# Definition:Extended Real Number Line

## Contents

## Definition

### Definition 1

The **extended real number line** $\overline{\R}$ is defined as:

- $\overline{\R} := \R \cup \left\{{+\infty, -\infty}\right\}$

that is, the set of real numbers together with two auxiliary symbols:

- $+\infty$,
**positive infinity** - $-\infty$,
**negative infinity**

such that:

- $\forall x \in \R: x < +\infty$
- $\forall x \in \R: -\infty < x$

### Definition 2

The **extended real number line** $\overline \R$ is the order completion of the set of real numbers $\R$.

The greatest element of $\overline \R$ is often denoted by $+\infty$ and its least element by $-\infty$.

## Structures on $\overline{\R}$

$\overline{\R}$ can be endowed with the following structures:

- An ordering extending the standard ordering on $\R$
- Extended Real Addition, extending real addition $+$
- Extended Real Subtraction, extending real subtraction $-$
- Extended Real Multiplication, extending real multiplication $\cdot$
- A topology extending the usual (Euclidean) topology on $\R$
- A $\sigma$-algebra extending the Borel $\sigma$-algebra $\mathcal B \left({\R}\right)$ on $\R$

## Also defined as

Some treatises define $\overline \R$ without the **negative infinity** $-\infty$, the Alexandroff extension of $\R$, isomorphic to the topological group of complex numbers with norm $1$ under multiplication..

This has the benefit that extended real addition is defined on all of $\overline \R$.

A drawback is that not all suprema and infima exist.

Depending on the context one may decide which form is most suitable.

## Also known as

This structure can be referred to as the **extended real line** or the **extended (set of) real numbers**.

Also, the notations $\left[{-\infty, +\infty}\right]$ and $\left[{-\infty \,.\,.\, +\infty}\right]$ can be encountered, mimicking the notation for real intervals.

## Also see

- Equivalence of Definitions of Extended Real Number Line
- Definition:Neighborhood of Infinity (Real Analysis)
- Definition:Alexandroff Extension
- Definition:Projective Real Line
- Definition:Extended Natural Number

- Results about
**extended real numbers**can be found here.