We have the ordered field of [[Rational Number|rational numbers]], but they aren’t complete there are holes everywhere, and to get to $\mathbb{R}$ we must fill in these gaps. There are several ways to do this [[@cummingsRealAnalysisLongform2021|(Cummings, 2021)]]. There is also a non-standard approach using so-called ultrafilters, another using hyperrationals, and several more beyond that. Cantor used Cauchy sequences of rationals: If you demand that each Cauchy sequence converges to something, and pairs of sequences whose difference is Cauchy must converge to the same thing, then by adding these limits into your set $\mathbb{Q}$ (by identifying them with an equivalence class of Cauchy sequences converging to them), you in effect complete $\mathbb{Q}$, giving $\mathbb{R}$. Another common method uses *Dedekind cuts.* Definitions of the real number and their arithmetic operations: 1. [[Real Number Axioms]]. 2. [[The set of real numbers is the set of Dedekind cuts]]. 3. [[The set of real numbers is the set of Cauchy sequences in the rationals]]. # Properties **Algebra**: (1) $(\mathbb{R},+,\times)$ is a [[Field (Algebra)|field]] more specifically and *ordered field*. (2) [[Real Power of Real Number]]. **Completeness**: Any [[Bound of Set of Reals|bounded]] subset of $\mathbb{R}$ has a tight bound which is known as [[Least Upper Bound Property|completeness of the real numbers]]. **Interleaving with other numbers**: The [[Archimedean Property of Real Numbers|Archimedean property]] of the reals is that for any real number, there is a [[Natural Numbers|natural number]] greater than it. The set of rational numbers is a [[Between two Different Real Numbers exists a Rational Number|dense subset]] of $\mathbb{R}$ and it follows that [[Between two Different Real Numbers exists an Irrational Number|a real number lies between any two non-equal reals]]. # Applications See [[Real Sequence]].