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.
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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
- Equivalence of Definitions of Extended Real Number Line
- Definition:Neighborhood of Infinity (Real Analysis)
- Definition:Alexandroff Extension of Real Number Line
- Definition:Projective Real Line
- Definition:Extended Natural Number
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 usual (Euclidean) topology on $\R$
- A $\sigma$-algebra extending the Borel $\sigma$-algebra $\map \BB \R$ on $\R$
- Results about extended real numbers can be found here.