nLab prealgebra real number

Contents

Contents

Idea

The real numbers as encountered in prealgebra.

Definition

Let RR be an Archimedean integral domain and let [0,1] R[0, 1]_R be the unit interval in RR. Let a:[0,9] a:\mathbb{N} \to [0, 9]_\mathbb{N} be a sequence of decimal digits, where n[0,9] n \in [0, 9]_\mathbb{N} is the set of all natural numbers 0n90 \leq n \leq 9, and let :[0,9] F\mathcal{I}:[0, 9]_\mathbb{N} \to F be the canonical embedding of [0,9] [0, 9]_\mathbb{N} into RR. The prealgebra real numbers \mathbb{R} is the initial Archimedean integral domain such that for every such sequence a:[0,9] a:\mathbb{N} \to [0, 9]_\mathbb{N}, the sequence

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

has a limit in the unit interval [0,1] [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

Last revised on June 9, 2022 at 20:15:16. See the history of this page for a list of all contributions to it.