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 calledplus 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 $\overline{ℝ}$ bydefining 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 ringalthough $ℝ$ 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 ismeasure 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  $[-1,\mathrm{\hspace{0.17em}1}]$, viathe map $x\mapsto (2/\pi )\arctan x$.Consequently, everycontinuous function $f:\overline{ℝ}\to \overline{ℝ}$ hasa minimum and maximum.

0.0.6 Examples

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

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