< [[nlab:rational numbers]] #Contents# * table of contents {:toc} ## Idea ## The rational numbers as familiar from school mathematics. ## Definition ## The rational numbers is the initial [[commutative ring]] $\mathbb{Q}$ with a function $(-)^{-1}:\mathbb{Z}_{+} \to \mathbb{Q}$ from the positive integers to the rational numbers such that for all positive integers $a:\mathbb{Z}_{+}$, $a \cdot a^{-1} = 1$. ## See also ## * [[natural numbers]] * [[integers]] * [[commutative ring]] * [[rational root theorem]] ## References ## * [[Frank Quinn]], *Proof Projects for Teachers* ([pdf](https://personal.math.vt.edu/fquinn/education/pfs4teachers0.pdf)) category: not redirected to nlab yet