This might sounds stupid, but I really don't know can I show Irrational numbers in proves? And if so, how to show it?
For example, when I want to show Rational numbers, I do this:
$\frac{m}{n}
$ , $m, n $ are integer $,$ $n\ne0$
Can I do something like that with Irrational numbers?
Best Answer
A real number $a$ is irrational iff its decimal (or binary) expansion doesn't become ultimately periodic. You can formulate this in the following way: $$a=a_0.a_1a_2a_3\ldots\qquad\bigl(a_0\in{\mathbb Z}, \ a_k\in\{0,1,\ldots,9\}\ (k\geq1)\bigr)$$ is irrational iff $\forall p\in{\mathbb N}_{\geq1}$, $\forall n\in{\mathbb N}_{\geq1}$ there is a $k>n$ with $a_k\ne a_{k+p}$.