extended real numbers

The extended real numbers are the real numbers together with $+\mathrm{\infty }$ (or simply $\mathrm{\infty }$) and $-\mathrm{\infty }$.  This set is usually denoted by $\overline{ℝ}$ or  $[-\mathrm{\infty },\mathrm{\infty }]$,  and the elements $+\mathrm{\infty }$ and $-\mathrm{\infty }$ are called plus and minus infinity, respectively.  (N.B.,  “$\overline{ℝ}$” may sometimes the algebraic closure of $ℝ$; see the special notations in algebra.)

The real numbers are in certain contexts called finite as contrast to $\mathrm{\infty }$.

0.0.1 Order on $\overline{ℝ}$

The order (http://planetmath.org/TotalOrder) relation on $ℝ$ extends to $\overline{ℝ}$ by defining that for any $x\in ℝ$, we have

$-\mathrm{\infty }$ $x,$

$x$ $\mathrm{\infty },$

and that .  For  $a\in ℝ$, let us also define intervals

$(a,\mathrm{\infty }]$ $=$ $\mathrm{\{}x\in ℝ:x>a\},$

$[-\mathrm{\infty },a)$ $=$

0.0.2 Addition

For any real number $x$, we define

$x+(\pm \mathrm{\infty })$ $=$ $(\pm \mathrm{\infty })+x=\pm \mathrm{\infty },$

and for $+\mathrm{\infty }$ and $-\mathrm{\infty }$, we define

$(\pm \mathrm{\infty })+(\pm \mathrm{\infty })$ $=$ $\pm \mathrm{\infty }.$

It should be pointed out that sums like $(+\mathrm{\infty })+(-\mathrm{\infty })$ are left undefined.  Thus $\overline{ℝ}$ is not an ordered ring although $ℝ$ is.

0.0.3 Multiplication

If $x$ is a positive real number, then

$x\cdot (\pm \mathrm{\infty })$ $=$ $(\pm \mathrm{\infty })\cdot x=\pm \mathrm{\infty }.$

Similarly, if $x$ is a negative real number, then

$x\cdot (\pm \mathrm{\infty })$ $=$ $(\pm \mathrm{\infty })\cdot x=\mp \mathrm{\infty }.$

Furthermore, for $\mathrm{\infty }$ and $-\mathrm{\infty }$, we define

$(+\mathrm{\infty })\cdot (+\mathrm{\infty })$ $=$ $(-\mathrm{\infty })\cdot (-\mathrm{\infty })=+\mathrm{\infty },$

$(+\mathrm{\infty })\cdot (-\mathrm{\infty })$ $=$ $(-\mathrm{\infty })\cdot (+\mathrm{\infty })=-\mathrm{\infty }.$

In many areas of mathematics, products like $0\cdot \mathrm{\infty }$ are left undefined.  However, a special case is measure theory, where it is convenient to define

$0\cdot (\pm \mathrm{\infty })$ $=$ $(\pm \mathrm{\infty })\cdot 0=0.$

0.0.4 Absolute value

For $\mathrm{\infty }$ and $-\mathrm{\infty }$, the absolute value is defined as

$$|\pm \mathrm{\infty }|=+\mathrm{\infty }.$$

0.0.5 Topology

The topology of $\overline{R}$ is given by the usual base of $ℝ$ together with with intervals of type  $[-\mathrm{\infty },a)$,  $(a,\mathrm{\infty }]$.  This makes $\overline{ℝ}$ into a compact topological space. $\overline{ℝ}$ can also be seen to be homeomorphic to the interval  $[-1,\mathrm{\hspace{0.17em}1}]$, via the map $x\mapsto (2/\pi )\arctan x$. Consequently, every continuous function $f:\overline{ℝ}\to \overline{ℝ}$ has a minimum and maximum.

0.0.6 Examples

1. By taking  $x=-1$  in the , we obtain the relations

   $(-1)\cdot (\pm \mathrm{\infty })$ $=$ $\mp \mathrm{\infty }.$

Title	extended real numbers
Canonical name	ExtendedRealNumbers
Date of creation	2013-03-22 13:44:44
Last modified on	2013-03-22 13:44:44
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	21
Author	matte (1858)
Entry type	Definition
Classification	msc 28-00
Classification	msc 12D99
Related topic	ImproperLimits
Related topic	IntermediateValueTheoremForExtendedRealNumbers
Related topic	ExampleOfNonCompleteLatticeHomomorphism
Defines	plus infinity
Defines	minus infinity
Defines	$\overline{ℝ}$
Defines	infinite
Defines	infinity
Defines	finite