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 $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory