## Definition ## Let $R$ be an [[Archimedean ordered integral domain]] and let $[0, 1]_R$ be the [[unit interval]] in $R$. The __infinite decimal representation__ of $[0, 1]_R$ is a function $\mathcal{I}:[0, 1]_R \to (\mathbb{N} \to [0, 9]_\mathbb{N})$ from the unit interval in $R$ to the type of sequences in the [[natural numbers]] that are bounded below by $0$ and bounded above by $9$, such that $r$ is equal to the [[limit of a sequence|limit of the following sequence]] $$\prod_{r:[0, 1]_R} r = \lim_{n \to \infty} \sum_{i=0}^{n} \frac{\mathcal{I}(r)_i}{10^i}$$ ## Properties ## * The infinite decimal representation of the [[rational numbers]] consist of all the eventually periodic sequences in the [[natural numbers]] that are bounded below by $0$ and bounded above by $9$. * The infinite decimal representation of the [[decimal numbers]] consist of all the sequences such that there is a natural number $N$ such that for all decimal number $i$ and natural numbers $n \geq N$, $\mathcal{I}(i)(n) = 0$ or $\mathcal{I}(i)(n) = 9$. ## See also ## * [[real numbers]] * [[pre-algebra real numbers]] * [[sequence]] * [[limit of a sequence]] * [[unit interval]]