Numbers as a Source of Mathematical Ideas

  • Israel Kleiner


Number systems have been a fruitful source of concepts, results, and theories in the evolution of mathematics. In fact, it has been suggested that much even of modern mathematics has its roots in the study of number and shape [78, 79]. This chapter offers suggestions for introducing various mathematical topics related to, and often originating in, the study of number systems. The material is organized around eight themes, which vary in detail and difficulty, and may serve as source material for courses or topics of varied degrees of sophistication and be addressed to various audiences – for example teachers, mathematics majors, and liberal-arts enthusiasts. The themes deal with algebraic, analytic, geometric, number-theoretic, set-theoretic, cultural, and philosophical issues. Although the themes are interconnected, they can be read independently. In many cases, we sketch the historical origin of the mathematical ideas involved. No attempt is made to be thorough, but references to an extensive bibliography are provided throughout. Readers are invited to come up with their own themes to suit their interests, needs, and objectives. The material in the next chapter may serve as an example of a possible theme.


Seventeenth Century Number System Algebraic Number Negative Number Division Ring 
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.


  1. 1.
    W. W. Adams and L. J. Goldstein, Introduction to N umber Theory, Prentice-Hall, 1976.Google Scholar
  2. 2.
    J. Agnew, Explorations in Number Theory, Wadsworth, 1972.Google Scholar
  3. 3.
    T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.Google Scholar
  4. 4.
    A. Arcavi, M. Bruckheimer, and R. Ben-Zvi, Maybe a mathematics teacher can profit from the study of the history of mathematics, For the Learning of Math. 3:1 (1982) 30–37.Google Scholar
  5. 5.
    B. Artman, The Concept of Number: From Quaternions to Monads and Topological Fields, Wiley, 1988.Google Scholar
  6. 6.
    S. Avital, Don’t be blue, number two, Arithm. Teacher 34 (Sept. 1986) 42–45.Google Scholar
  7. 7.
    G. Bachman, Introduction to p-adic Numbers and Valuation Theory, Academic Press, 1964.Google Scholar
  8. 8.
    I. G. Bashmakova, Diophantus and Diophantine Equations, Math. Assoc. of Amer., 1997. (Translated from the Russian by A. Shenitzer.)Google Scholar
  9. 9.
    I. G. Bashmakova, Arithmetic of algebraic curves from Diophantus to Poincaré, Historia Math. 8:4 (1981) 393–416.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    I. G. Bashmakova and G. Smirnova, The Beginnings and Evolution of Algebra, Math. Assoc. of America, 2000. (Translated from the Russian by A. Shenitzer.)Google Scholar
  11. 11.
    P. Beckmann, A History of π, St. Martin’s Press, 1971.Google Scholar
  12. 12.
    A. H. Beiler, Recreations in the Theory of Numbers, Dover, 1964.Google Scholar
  13. 13.
    W. P. Berlinghoff and F. Q. Gouvea, Math Through the Ages: A Gentle History for Teachers and Others, expanded ed., Math. Assoc. of Amer., 2004.Google Scholar
  14. 14.
    L. Berggren, J. Borwein, and P. Borwein, Pi: A Source Book, Springer, 1997.Google Scholar
  15. 15.
    L. M. Blumenthal, A Modern View of Geometry, W. H. Freeman, 1961.MATHGoogle Scholar
  16. 16.
    Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, 1966.Google Scholar
  17. 17.
    U. Bottazzini, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag, 1986.Google Scholar
  18. 18.
    C. B. Boyer, A History of Mathematics, revised by U. C. Merzbach, Wiley & Sons, 1989.Google Scholar
  19. 19.
    C. B. Boyer, Fundamental steps in the development of numeration, Isis 35:2 (1944) 153–168.CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    F. E. Browder (ed.), Mathematical Developments Arising from Hilbert Problems, 2 Vols., Amer. Math. Soc, 1976.Google Scholar
  21. 21.
    D. M. Burton, Elementary Number Theory, 2nd ed., Wm. C. Brown, 1989.Google Scholar
  22. 22.
    D. M. Burton., A First Course in Rings and Ideals, Addison-Wesley, 1970.Google Scholar
  23. 23.
    D. Castellanos, The ubiquitous π, Math. Mag. 61 (1988) 67–98 and 148–163.Google Scholar
  24. 24.
    B. Cipra, The circle has been squared, Science 244: 4904 (May 5 1989) 528.Google Scholar
  25. 25.
    M. P. Closs (ed.), Native American Mathematics, Univ. of Texas Press, 1986.Google Scholar
  26. 26.
    P. J. Cohen and R. Hersh, Non-Cantorian set theory, Scientific Amer. 217 (Dec. 1967) 104–116.CrossRefMathSciNetGoogle Scholar
  27. 27.
    J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, 1996.Google Scholar
  28. 28.
