A Geometric Proof to Cantor's Theorem and an Irrationality Measure for Some Cantor's Series

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

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.

特定整数平方根无理性的几何证明

Geometric Proof of the Irrationality of Square-Roots for Select Integers

本文通过几何构造扩展经典方法，证明√6等特定整数平方根的无理性，并探讨该方法对三角数等情形的适用边界与推广可能。

Irrationality From The Book

We generalize Tennenbaum's geometric proof of the irrationality of sqrt(2) to sqrt(n) for n = 3, 5, 6 and 10. Modified version published in Mathematics Magazine \textbf{85} (2012), no. 2, 110--114.