**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 π^{2}, π^{3}, and π^{4}
are irrational. The transcendence of π and e are easy generalizations
from these irrational cases.

**Comments:**
17 pages.

**Download:** **PDF**

[v1] 28 Feb 2011

**Unique-IP document downloads:** 903 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful. *