The Expanding Universe of Numbers

  • W. A. Coppel
Part of the Universitext book series (UTX)


For many people, numbers must seem to be the essence of mathematics. Number theory, which is the subject of this book, is primarily concerned with the properties of one particular type of number, the ‘whole numbers’ or integers. However, there are many other types, such as complex numbers and p-adic numbers. Somewhat surprisingly, a knowledge of these other types turns out to be necessary for any deeper understanding of the integers.

In this introductory chapter we describe several such types (but defer the study of p-adic numbers to Chapter VI). To embark on number theory proper the reader may proceed to Chapter II now and refer back to the present chapter, via the Index, only as occasion demands.


Vector Space Rational Number Nonempty Subset Identity Element Associative Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Selected References

  1. [1]
    L.V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill, New York, 1978.Google Scholar
  2. [2]
    N.I. Akhiezer and I.M. Glazman, Theory of linear operators in Hilbert space, English transl. by E.R. Dawson based on 3rd Russian ed., Pitman, London, 1981.Google Scholar
  3. [3]
    D. Amir, Characterizations of inner product spaces, Birkhäuser, Basel, 1986.MATHGoogle Scholar
  4. [4]
    M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, Mass., 1969.MATHGoogle Scholar
  5. [5]
    F.V. Atkinson, The reversibility of a differentiable mapping, Canad. Math. Bull. 4 (1961), 161–181.MATHMathSciNetGoogle Scholar
  6. [6]
    G. Birkhoff, Lattice theory, corrected reprint of 3rd ed., American Mathematical Society, Providence, R.I., 1979.MATHGoogle Scholar
  7. [7]
    G. Birkhoff and S. MacLane, A survey of modern algebra, 3rd ed., Macmillan, New York, 1965.Google Scholar
  8. [8]
    F. van der Blij, History of the octaves, Simon Stevin 34 (1961), 106–125.MATHMathSciNetGoogle Scholar
  9. [9]
    H. Bohr, Almost periodic functions, English transl. by H. Cohn and F. Steinhardt, Chelsea, New York, 1947.Google Scholar
  10. [10]
    G. Boole, An investigation of the laws of thought, on which are founded the mathematical theories of logic and probability, reprinted, Dover, New York, 1957. [Original edition, 1854]Google Scholar
  11. [11]
    C. Caratheodory, Theory of functions of a complex variable, English transl. by F. Steinhardt, 2 vols., 2nd ed., Chelsea, New York, 1958/1960.Google Scholar
  12. [12]
    G. Cardano, The great art or the rules of algebra, English transl. by T.R. Witmer, M.I.T. Press, Cambridge, Mass., 1968. [Latin original, 1545]Google Scholar
  13. [13]
    E. Cartan, Nombres complexes, Encyclopédie des sciences mathématiques, Tome I, Fasc. 4, Art. I.5, Gauthier-Villars, Paris, 1908. [Reprinted in Oeuvres complètes, Partie II, Vol. 1, pp. 107–246.]Google Scholar
  14. [14]
    G. Chichilnisky, Topology and invertible maps, Adv. inAppl. Math. 21 (1998), 113–123.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    W.K. Clifford, Mathematical Papers, reprinted, Chelsea, New York, 1968.Google Scholar
  16. [16]
    E.A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955.MATHGoogle Scholar
  17. [17]
    C. Corduneanu, Almost periodic functions, English transl. by G. Berstein and E. Tomer, Interscience, New York, 1968.Google Scholar
  18. [18]
    T. Dantzig, Number: The language of science, 4th ed., Pi Press, Indianapolis, IN, 2005.Google Scholar
  19. [19]
    R. Dedekind, Essays on the theory of numbers, English transl. by W.W. Beman, reprinted, Dover, New York, 1963.Google Scholar
  20. [20]
    R. Deheuvels, Formes quadratiques et groupes classiques, Presses Universitaires de France, Paris, 1981.MATHGoogle Scholar
