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Ordered Ring Equivalence Relation Ordering Relation Integral Domain Representation Lie Algebra Operation Multiplication Equivalence Class Valuation Function Class Integral Closure Addition Natural Number Positive Euclidean Ring
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Integer
The set of integers, denoted by the symbol
, is the set
consisting of the natural numbers
and their negatives.
Mathematically,
is defined to be the set of equivalence classes
of pairs of natural numbers
under the equivalence relation
if 
.
Addition and multiplication of integers are defined as follows:
is denoted by symbol 
if 
(resp. 
if 
), where 
is the unique natural number such that 
(resp. 
). Under this notation, we recover the familiar representation
of the integers as
. Here are some examples:
equivalence class of 
equivalence class of
equivalence class of 
equivalence class of
equivalence class of 
equivalence class of
under the addition and multiplication operations
defined above form an integral domain. The integers admit the following ordering relation
making
into an ordered ring:
in
if
in
.
The ring of integers is also a Euclidean domain, with valuation given by the absolute value function.