# Numbers as a Source of Mathematical Ideas

• Israel Kleiner
Chapter

## Abstract

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.

## Keywords

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.

## References

1. 1.
W. W. Adams and L. J. Goldstein, Introduction to N umber Theory, Prentice-Hall, 1976.Google Scholar
2. 2.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
46. 46.
R. M. French, The Banach-Tarski theorem, Math. Intell. 10:4 (1988) 21–28.
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.
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.
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.
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.
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.
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.
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.
72. 72.
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.
77. 77.
C. C. MacDuffee, The p-adic numbers of Hensel, Amer. Math. Monthly 45 (1938) 500–508.
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.
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.
83. 83.
K. O. May, The impossibility of a division algebra of vectors in three dimensional space, Amer. Math. Monthly 73 (1966) 289–291.
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.
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.
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.
92. 92.
I. Niven, Equations in quaternions, Amer. Math. Monthly 48 (1941) 654–661.
93. 93.
I. Niven, The transcendence of π, Amer. Math. Monthly 46 (1939) 469–471.
94. 94.
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford Univ. Press, 1966.
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.
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.
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.
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.
109. 108.
I. Stewart and D. Tall, The Foundations of Mathematics, Oxford Univ. Press, 1977.
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.
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.
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.
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.
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