    J. H. Conway and R. K. Guy, Surreal numbers, Math Horizons (November 1996) 26–31.Google Scholar
  29. 29.
    R. Courant and H. Robbins, What is Mathematics? Oxford Univ. Press, 1941.Google Scholar
  30. 30.
    J. Crossley, The Emergence of Number, World Scientific, 1987.Google Scholar
  31. 31.
    M. J. Crowe, A History of Vector Analysis, Univ. of Notre Dame Press, 1968.Google Scholar
  32. 32.
    T. Dantzig, Number: The Language of Science, 4th ed., Free Press, 1967.Google Scholar
  33. 33.
    J. W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard Univ. Press, 1979.MATHGoogle Scholar
  34. 34.
    P. J. Davis, Number, Sc. Amer. 211 (Sept. 1964) 51–59.Google Scholar
  35. 35.
    P. J. Davis, The Lore of Large Numbers, Random House, 1961.Google Scholar
  36. 36.
    M. Davis and R. Hersh, Nonstandard analysis, Sc. Amer. 226 (1972) 78–86.CrossRefMathSciNetGoogle Scholar
  37. 37.
    P. J. Davis, R. Hersh, and E. A. Marchisotto, The Mathematical Experience, Study Edition, Birkhäuser, 1995 (orig. 1981).Google Scholar
  38. 38.
    U. Dudley, Numerology, or What Pythagoras Wrought, Math. Assoc. of Amer., 1997.Google Scholar
  39. 39.
    U. Dudley, Formulas for primes, Math. Mag. 56 (1983) 17–22.CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    H. D. Ebbinghaus et al, Numbers, Springer-Verlag, 1990.Google Scholar
  41. 41.
    A. W. F. Edwards, Pascal’s Arithmetical Triangle, Oxford Univ. Press, 1987.MATHGoogle Scholar
  42. 42.
    C. H. Edwards, The Historical Development of the Calculus, Springer-Verlag, 1979.Google Scholar
  43. 43.
    H. Eves, Great Moments in Mathematics: (a) before 1650 and (b) after 1650, Math. Assoc. of Amer., 1983.Google Scholar
  44. 44.
    G. Flegg, Numbers: Their History and Meaning, Andre Deutsch, 1983.Google Scholar
  45. 45.
    C. G. Fraser, Some observations on mathematical analysis in the 18th century, Arch. Hist. Exact Sci., 39:4 (1989) 317–335.MATHMathSciNetGoogle Scholar
  46. 46.
    R. M. French, The Banach-Tarski theorem, Math. Intell. 10:4 (1988) 21–28.CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    A. Gardiner, Infinite Processes: Background to Analysis, Springer-Verlag, 1982.Google Scholar
  48. 48.
    M. Gardner, The Magic Numbers of Dr. Matrix, Prometheus Books, 1985.Google Scholar
  49. 49.
    M. Gardner, The concept of negative numbers and the difficulty of grasping it, Scientific Amer. 236 (1977) 131.CrossRefGoogle Scholar
  50. 50.
    J. Gardner and S. Wagon, At long last, the circle has been squared, Notices of the Amer. Math. Soc. 36 (1989) 1338–1343.MATHMathSciNetGoogle Scholar
  51. 51.
    A. Gillies, Frege, Dedekind, and Peano on the Foundations of Arithmetic, Van Gorcum, 1982.Google Scholar
  52. 52.
    H. Goldstine, The Computer from Pascal to Von Neumann, Princeton Univ. Press, 1972.Google Scholar
  53. 53.
    I. Grattan-Guinness, From Calculus to Set Theory, 1630–1910: An Introductory History, Princeton Univ. Press, 2000. MATHGoogle Scholar
  54. 54.
    E. Grosswald, Topics from the Theory of Numbers, 2nd ed., Birkhäuser, 1984.Google Scholar
  55. 55.
    P. R. Halmos, Naive Set Theory, Springer-Verlag, 1974 (orig. 1960).Google Scholar
  56. 56.
    T. L. Hankins, Sir William Rowan Hamilton, The Johns Hopkins Univ. Press, 1980.MATHGoogle Scholar
  57. 57.
    V. Harnik, Infinitesimals from Leibniz to Robinson: time to bring them back to school, Math. Intell. 8:2 (1986) 41–47, 63.Google Scholar
  58. 58.
    M. E. Hellman, The math of public key cryptography, Scientific Amer. 241:2 (Aug. 1979) 146–157.CrossRefMathSciNetGoogle Scholar
  59. 59.
    B. Henry, Every Number is Special, Dale Seymour, 1985.Google Scholar
  60. 60.
    D. Hilbert, The Foundations of Geometry, Open Court, 1959.Google Scholar
  61. 61.
    A. P. Hillman and G. L. Alexanderson, A First Undergraduate Course inAbstract Algebra, 4th ed., Wadsworth, 1983.Google Scholar
  62. 62.
