Numbers pp 9-26 | Cite as

Natural Numbers, Integers, and Rational Numbers

  • K. Mainzer
Part of the Graduate Texts in Mathematics book series (GTM, volume 123)


1. Egyptians and Babylonians. Symbols for numbers are found in the earliest remains of human writing. Even in the early stone age we find them in the form of notches in bones or as marks on the walls of caves. It was the age when man lived as a hunter and today we can only speculate as to whether ∥∥ for example was intended to represent the size of the kill. Number systems mark the beginning of arithmetic. The first documents go back to the earliest civilizations in the valley of the Nile, Euphrates and Tigris. Hieroglyphs for the numbers 10 000, 100 000 and 1 000 000 are to be found on a mace of King Narmer, of the first Egyptian dynasty (circa 3000 BC).


Natural Number Integral Domain Successor Function Unit Fraction Number Symbol 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aristoteles: Metaphysik, Aristotelis Opera, ed. I. Bekker, Berlin 1831, repr. Darmstadt 1960.Google Scholar
  2. [2]
    Aristoteles: Physik, Aristotelis Opera, ed. I. Bekker, Berlin 1831, repr. Darmstadt 1960.Google Scholar
  3. [3]
    Becker, O.: Grundlagen der Mathem. in geschichtlicher Entwick-lung, Freiburg/München 1954, 21964, Frankfurt 1975.Google Scholar
  4. [4]
    Bolzano, B.: Paradoxien des Unendlichen, ed. F. Přihonský, Leipzig 1851, Berlin 21889, ed. A. Höfler, Leipzig 1920, mit Einl., Anm., Reg. u. Bibliographie ed. B. van Rootselaar, Hamburg 1955, 21975.Google Scholar
  5. [5]
    Bolzano B.: Reine Zahlenlehre, in: B. Bolzano — Gesamtausgabe (eds. E. Winter, J. Berg, F. Kambartel, J. Loužil, B. v. Rootselaar), Reihe II Nachlaß, A. Nachgelassene Schriften, Bd. 8 Größenlehre II, Reine Zahlenlehre, Stuttgart/Bad Cannstatt 1976.Google Scholar
  6. [6]
    Bourbaki, N.: Elements de mathématique, Paris, 1939 and later.Google Scholar
  7. [7]
    Bruins, E.M., Rutten, M.: Textes mathématiques de Suse, Mém-oires de la Mission Archéologique en Iran, Tome 34, Paris 1961.Google Scholar
  8. [8]
    Cantor, G.: Gesam. Abh. mathem. u. philos. Inhalts, Berlin 1932, repr. Berlin 1980.Google Scholar
  9. [9]
    Dedekind, R.: Was sind und was sollen die Zahlen? Braunschweig 1888, 101965, repr. 1969.Google Scholar
  10. [10]
    Dedekind, R.: Mathem. Werke Bd. 3, Braunschweig 1932, repr. New York 1969.Google Scholar
  11. [11]
    Frege, G.: Die Grundlagen der Arithmetik. Eine logisch mathema-tische Untersuchung über den Begriff der Zahl, Breslau 1884, repr. Darmstadt/Hildesheim 1961.Google Scholar
  12. [12]
    Gauss, C.F.: Werke Bd. 3, Göttingen 1876.Google Scholar
  13. [13]
    Hankel, H.: Theorie der complexen Zahlensysteme, Leipzig 1867.Google Scholar
  14. [14]
    Hilbert, D.: Über den Zahlbegriff, in: Jahresber. d. Deutschen Math. Verein. 1900, 180–184.Google Scholar
  15. [15]
    Juschkewitsch, A.P.: Geschichte der Mathematik im Mittelalter, dt. Leipzig 1964.Google Scholar
  16. [16]
    Kronecker, L.: Über den Zahlbegriff, in: Journ. f. d. reine u. angew. Mathem. 101 1887, 339, in: Math. Werke Bd. 3, Leipzig 1899/1931, repr. New York 1968, 249-274.Google Scholar
  17. [17]
    Kronecker, L.: Grundzüge einer arithmetischen Theorie der algebraischen Größen, in: J. f. d. reine u. angew. Mathem. 1882, 1–122, in: Math. Werke Bd. 2, Leipzig 1897, repr. New York 1968, 237-287.Google Scholar
  18. [18]
    Landau, E.: Grundlagen der Analysis, Leipzig 1930, repr. Darmstadt 1963.Google Scholar
  19. [19]
    Lepsius, R.: Über eine Hieroglyphische Inschrift am Tempel in Edfu, in: Abh. d. Kgl. Akad. d. Wiss., Berlin 1855, 69–111.Google Scholar
  20. [20]
    Neugebauer, O.: Mathem. Keilschrifttexte, Quellen u. Studien A3, Berlin I, II 1935, III 1937.Google Scholar
  21. [21]
    Neumann, J.v.: Zur Einführung der transfiniten Zahlen, in: Acta Szeged 1 1923, 199–202, repr. in: A.H. Taub (ed.), Collected Works, Oxford/London/Paris Bd. 1 1961, 24-33.MATHGoogle Scholar
  22. [22]
    Ohm, M.: Die reine Elementarmathematik Bd. 1, Berlin 21834.Google Scholar
  23. [23]
    Papyrus Rhind, (Hrsg. A. Eisenlohr) Leipzig 1877; A.B. Chace, The Rhind Mathem. Papyrus, Oberlin I 1927, II 1929.Google Scholar
  24. [24]
    Peano, G.: Arithmetices principia nova exposita, in: Opere scelte Bd. II, Rom 1958, 20–55.Google Scholar
  25. [25]
    Russell, B.: The principles of mathematics, London 1903, 71956.Google Scholar
  26. [26]
    Steinitz, E.: Algebraische Theorie der Körper, in: J. f. d. reine u. angew. Math. 137 1910, 167–309.MATHGoogle Scholar
  27. [27]
    Stifel, M.: Arithmetica integra, Nürnberg 1544.Google Scholar
  28. [28]
    Struwe, W.W.: Papyrus des staatl. Museums der schönen Künste in Moskau, Quellen u. Studien A1 1930.Google Scholar
  29. [29]
    Tropfke, J.: Geschichte der Elementarmathematik, Bd. 1 Arith-metik und Algebra, vollst. neu bearb. von H. Gericke, K. Reich u. K. Vogel, Berlin 41980.Google Scholar
  30. [30]
    Weber, H.: Lehrbuch der Algebra Bd. 1 1895, repr. der 3. Aufl. New York 1961.Google Scholar

