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 the extended real number line $\overline \R$ 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
The extended real number line can be referred to as:
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.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): infinity $(1)$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): infinity $(1)$