# Real Number is not necessarily Rational Number

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## Theorem

Let $x$ be a real number.

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

## Proof

Let $x = \sqrt 2$.

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.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $1$: Algebraic Structures: $\S 1$: The Language of Set Theory