Definition:Extended Real Number Line

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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.