Transcendence of e and π
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.
KeywordsPrime Number Rational Number Symmetric Function Rational Coefficient Fundamental Theorem
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Additional Reading for Chapter 7
- [JA]J. Archbold, Algebra, 4th edition, Pitman, London, 1970.Google Scholar
- [AC]A. Clark, Elements of Abstract Algebra, Wadsworth, Belmont, California, 1971.Google Scholar
- [WF]W.L. Ferrar, Higher Algebra, Clarendon, Oxford, 1958.Google Scholar
- [CH]C.R. Hadlock, Field Theory and its Classical Problems, Carus Mathematical Monographs, No. 19, Mathematical Association of America, 1978.Google Scholar
- [EH]E.W. Hobson, Squaring the Circle, Cambridge University Press, 1913; reprinted in Squaring the Circle and Other Monographs, Chelsea, 1953.Google Scholar
- [FK1]F. Klein, Elementary Mathematics from an Advanced Standpoint, (vol 1: Arithmetic, Algebra and Analysis), Dover, New York, 1948.Google Scholar
- [FK]F. Klein, Famous Problems of Elementary Geometry; reprinted in Famous Problems and Other Monographs, Chelsea, 1962.Google Scholar
- [IN2]I. Niven, Irrational Numbers, Carus Mathematical Monographs, No.11, Mathematical Association of America, 1963.Google Scholar
- [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