# Definition:Extended Real Number Line

## 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:

## 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

• Results about extended real numbers can be found here.