PlanetMath: extended real numbers

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extended real numbers

(Definition)

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

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

Order on $\overline{\mathbb{R}}$

The order relation on $\mathbb{R}$ extends to $\overline{\mathbb{R}}$ by defining that for any $x\in \mathbb{R}$ , we have

$\displaystyle -\infty$	$\displaystyle <$	$\displaystyle x,$
$\displaystyle x$	$\displaystyle <$	$\displaystyle \infty,$

and that $-\infty < \infty$ . For $a\in\mathbb{R}$ , let us also define intervals

$\displaystyle (a,\,\infty{]}$	$\displaystyle =$	$\displaystyle \{x\in \mathbb{R}: x>a \},$
$\displaystyle {[}{-\infty},\,a)$	$\displaystyle =$	$\displaystyle \{x\in \mathbb{R}: x<a \}.$

Addition

For any real number

, we define

$\displaystyle x + (\pm\infty)$

$\displaystyle =$

$\displaystyle (\pm\infty) + x = \pm\infty,$

and for $+\infty$ and $-\infty$ , we define

$\displaystyle (\pm \infty) + (\pm \infty)$

$\displaystyle =$

$\displaystyle \pm \infty.$

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

Multiplication

is a positive real number, then

$\displaystyle x \cdot (\pm \infty)$

$\displaystyle =$

$\displaystyle (\pm\infty)\cdot x = \pm\infty.$

Similarly, if

is a negative real number, then

$\displaystyle x \cdot (\pm \infty)$

$\displaystyle =$

$\displaystyle (\pm \infty)\cdot x = \mp\infty.$

Furthermore, for $\infty$ and $-\infty$ , we define

$\displaystyle (+\infty) \cdot(+\infty)$	$\displaystyle =$	$\displaystyle (-\infty)\cdot (-\infty) = +\infty,$
$\displaystyle (+\infty) \cdot (- \infty)$	$\displaystyle =$	$\displaystyle (-\infty)\cdot (+\infty) = -\infty.$

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

$\displaystyle 0\cdot (\pm \infty)$

$\displaystyle =$

$\displaystyle (\pm \infty) \cdot 0 = 0.$

Absolute value

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

$\displaystyle \vert\pm \infty\vert = +\infty.$

Topology

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

Examples

By taking in the product rule, we obtain the relations

$\displaystyle (-1)\cdot (\pm\infty)$ $\displaystyle =$ $\displaystyle \mp \infty.$

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"extended real numbers" is owned by matte. [ full author list (6) | owner history (2) ]

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Also defines:

plus infinity, minus infinity, $\overline{\mathbb{R}}$ , infinite, infinity, finite

This object's parent.

Attachments:: continuous functions on the extended real numbers (Theorem) by Wkbj79

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Cross-references: continuous function, map, homeomorphic, compact, type, base, topology, absolute value, theory, measure, products, negative, positive, ordered ring, sums, intervals, relation, special notations in algebra, algebraic closure, plus, real numbers
There are 77 references to this entry.

This is version 18 of extended real numbers, born on 2003-07-12, modified 2007-04-27.
Object id is 4441, canonical name is ExtendedRealNumbers.
Accessed 17234 times total.

Classification:

AMS MSC:	28-00 (Measure and integration :: General reference works )
	12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)

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