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:

• $(a,b)+(c,d):=(a+c,b+d)$

• $(a,b)\cdot (c,d):=(ac+bd,ad+bc)$

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:

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

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

• $-1=$ 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.