures," Proceedings of the National Academy of Sciences 33 (February 1947):34-41.