#math/analysis Elements of $\mathbb{R}\setminus \mathbb{Q}$ are irrational numbers. If $x$ irrational and $r$ is rational, then $ y=x+r\in \mathbb{R}\setminus \mathbb{Q}. $ Suppose $y$ was in fact rational. Then $y-r=x$ is rational where the difference of rationals is rational. Similarly, if $r\neq{}0$ then $ rx\in \mathbb{R}\setminus \mathbb{Q}\quad\quad\forall \:{}r\in \mathbb{Q}\text{ where }r\neq{}0. $ ## Theorem Every interval $(a,b)$, no matter how small, contains both rational and irrational numbers. In fact it contains infinitely many rational numbers and infinitely many irrational numbers. #TODO Requires knowledge on cuts.