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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!

## A bit of history

Before giving further comments on Fermat’s theorem and related matters let me give a bit of history: 1640 Fermat’s theorem 1740(circa) Euler’s generalisation of FT 2004 Euler’s generalisation of FT - a further generalisation (Devaraj)) 2006 Minimum Universal

^{}exponent generalisation of Fermat’s T. (Devaraj).2012 Ultimate generalisation of FT -Pahio and Devaraj

My paper ” Euler’s generalisation…….” freed FT of the requirement of base and exponent to be coprime. Secondly we can identify small factors of very large numbers by merely operating on the exponents. Before concluding this message I would like to thank Pahio for enabling ültimate generalisation of FT.