nLab prealgebra real number

Idea

The real numbers as encountered in prealgebra.

Definition

Let $R$ be an Archimedean integral domain and let $[0, 1]_R$ be the unit interval in $R$. Let $a:\mathbb{N} \to [0, 9]_\mathbb{N}$ be a sequence of decimal digits, where $n \in [0, 9]_\mathbb{N}$ is the set of all natural numbers $0 \leq n \leq 9$, and let $\mathcal{I}:[0, 9]_\mathbb{N} \to F$ be the canonical embedding of $[0, 9]_\mathbb{N}$ into $R$. The prealgebra real numbers $\mathbb{R}$ is the initial Archimedean integral domain such that for every such sequence $a:\mathbb{N} \to [0, 9]_\mathbb{N}$, the sequence

$$b(n) \coloneqq \sum_{i=0}^{n} \frac{\mathcal{I}(a_i)}{10^{i+1}}$$

has a limit in the unit interval $[0, 1]_\mathbb{R}$.

Properties

If the limited principle of omniscience is true, then every Cauchy real number is a prealgebra real number.

See also

• real numbers
• limited principle of omniscience
• infinite decimal representation of a unit interval