  21. [21]
    J. Dieudonné, Foundations of modern analysis, enlarged reprint, Academic Press, New York, 1969.MATHGoogle Scholar
  22. [22]
    J. Dieudonné et al.,Abrégé d'histoire des mathématiques 1700–1900, reprinted, Hermann, Paris, 1996.Google Scholar
  23. [23]
    J. Dugundji and A. Granas, Fixed point theory I, PWN, Warsaw, 1982.MATHGoogle Scholar
  24. [24]
    H.-D. Ebbinghaus et al., Numbers, English transl. of 2nd German ed. by H.L.S. Orde, Springer-Verlag, New York, 1990.Google Scholar
  25. [25]
    L. Euler, Introduction to analysis of the infinite, Book I, English transl. by J.D. Blanton, Springer-Verlag, New York, 1988.Google Scholar
  26. [26]
    W. Fleming, Functions of several variables, 2nd ed., Springer-Verlag, New York, 1977.MATHGoogle Scholar
  27. [27]
    L.R. Ford Jr. and D.R. Fulkerson, Flows in networks, Princeton University Press, Princeton, N.J., 1962.MATHGoogle Scholar
  28. [28]
    F.R. Gantmacher, The theory of matrices, English transl. by K.A. Hirsch, 2 vols., Chelsea, New York, 1959.Google Scholar
  29. [29]
    M. Hall, The theory of groups, reprinted, Chelsea, New York, 1976.MATHGoogle Scholar
  30. [30]
    P.R. Halmos, Lectures on Boolean algebras, Van Nostrand, Princeton, N.J., 1963.MATHGoogle Scholar
  31. [31]
    P.R. Halmos, Finite-dimensional vector spaces, 2nd ed., reprinted, Springer-Verlag, New York, 1974.MATHGoogle Scholar
  32. [32]
    G.G. Hamedani and G.G. Walter, A fixed point theorem and its application to the central limit theorem, Arch. Math. 43 (1984), 258–264.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    E. Hellinger and O. Toeplitz, Integralgleichungen und Gleichungen mit unendlichvielen Unbekannten, reprinted, Chelsea, New York, 1953. [Original edition, 1928]Google Scholar
  34. [34]
    L. Henkin, On mathematical induction, Amer. Math. Monthly 67 (1960), 323–338.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    I.N. Herstein, Topics in algebra, reprinted, Wiley, London, 1976.Google Scholar
  36. [36]
    I.N. Herstein, Noncommutative rings, reprinted, Mathematical Association of America, Washington, D.C., 1994.Google Scholar
  37. [37]
    R.A. Horn and C.A. Johnson, Matrix analysis, corrected reprint, Cambridge University Press, 1990.Google Scholar
  38. [38]
    J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1990.Google Scholar
  39. [39]
    E.V. Huntingdon, Boolean algebra: A correction, Trans. Amer. Math. Soc. 35 (1933), 557–558.MathSciNetGoogle Scholar
  40. [40]
    J. Jachymski, A short proof of the converse to the contraction principle and some related results, Topol. Methods Nonlinear Anal. 15 (2000), 179–186.MATHMathSciNetGoogle Scholar
  41. [41]
    N. Jacobson, Basic Algebra I,II, 2nd ed., Freeman, New York, 1985/1989.Google Scholar
  42. [42]
    F. Kasch, Modules and rings, English transl. by D.A.R. Wallace, Academic Press, London, 1982.Google Scholar
  43. [43]
    B.M. Kiernan, The development of Galois theory from Lagrange to Artin, Arch. Hist. Exact Sci. 8 (1971), 40–154.MATHCrossRefMathSciNetGoogle Scholar
  44. [44]
    W. Kulpa, The Poincaré–Miranda theorem, Amer. Math. Monthly 104 (1997), 545–550.MATHCrossRefMathSciNetGoogle Scholar
  45. [45]
    E. Kunz, Introduction to commutative algebra and algebraic geometry, English transl. by M. Ackerman, Birkhäuser, Boston, Mass., 1985.Google Scholar
  46. [46]
    T.Y. Lam, A first course in noncommutative rings, Springer-Verlag, New York, 1991.MATHGoogle Scholar
  47. [47]
    E. Landau, Foundations of analysis, English transl. by F. Steinhardt, 3rd ed., Chelsea, New York, 1966. [German original, 1930]Google Scholar
  48. [48]
    S. Lang, Algebra, corrected reprint of 3rd ed., Addison-Wesley, Reading, Mass., 1994.Google Scholar
