Authors: Tim Jones
Transcendence of a number implies the irrationality of powers of a number, but in the case of π there are no separate proofs that powers of π are irrational. We investigate this curiosity. Transcendence proofs for e involve what we call Hermite's technique; for π's transcendence Lindemann's adaptation of Hermite's technique is used. Hermite's technique is presented and its usage is demonstrated with irrationality proofs of e and powers of e. Applying Lindemann's adaptation to a complex polynomial, π is shown to be irrational. This use of a complex polynomial generalizes and powers of π are shown to be irrational. The complex polynomials used involve roots of i and yield regular polygons in the complex plane. One can use graphs of these polygons to visualize various mechanisms used to proof π2, π3, and π4 are irrational. The transcendence of π and e are easy generalizations from these irrational cases.
Comments: 17 pages.
[v1] 28 Feb 2011
Unique-IP document downloads: 851 times
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