integer
The set of integers, denoted by the symbol $\mathbb{Z}$, is the set $\{\mathrm{\dots}3,2,1,0,1,2,3,\mathrm{\dots}\}$ consisting of the natural numbers^{} and their negatives.
Mathematically, $\mathbb{Z}$ is defined to be the set of equivalence classes^{} of pairs of natural numbers $\mathbb{N}\times \mathbb{N}$ under the equivalence relation $(a,b)\sim (c,d)$ if $a+d=b+c$.
Addition^{} and multiplication of integers are defined as follows:

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$(a,b)+(c,d):=(a+c,b+d)$

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$(a,b)\cdot (c,d):=(ac+bd,ad+bc)$
Typically, the class of $(a,b)$ is denoted by symbol $n$ if $b\le a$ (resp. $n$ if $a\le b$), where $n$ is the unique natural number such that $a=b+n$ (resp. $a+n=b$). Under this notation, we recover the familiar representation of the integers as $\{\mathrm{\dots},3,2,1,0,1,2,3,\mathrm{\dots}\}$. Here are some examples:

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$0=$ equivalence class of $(0,0)=$ equivalence class of $(1,1)=\mathrm{\dots}$

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$1=$ equivalence class of $(1,0)=$ equivalence class of $(2,1)=\mathrm{\dots}$

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$1=$ equivalence class of $(0,1)=$ equivalence class of $(1,2)=\mathrm{\dots}$
The set of integers $\mathbb{Z}$ under the addition and multiplication operations^{} defined above form an integral domain^{}. The integers admit the following ordering relation making $\mathbb{Z}$ into an ordered ring: $(a,b)\le (c,d)$ in $\mathbb{Z}$ if $a+d\le b+c$ in $\mathbb{N}$.
The ring of integers^{} is also a Euclidean domain^{}, with valuation^{} given by the absolute value^{} function.
Title  integer 

Canonical name  Integer 
Date of creation  20130322 11:50:39 
Last modified on  20130322 11:50:39 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 1100 
Classification  msc 0300 
Synonym  rational integer 
Synonym  $\mathbb{Z}$ 
Related topic  Irrational 