  49. [49]
    W. Maak, Fastperiodische Funktionen, Springer-Verlag, Berlin, 1950.MATHGoogle Scholar
  50. [50]
    A.I. Mal'cev, Foundations of linear algebra, English transl. by T.C. Brown, Freeman, San Francisco, 1963.Google Scholar
  51. [51]
    B.H. Matzat, Über das Umkehrproblem der Galoisschen Theorie, Jahresber. Deutsch. Math.-Verein. 90 (1988), 155–183.MATHMathSciNetGoogle Scholar
  52. [52]
    K. Menninger, Number words and number symbols, English transl. by P. Broneer, M.I.T. Press, Cambridge, Mass., 1969.Google Scholar
  53. [53]
    L. Mirsky, Transversal theory, Academic Press, London, 1971.MATHGoogle Scholar
  54. [54]
    P. Morandi, Field and Galois theory, Springer, New York, 1996.MATHGoogle Scholar
  55. [55]
    T. Nagahara and H. Tominaga, Elementary proofs of a theorem of Wedderburn and a theorem of Jacobson, Abh. Math. Sem. Univ. Hamburg 41 (1974), 72–74.MATHCrossRefMathSciNetGoogle Scholar
  56. [56]
    R. Narasimhan, Complex analysis in one variable, Birkhäuser, Boston, Mass., 1985.MATHGoogle Scholar
  57. [57]
    A.H. Read, Higher derivatives of analytic functions from the standpoint of functional analysis, J. London Math. Soc. 36 (1961), 345–352.MATHCrossRefMathSciNetGoogle Scholar
  58. [58]
    F. Riesz and B. Sz.-Nagy, Functional analysis, English transl. by L.F. Boron of 2nd French ed., F. Ungar, New York, 1955.Google Scholar
  59. [59]
    H. Rothe, Systeme geometrischer Analyse, Encyklopädie der Mathematischen Wissenschaften III 1.2, pp. 1277–1423, Teubner, Leipzig, 1914–1931.Google Scholar
  60. [60]
    J.J. Rotman, An introduction to the theory of groups, 4th ed., Springer-Verlag, New York, 1995.MATHGoogle Scholar
  61. [61]
    J. Rotman, Galois theory, 2nd ed., Springer-Verlag, New York, 1998.MATHGoogle Scholar
  62. [62]
    S. Rudeanu, Boolean functions and equations, North-Holland, Amsterdam, 1974.MATHGoogle Scholar
  63. [63]
    W. Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill, New York, 1976.MATHGoogle Scholar
  64. [64]
    N.A. Salingaros and G.P. Wene, The Clifford algebra of differential forms, Acta Appl. Math. 4 (1985), 271–292.MATHCrossRefMathSciNetGoogle Scholar
  65. [65]
    D.B. Shapiro, Products of sums of squares, Exposition. Math. 2 (1984), 235–261.MATHMathSciNetGoogle Scholar
  66. [66]
    R. Sikorski, Boolean algebras, 3rd ed., Springer-Verlag, New York, 1969.MATHGoogle Scholar
  67. [67]
    T.A. Springer and F.D. Veldkamp, Octonions, Jordan algebras, and exceptional groups, Springer, Berlin, 2000.MATHGoogle Scholar
  68. [68]
    E. Steinitz, Algebraische Theorie der Körper, reprinted, Chelsea, New York, 1950.Google Scholar
  69. [69]
    M.H. Stone, The representation of Boolean algebras, Bull. Amer. Math. Soc. 44 (1938), 807–816.CrossRefMathSciNetGoogle Scholar
  70. [70]
    K.D. Stroyan and W.A.J. Luxemburg, Introduction to the theory of infinitesimals, Academic Press, New York, 1976.MATHGoogle Scholar
  71. [71]
    A. Sudbery, Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), 199–225.MATHCrossRefMathSciNetGoogle Scholar
  72. [72]
    C.T.C. Wall, A geometric introduction to topology, reprinted, Dover, New York, 1993.Google Scholar
  73. [73]
    A. Zygmund, Trigonometric series, 3rd ed., Cambridge University Press, 2003.Google Scholar

Additional References

  1. J.C. Baez, The octonions, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 145–205.MATHCrossRefMathSciNetGoogle Scholar
  2. J.H. Conway and D.A. Smith, On quaternions and octonions: their geometry, arithmetic and symmetry, A.K. Peters, Natick, Mass., 2003.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • W. A. Coppel
    • 1
  1. 1.GriffithAustralia

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