Transcendence of e and π

  • Arthur Jones
  • Kenneth R. Pearson
  • Sidney A. Morris
Part of the Universitext book series (UTX)


The main purpose of this chapter is to prove that the number π is transcendental, thereby completing the proof of the impossibility of squaring the circle (Problem III of the Introduction). We first give the proof that e is a transcendental number, which is somewhat easier. This is of considerable interest in its own right, and its proof introduces many of the ideas which will be used in the proof for π. With the aid of some more algebra — the theory of symmetric polynomials — we can then modify the proof for e to give the proof for π.


Prime Number Rational Number Symmetric Function Rational Coefficient Fundamental Theorem 
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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Additional Reading for Chapter 7

  1. [JA]
    J. Archbold, Algebra, 4th edition, Pitman, London, 1970.Google Scholar
  2. [AB]
    A. Baker, Transcendental Number Theory, Cambridge University Press, Cambridge, 1975.CrossRefzbMATHGoogle Scholar
  3. [FB]
    F. Beukers, J.P. Bezivia and P. Robba, “An Alternative Proof of the Lindemann-Weierstrass Theorem”, American Mathematical Monthly, 97 (1990), 193–197.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [GB]
    G. Birkhoff and S. MacLane, A Survey of Modern Algebra, Macmillan, New York, 1953.zbMATHGoogle Scholar
  5. [AC]
    A. Clark, Elements of Abstract Algebra, Wadsworth, Belmont, California, 1971.Google Scholar
  6. [WF]
    W.L. Ferrar, Higher Algebra, Clarendon, Oxford, 1958.Google Scholar
  7. [PG]
    P. Gordan, “Transcendenz von e und π”, Mathematische Annalen, 43 (1893), 222–224.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [AH]
    A. Hurwitz, “Beweis der Transcendenz der Zahl e”, Mathematische Annalen, 43 (1893), 220–222.MathSciNetCrossRefGoogle Scholar
  9. [CH]
    C.R. Hadlock, Field Theory and its Classical Problems, Carus Mathematical Monographs, No. 19, Mathematical Association of America, 1978.Google Scholar
  10. [ChH]
    Ch. Hermite, “Sur la fonction exponentielle”, Comptes Rendus des Séances de l’Académie des Sciences Paris, 77 (1873), 18–24.zbMATHGoogle Scholar
  11. [DH]
    D. Hilbert, “Über die Transcendenz der Zahlen e und π”, Mathematische Annalen, 43 (1893), 216–219; reprinted in Gesammelte Abhandlungen Vol.1, Chelsea, 1965.MathSciNetCrossRefGoogle Scholar
  12. [EH]
    E.W. Hobson, Squaring the Circle, Cambridge University Press, 1913; reprinted in Squaring the Circle and Other Monographs, Chelsea, 1953.Google Scholar
  13. [GH1]
    G.H. Hardy, A Course of Pure Mathematics, 10th edition, Cambridge University Press, Cambridge, 1952.zbMATHGoogle Scholar
  14. [GH]
    G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 3rd edition, Clarendon, Oxford, 1954.zbMATHGoogle Scholar
  15. [FK1]
    F. Klein, Elementary Mathematics from an Advanced Standpoint, (vol 1: Arithmetic, Algebra and Analysis), Dover, New York, 1948.Google Scholar
  16. [FK]
    F. Klein, Famous Problems of Elementary Geometry; reprinted in Famous Problems and Other Monographs, Chelsea, 1962.Google Scholar
  17. [FL]
    F.L. Lindemann, “Über die Zahl π”, Mathematische Annalen, 20 (1882), 213–225.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [IN2]
    I. Niven, Irrational Numbers, Carus Mathematical Monographs, No.11, Mathematical Association of America, 1963.Google Scholar
  19. [IN3]
    I. Niven, “The transcendence of π”, American Mathematical Monthly, 46 (1939), 469–471.MathSciNetCrossRefGoogle Scholar
  20. [DS]
    D.E. Smith, The History and Transcendence of π; reprinted in W.A. Young, Monographs on Topics of Modern Mathematics Relevant to the Elementary Field, Dover, 1955.Google Scholar
  21. [MS]
    M. Spivak, Calculus, Benjamin, New York, 1967.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Arthur Jones
    • 1
  • Kenneth R. Pearson
    • 1
  • Sidney A. Morris
    • 2
  1. 1.Department of MathematicsLa Trobe UniversityBundooraAustralia
  2. 2.Faculty of InformaticsUniversity of WollongongWollongongAustralia

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