A geometric proof that $e$ is irrational and a new measure of its irrationality

Details A geometric proof that $e$ is irrational and a new measure of its irrationality A geometric proof that $e$ is irrational and a new measure of its irrationality

2007-04-10

arxiv.org/abs/0704.1282v2

Amer. Math. Monthly 113 (2006) 637-641 and 114 (2007) 659

Mathematics

History and Overview

7 pages, 1 figure, Addendum gives details on why the intersection of the intervals is $e$

Scientific paper

We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q > 1$, then $|e - p/q| > 1/(S(q)+1)!$, where $S(q)$ is the smallest positive integer such that $S(q)!$ is a multiple of $q$. We relate this measure for $e$ to a known one and to the greatest prime factor of an integer. We make two conjectures and recall a theorem of Cantor that can be proved by a similar construction.

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