# If the imaginary unit i rational or irrational ?

##### 1 Answer
Feb 19, 2017

The set of rational numbers is a subset of the real numbers

$\mathbb{Q} \subset \mathbb{R}$

The set of rational numbers are those that are real but not rational

$P = \mathbb{R} - \mathbb{Q}$, (or $P = \mathbb{R} / \mathbb{Q}$)
$P \subset \mathbb{R}$

And the real numbers are a subset of the complex number:

$\mathbb{R} \subset \mathbb{C}$

And so $i = \sqrt{- 1} \in \mathbb{C} \cancel{\in} \mathbb{R} \implies i \cancel{\in} P$

And so $i$ is neither rational nor irrational, as these are reserved for real numbers only.

$i$ is complex!