# extended real numbers

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 the algebraic closure of $\mathbb{R}$; see the special notations in algebra.)

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

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

The order (http://planetmath.org/TotalOrder) 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

For any real number $x$, 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.

## 0.0.3 Multiplication

If $x$ is a positive real number, then

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

Similarly, if $x$ 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

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

## 0.0.4 Absolute value

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

 $|\pm\infty|=+\infty.$

## 0.0.5 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.

## 0.0.6 Examples

1. 1.

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

 $\displaystyle(-1)\cdot(\pm\infty)$ $\displaystyle=$ $\displaystyle\mp\infty.$
Title extended real numbers ExtendedRealNumbers 2013-03-22 13:44:44 2013-03-22 13:44:44 matte (1858) matte (1858) 21 matte (1858) Definition msc 28-00 msc 12D99 ImproperLimits IntermediateValueTheoremForExtendedRealNumbers ExampleOfNonCompleteLatticeHomomorphism plus infinity minus infinity $\overline{\mathbb{R}}$ infinite infinity finite