The Transcendence of π
📜 Abstract
The proof that π is transcendental is not well-known despite the fact that it isn’t too difficult for a university mathematics student to follow. The purpose of this paper is to make the proof more widely available. A bonus is that the proof also shows that e is transcendental as well. The material in these notes are not mine; it is taken from a supplement issued by Ian Stewart as an adjunct to a Rings and Fields course in 1970 at the University of Warwick.
✨ Summary
The paper is an expository presentation of elementary-analysis proofs that π is irrational, π² is irrational, e is transcendental, and finally π is transcendental. Its central argument follows the Hermite–Lindemann method: construct auxiliary polynomials, derive integer-valued expressions, and show that corresponding integrals tend to zero, producing contradictions. The author explicitly identifies Ian Stewart’s 1970 University of Warwick course supplement as the source of the notes.
The paper appears to have had primarily educational and expository influence rather than documented industrial impact. A 2025 expository article on elementary proofs of π’s transcendence identifies Mayer’s 2006 note as one of its principal antecedents and develops the same proof strategy further. (mathscholar.org) The note is also discussed in later mathematical question-and-answer literature as a source for understanding Lindemann’s proof and its use of symmetric-function arguments. (math.stackexchange.com) It is catalogued in mathematical bibliographies as a November 2006 web manuscript by Steve Mayer. (ftp.math.utah.edu) No specific industrial application or product use of this paper itself was identified in the search.