Further Readings

  1. [31]
    Aaboe, Asger: Episodes from the Early History of Mathematics (New Mathematical Library, No. 13) New York: Random House and L.W. Singer, 1964.MATHGoogle Scholar
  2. [32]
    Cajori, Florien: A History of Mathematical Notations, 2 vols. Chicago: Open Court Publishing, 1928–1929.Google Scholar
  3. [33]
    Kramer, Edna: The Nature and Growth of Modern Mathematics, Princeton: Princeton University Press, 1970.Google Scholar
  4. [34]
    Menninger, Karl: Number Words and Number Symbols: A Cultural History of Numbers, Cambridge, Mass.: The M.I.T. Press, 1969.MATHGoogle Scholar
  5. [35]
    Neugebauer, Otto: The Exact Sciences in Antiquity, 2nd ed. New York: Harper and Row, 1962.Google Scholar
  6. [36]
    Ore, Oystein: Number Theory and Its History, New York: McGraw-Hill, 1948.MATHGoogle Scholar
  7. [37]
    Resnikoff, H.L. and Wells, R.O. Jr.: Mathematics in Civilization, New York: Dover, 1984.Google Scholar
  8. [38]
    Sondheimer, Ernst and Rogerson, Alan: Numbers and Infinity: A Historical Account of Mathematical Concepts, Cambridge: Cambridge University Press, 1981.MATHGoogle Scholar
  9. [39]
    Van der Waerden, B.L.: Science Awakening, tr. by Arnold Dresden, New York: Oxford University Press, 1961; New York: John Wiley, 1963 (paperback ed.).Google Scholar

Copyright information

© Springer Science+Business Media New York  1991

Authors and Affiliations

  • K. Mainzer

There are no affiliations available

Personalised recommendations