Hyperreal number

Builds on Real numbers Ultrafilter Sequence Equivalence relation Ordered field Ultraproduct Archimedean ordered field

Definition

Let $\mathcal{U}$ be a nonprincipal ultrafilter on $\N$ . For sequences $a = (a_n)$ and $b = (b_n)$ of real numbers, put $a \sim b$ if $\set{n \in \N : a_n = b_n} \in \mathcal{U}$ . This is an equivalence relation; write $[a]$ for the class of $a$ . The set of hyperreal numbers is ${}^*\R = \R^{\N} / {\sim}$ , with the following structure.

$[a] + [b] = [(a_n + b_n)]$ and $[a] \cdot [b] = [(a_n b_n)]$ .
$[a] < [b]$ if $\set{n \in \N : a_n < b_n} \in \mathcal{U}$ .
Each real number $r$ is identified with the class of the constant sequence $(r, r, r, \dots)$ .

A hyperreal $x$ is infinitesimal if $\abs{x} < r$ for every real $r > 0$ , and infinite if $\abs{x} > r$ for every real $r$ .

Remarks

The operations and the order are well defined, and ${}^*\R$ is an ordered field containing $\R$ ; it is the ultrapower of $\R$ by $\mathcal{U}$ .

Example: $\varepsilon = [(1/(n+1))]$ is a positive infinitesimal. For a real $r > 0$ , the set $\set{n : 1/(n+1) < r}$ is cofinite, and a nonprincipal ultrafilter contains every cofinite set, so $\varepsilon < r$ . Also $\varepsilon > 0$ because every term is positive. So ${}^*\R$ is not Archimedean.

Non-example: if $\mathcal{U}$ is the principal ultrafilter of the sets containing a fixed $k \in \N$ , then $[a] \mapsto a_k$ is an isomorphism onto $\R$ , and there are no nonzero infinitesimals.

Under the continuum hypothesis, the ordered field ${}^*\R$ does not depend on the choice of $\mathcal{U}$ , up to isomorphism.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Archimedean ordered field concept
Equivalence relation concept
Ordered field concept
Real numbers object
Sequence concept
Ultrafilter concept
Ultraproduct concept
Hyperreal number definition

Tag	`TK6R2C9KRE`
URI	`m2://TK6R2C9KRE`
Kind	definition
Writing	W0 stub
MSC	26E35, 03H05, 03C20
Defines	Hyperreal number
Uses	Real numbers, Ultrafilter, Sequence, Equivalence relation, Ordered field, Ultraproduct, Archimedean ordered field
Changed	2 November 2026

Hyperreal number

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here