---
tag: 421QMTFG9T
kind: concept
name: Hyperreal number
msc: ["26E35", "03H05", "03C20"]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperreal_number
same_as:
  wikidata: Q5958543
  wikipedia: Hyperreal number
---

## Summary

The hyperreal numbers form an [[ordered field|0P3VCBDEE2]] ${}^*\R$ that contains the [[real numbers|08J2M1JYAG]] together with infinitesimal and infinite numbers.
The usual construction is an [[ultrapower|AX9Q86GB8M]] of $\R$: sequences of reals modulo agreement on a set in a nonprincipal [[ultrafilter|J64Y3JC0RE]] on $\N$.
By [[Łoś's theorem|AGYBSTD7BG]], ${}^*\R$ satisfies the same first-order statements as $\R$, which makes it the basic number system of [[nonstandard analysis|WM62GKCBWN]].
It is not [[Archimedean|0XH79S06J3]]: an infinite hyperreal is larger than every natural number.

## Explanations

### L1 human

Think of a hyperreal number as a sequence of real numbers, where two sequences count as equal if they agree "almost everywhere" in a precise sense.
The constant sequence $(3, 3, 3, \dots)$ is the real number $3$.
The sequence $(1, 1/2, 1/3, \dots)$ is positive but eventually below any positive real, so it is an infinitesimal.
The sequence $(1, 2, 3, \dots)$ is larger than every real number, so it is infinite.