TITLE: Proving Irrationality
QUESTION [45 upvotes]: How is it possible to prove a number is irrational?
First part of that question: How it possible to know that a number will go on infinitely?
Second part: How is it possible to know that no repetition will occur during the infinite sequence of digits?
Any examples of proofs of irrationality?

REPLY [5 votes]: An elegant proof that $\sqrt[n]{2}$ is irrational for integers $n \ge 2$.
There are many solutions when $n = 2$. An example is this (from wiki)
When $n > 2$, we can use Fermat's Last Theorem (!)
Suppose $\sqrt[n]{2}$ is rational. Then, $\sqrt[n]{2} = \frac{a}{b}$ for some relatively prime positive integers $a, b$. Then, $a^n = 2b^n \Rightarrow a^n = b^n + b^n$, contradicting Fermat's Last Theorem!