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
By Proof by Counterexample:

Let $x = \sqrt 2$.

From Square Root of 2 is Irrational:
 * $\sqrt 2$ 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.