**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:** 527 times

**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. *