Definition:Extended Real Number Line

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Definition

Definition 1

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

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

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.