    H. E. Huntley, The Divine Proportion, Dover, 1970.Google Scholar
  63. 63.
    G. Ifrah, From One to Zero, Penguin, 1985.Google Scholar
  64. 64.
    M. C. Irwin, Geometry of continued fractions, Amer. Math. Monthly 96 (1989) 696–703.CrossRefMATHMathSciNetGoogle Scholar
  65. 65.
    I. Kantor and A. S. Solodovnikov, Hypercomplex Numbers, Springer-Verlag, 1989. (Translated from the Russian by A. Shenitzer.)Google Scholar
  66. 66.
    L. C. Karpinski, The History of Arithmetic, Russell and Russell, 1965.Google Scholar
  67. 67.
    E. Kasner and J. R. Newman, Mathematics and the Imagination, Simon & Schuster, 1967.Google Scholar
  68. 68.
    V. J. Katz, A History of Mathematics: An Introduction, 3rd. ed., Addison-Wesley, 2009.Google Scholar
  69. 69.
    J. Keisler, Elementary Calculus: An Infinitesimal Approach, 2nd ed., Prindle, Weber & Schmidt, 1986.Google Scholar
  70. 70.
    I. Kleiner, A History of Abstract Algebra, Birkhäuser, 2007.Google Scholar
  71. 71.
    M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford Univ. Press, 1972.MATHGoogle Scholar
  72. 72.
    D. E. Knuth, Surreal Numbers, Addison-Wesley, 1974.Google Scholar
  73. 73.
    F. Le Lionais, Les Nombres Remarquables, Hermann, 1983.Google Scholar
  74. 74.
    W. J. LeVeque, Topics in Number Theory, 2 Vols., Addison- Wesley, 1965.Google Scholar
  75. 75.
    D. J. Lewis, Diophantine equations and p-adic methods. In Studies in Number Theory, ed. by W. J. LeVeque, Math. Assoc. of Amer., 1969, pp. 25–75.Google Scholar
  76. 76.
    C. C. MacDuffee, Algebra’s debt to Hamilton, Scripta Math. 10 (1944) 25–35.MATHMathSciNetGoogle Scholar
  77. 77.
    C. C. MacDuffee, The p-adic numbers of Hensel, Amer. Math. Monthly 45 (1938) 500–508.CrossRefMathSciNetGoogle Scholar
  78. 78.
    S. Mac Lane, Mathematics: Form and Function, Springer-Verlag, 1986.Google Scholar
  79. 79.
    S. Mac Lane, Mathematical models: a sketch for the philosophy of mathematics, Amer. Math. Monthly 88 (1981) 462–472.CrossRefMathSciNetGoogle Scholar
  80. 80.
    E. Maor, e: The Story of a Number, Princeton Univ. Press, 1994.Google Scholar
  81. 81.
    E. Maor, To Infinity and Beyond: A Cultural History of the Infinite, Birkhäuser, 1987.Google Scholar
  82. 82.
    W. Massey, Cross products of vectors in higher dimensional euclidean spaces, Amer. Math. Monthly 90 (1983) 697–701.CrossRefMATHMathSciNetGoogle Scholar
  83. 83.
    K. O. May, The impossibility of a division algebra of vectors in three dimensional space, Amer. Math. Monthly 73 (1966) 289–291.CrossRefMathSciNetGoogle Scholar
  84. 84.
    B. Mazur, Imagining Numbers, Farrar Straus Giroux, 2003.Google Scholar
  85. 85.
    K. Menninger, Number Words and Number Symbols: A Cultural History of Numbers, M.I.T. Press, 1969.MATHGoogle Scholar
  86. 86.
    G. H. Moore, Zermelo’s Axiom of Choice: Its Origins, Development, and Influence, Springer- Verlag, 1982.Google Scholar
  87. 87.
    P. J. Nahin, An Imaginary Tale: The Story of \(\sqrt{-1}\), Princeton Univ. Press, 1998.Google Scholar
  88. 88.
    M. B. Nathanson, A short proof of Cauchy’s polygonal theorem, Proc. Amer. Math. Soc. 99 (1987) 22–24.MATHMathSciNetGoogle Scholar
  89. 89.
    I. Niven, Numbers: Rational and Irrational, Random House, 1961.Google Scholar
  90. 90.
    I. Niven, Irrational Numbers, Math. Assoc. of America, 1956.Google Scholar
  91. 91.
    I. Niven, The roots of a quaternion, Amer. Math. Monthly 49 (1942) 386–388.CrossRefMATHMathSciNetGoogle Scholar
  92. 92.
    I. Niven, Equations in quaternions, Amer. Math. Monthly 48 (1941) 654–661.CrossRefMATHMathSciNetGoogle Scholar
  93. 93.
