## FANDOM

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Axiom 1: All rational numbers can be expressed as a fraction of two integers.

$\forall x\in \mathbb{Q} ,\exists \left\{ a,b\right\} \subset \mathbb{Z} :x=\dfrac {a}{b}$

Axiom 2 Let $l$ be the property of a fraction being in lowest terms. If both the denominator and numerator are even then the fraction is not in its lowest terms because $2$ would be a common factor.

$\mathbb{E} \left\{ 2k: k\in\mathbb{Z}\right\}$
$l\Leftrightarrow a\notin\mathbb{E}\bigvee b\notin\mathbb{E}$

Assume it the square root is rational.
$\sqrt{2} \in \mathbb{R}$
$\Rightarrow \sqrt{2}= \frac{a}{b} \because Ax. 1$
$\Rightarrow 2= \frac{a^2}{b^2}$
$\Rightarrow 2b^2= a^2$
$\Rightarrow a^2\in \mathbb{E}$