# Real Number is not necessarily Rational Number

## Theorem

Let $x$ be a real number.

Then it is not necessarily the case that $x$ is also a rational number.

## Proof

Let $x = \sqrt 2$.

$\sqrt 2$ is an irrational number.

By definition:

$x \in \R \setminus \Q$

where:

$\R$ is the set of real numbers
$\Q$ is the set of rational numbers
$\setminus$ denotes the set difference.

Thus $x$, while being a real number, is not also a rational number.

Hence the result.

$\blacksquare$