An ultrafilterMathworldPlanetmath on a set I is called nonprincipal if no finite subsets of I are in .

Fix once and for all a nonprincipal ultrafilter on the set of natural numbersMathworldPlanetmath. Let be the equivalence relationMathworldPlanetmath on the set of sequences of real numbers given by


Let * be the set of equivalence classesMathworldPlanetmath of under the equivalence relation . The set * is called the set of hyperreals. It is a field under coordinatewise additionPlanetmathPlanetmath and multiplication:

{an}+{bn} = {an+bn}
{an}{bn} = {anbn}

The field * is an ordered field under the ordering relation


The real numbers embed into * by the map sending the real number x to the equivalence class of the constant sequence given by xn:=x for all n. In what follows, we adopt the convention of treating as a subset of * under this embeddingPlanetmathPlanetmath.

A hyperreal x* is:

  • limited if a<x<b for some real numbers a,b

  • positive unlimited if x>a for all real numbers a

  • negative unlimited if x<a for all real numbers a

  • unlimited if it is either positive unlimited or negative unlimited

  • positive infinitesimalMathworldPlanetmath if 0<x<a for all positive real numbers a+

  • negative infinitesimal if a<x<0 for all negative real numbers a-

  • infinitesimal if it is either positive infinitesimal or negative infinitesimal

For any subset A of , the set A* is defined to be the subset of * consisting of equivalence classes of sequences {an} such that


The sets *, *, and * are called hypernaturals, hyperintegers, and hyperrationals, respectively. An element of * is also sometimes called hyperfinite.

Title hyperreal
Canonical name Hyperreal
Date of creation 2013-03-22 12:35:45
Last modified on 2013-03-22 12:35:45
Owner djao (24)
Last modified by djao (24)
Numerical id 4
Author djao (24)
Entry type Definition
Classification msc 26E35
Synonym nonstandard real
Synonym non-standard real
Related topic Infinitesimal2
Defines nonprincipal ultrafilter
Defines infinitesimal
Defines hypernatural
Defines hyperinteger
Defines hyperrational
Defines hyperfinite