%%% This file is part of PlanetMath snapshot of 2009-01-12
%%% Primary Title: integer
%%% Primary Category Code: 03-00
%%% Filename: Integer.tex
%%% Version: 8
%%% Owner: CWoo
%%% Author(s): CWoo, djao
%%% PlanetMath is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.        
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}

\usepackage{html}

\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}

\begin{document}

 The set of integers, denoted by the symbol $\mathbb{Z}$, is the set $\{\dots -3, -2, -1, 0, 1, 2, 3, \dots\}$ consisting of the \htmladdnormallink{natural numbers}{http://planetmath.org/encyclopedia/NaturalNumber.html} and their \htmladdnormallink{negatives}{http://planetmath.org/encyclopedia/Negative.html}.

Mathematically, $\mathbb{Z}$ is defined to be the set of \htmladdnormallink{equivalence classes}{http://planetmath.org/encyclopedia/EquivalenceClass.html} of pairs of natural numbers $\mathbb{N} \times \mathbb{N}$ under the \htmladdnormallink{equivalence relation}{http://planetmath.org/encyclopedia/EquivalenceClass2.html} $(a,b) \sim (c,d)$ if $a+d = b+c$.

\htmladdnormallink{Addition}{http://planetmath.org/encyclopedia/Addition.html} and \htmladdnormallink{multiplication}{http://planetmath.org/encyclopedia/Multiplication.html} of integers are defined as follows:
\begin{itemize}
\item $(a,b)+(c,d) := (a+c,b+d)$
\item $(a,b)\cdot(c,d) := (ac+bd,ad+bc)$
\end{itemize}
Typically, the \htmladdnormallink{class}{http://planetmath.org/encyclopedia/LimitationOfSizePrinciple.html} of $(a,b)$ is denoted by symbol $n$ if $b \leq a$ (resp. $-n$ if $a \leq b$), where $n$ is the unique natural number such that $a=b+n$ (resp. $a+n=b$). Under this notation, we recover the familiar \htmladdnormallink{representation}{http://planetmath.org/encyclopedia/InfiniteDimensional3.html} of the integers as $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. Here are some examples:
\begin{itemize}
\item $0 = $ equivalence class of $(0,0) = $ equivalence class of $(1,1) = \dots$
\item $1 = $ equivalence class of $(1,0) = $ equivalence class of $(2,1) = \dots$
\item $-1 = $ equivalence class of $(0,1) = $ equivalence class of $(1,2) = \dots$
\end{itemize}
The set of integers $\mathbb{Z}$ under the addition and multiplication \htmladdnormallink{operations}{http://planetmath.org/encyclopedia/Operation.html} defined above form an \htmladdnormallink{integral domain}{http://planetmath.org/encyclopedia/IntegralDomain.html}. The integers admit the following \htmladdnormallink{ordering relation}{http://planetmath.org/encyclopedia/OrderingRelation.html} making $\mathbb{Z}$ into an \htmladdnormallink{ordered ring}{http://planetmath.org/encyclopedia/OrderedField.html}: $(a,b) \leq (c,d)$ in $\mathbb{Z}$ if $a+d \leq b+c$ in $\mathbb{N}$.

The \htmladdnormallink{ring of integers}{http://planetmath.org/encyclopedia/IntegralClosure.html} is also a \htmladdnormallink{Euclidean domain}{http://planetmath.org/encyclopedia/EuclideanRing2.html}, with \htmladdnormallink{valuation}{http://planetmath.org/encyclopedia/NonArchimedean.html} given by the absolute value \htmladdnormallink{function}{http://planetmath.org/encyclopedia/Range2.html}.

\end{document}