# Definition:Extended Real Number Line

## Definition

### Definition 1

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

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

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 $-\infty$.

## Also defined as

Some sources define $\overline \R$ as $\R \cup \set \infty$, that is, without the negative infinity $-\infty$.

This is the Alexandroff extension of $\R$.

This is 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
the extended (set of) real numbers

Also, the notations $\sqbrk {-\infty, +\infty}$ and $\closedint {-\infty} {+\infty}$ can be encountered, extending the notation for real intervals.

## Also see

### Structures on $\overline \R$

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

• Results about extended real numbers can be found here.