Definition:Extended Real Number Line

Definition 1
The set of extended real numbers $\overline{\R}$ 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

Definition 2
The extended real line, $\overline \R$, is the order completion of the real line, $\R$.

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

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 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 $\mathbb R$, 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
Some authors call this the extended real line.

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