The set of integers, denoted by the symbol , is the set {-3,-2,-1,0,1,2,3,} consisting of the natural numbersMathworldPlanetmath and their negatives.

Mathematically, is defined to be the set of equivalence classesMathworldPlanetmathPlanetmath of pairs of natural numbers × under the equivalence relation (a,b)(c,d) if a+d=b+c.

AdditionPlanetmathPlanetmath and multiplication of integers are defined as follows:

  • (a,b)+(c,d):=(a+c,b+d)

  • (a,b)(c,d):=(ac+bd,ad+bc)

Typically, the class of (a,b) is denoted by symbol n if ba (resp. -n if ab), 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 {,-3,-2,-1,0,1,2,3,}. Here are some examples:

  • 0= equivalence class of (0,0)= equivalence class of (1,1)=

  • 1= equivalence class of (1,0)= equivalence class of (2,1)=

  • -1= equivalence class of (0,1)= equivalence class of (1,2)=

The set of integers under the addition and multiplication operationsMathworldPlanetmath defined above form an integral domainMathworldPlanetmath. The integers admit the following ordering relation making into an ordered ring: (a,b)(c,d) in if a+db+c in .

The ring of integersMathworldPlanetmath is also a Euclidean domainMathworldPlanetmath, with valuationMathworldPlanetmathPlanetmath given by the absolute valueMathworldPlanetmathPlanetmathPlanetmath function.

Title integer
Canonical name Integer
Date of creation 2013-03-22 11:50:39
Last modified on 2013-03-22 11:50:39
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 11-00
Classification msc 03-00
Synonym rational integer
Related topic Irrational