    I. Niven, The transcendence of π, Amer. Math. Monthly 46 (1939) 469–471.CrossRefMathSciNetGoogle Scholar
  94. 94.
    C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford Univ. Press, 1966.MATHGoogle Scholar
  95. 94a.
    C. D. Olds, A. Lax, and G. Davidoff, The Geometry of Numbers, Math. Assoc. of Amer., 2000.Google Scholar
  96. 95.
    O. Ore, Number Theory and its History, McGraw-Hill, 1948.Google Scholar
  97. 96.
    O. O’Shea and U. Dudley, The Magic Numbers of the Professor, Math. Assoc. of Amer., 2007.Google Scholar
  98. 97.
    H. Pollard and H. G. Diamond, The Theory of Algebraic Numbers, 2nd ed., Math. Assoc. of Amer., 1975.Google Scholar
  99. 98.
    C. Pomerance, The search for prime numbers, Scientific Amer. 247:6 (1982) 136–147.CrossRefGoogle Scholar
  100. 99.
    P. Ribenboim, The Book of Prime Number Records, 2nd ed., Springer-Verlag, 1989.Google Scholar
  101. 100.
    S. P. Richards, A Number for Your Thoughts, S. P. Richards Publ., 1982.Google Scholar
  102. 101.
    R. Rucker, Infinity and the Mind: The Science and Philosophy of the Infinite, Birkhäuser, 1982.Google Scholar
  103. 102.
    H. Schwerdtfeger, Geometry of Complex Numbers, Dover, 1979.Google Scholar
  104. 103.
    J. Sesiano, The appearance of negative solutions in mediaeval mathematics, Arch. Hist. Exact Sc. 32:2 (1985) 105–150.CrossRefMATHMathSciNetGoogle Scholar
  105. 104.
    G. F. Simmons, Calculus with Analytic Geometry, McGraw-Hill, 1985.Google Scholar
  106. 105.
    E. Sondheimer and A. Rogerson, Numbers and Infinity: A Historical Account of Mathematical Concepts, Cambridge Univ. Press, 1981.MATHGoogle Scholar
  107. 106.
    H. Stark, An Introduction to Number Theory, M.I.T. Press, 1978.Google Scholar
  108. 107.
    L. A. Steen, New models of the real-number line, Scientific Amer. 225 (1971) 92–99.CrossRefMathSciNetGoogle Scholar
  109. 108.
    I. Stewart and D. Tall, The Foundations of Mathematics, Oxford Univ. Press, 1977.MATHGoogle Scholar
  110. 109.
    J. Stillwell, Mathematics and its History, 2nd ed., Springer-Verlag, 2002.Google Scholar
  111. 109a.
    J. Stillwell, Roads to Infinity: The Mathematics of Truth and Proof, A K Peters, 2010.Google Scholar
  112. 110.
    F. J. Swetz., Capitalism and Arithmetic, Open Court, 1987.Google Scholar
  113. 110.
    O. Toeplitz, The Calculus: A Genetic Approach, Univ. of Chicago Press, 1963.Google Scholar
  114. 111.
    B. L. Van der Waerden, A History of Algebra, Springer-Verlag, 1985.Google Scholar
  115. 112.
    B. L. Van der Waerden, The discovery of quaternions, Math. Mag. 49 (1976) 227–234.CrossRefMATHMathSciNetGoogle Scholar
  116. 113.
    D. H. Van Osdol, Truth with respect to an ultrafilter or how to make intuition rigorous, Amer. Math. Monthly 79 (1972) 355–363.CrossRefMATHMathSciNetGoogle Scholar
  117. 114.
    N. Ya. Vilenkin, In Search of Infinity, Birkhäuser, 1995. (Translated from the Russian by A. Shenitzer.)Google Scholar
  118. 115.
    N. N. Vorobov, Fibonacci Numbers, Blaisdell, 1961.Google Scholar
  119. 116.
    S. Wagon, The Banach -Tarski Paradox, Cambridge Univ. Press, 1985.MATHGoogle Scholar
  120. 117.
    D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin, 1986.Google Scholar
  121. 118.
    R. L. Wilder, Mathematics as a Cultural System, Pergamon Press, 1981.Google Scholar
  122. 119.
    R. L. Wilder, Evolution of Mathematical Concepts: An Elementary Study, Wiley, 1968.Google Scholar
  123. 120.
    M. Yaglom, Complex Numbers in Geometry, Academic Press, 1968.Google Scholar
  124. 121.
    B. H. Yandell, The Honors Class: Hilbert’s Problems and their Solvers, A K Peters, 2002.Google Scholar
  125. 122.
    S. Yeshurun, Commonly known and less commonly known numbers, Theta 3:1 (1989) 28–34.Google Scholar
  126. 123.
    C. Zaslavsky, Africa Counts, Prindle, Weber and Schmidt, 1973.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations