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# 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 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 $\infty$.

# 0.0.1 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

$\displaystyle(a,\,\infty{]}$ | $\displaystyle=$ | $\displaystyle\{x\in\mathbb{R}:x>a\},$ | ||

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

# 0.0.2 Addition

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

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

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

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

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

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## Comments

## What is wrong with this latex code

PM reports that an error in this file:

---------------------------------------------

66: \begin{eqnarray*}

67: (a,\infty] &=& \{ x\in \sR : x>a \}, \\

68: [-\infty,a) &=& \{ x\in \sR : x>a \}.

69: \end{eqnarray*}

70:

!!! Missing $$ inserted

\sR is defined as

\newcommand{\sR}[0]{\mathbb{R}}

and that works fine elsewhere in the entry.

---------------------------------------------

I can't see any problem, and the entry renders fine without

these lines.

Any ideas?

## Re: What is wrong with this latex code

Array enviroments have a problem with having a [ as the first character of a new line as you can use [ <measurement> ] to set the gap between lines (or something). You might try replacing line 68 with

68: {[}-\infty, a)........

## Re: What is wrong with this latex code

silverfish wrote:

> 68: {[}-\infty, a)........

I think you also need {-\infty} rather than -\infty in order for the spacing to come out correctly.

## Re: What is wrong with this latex code

That does it. Thanks!