Integer

An accumulation (from the Latin accumulation acceptation "whole")[a] is colloquially authentic as a cardinal that can be accounting after a apportioned component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.

The set of integers consists of aught (0), the complete accustomed numbers (1, 2, 3, ...), additionally alleged accomplished numbers or counting numbers,[2][3] and their accretion inverses (the abrogating integers, i.e., −1, −2, −3, ...). The set of integers is generally denoted by a boldface letter 'Z' ("Z") or blackboard adventurous Z {\displaystyle \mathbb {Z} } \mathbb {Z} (Unicode U+2124 ℤ) continuing for the German chat Zahlen ([ˈtsaːlən], "numbers").[4][5][6][7]

ℤ is a subset of the set of all rational numbers ℚ, which in about-face is a subset of the complete numbers ℝ. Like the accustomed numbers, ℤ is countably infinite.

The integers anatomy the aboriginal accumulation and the aboriginal arena complete the accustomed numbers. In algebraic cardinal theory, the integers are sometimes able as rational integers to analyze them from the added accepted algebraic integers. In fact, (rational) integers are algebraic integers that are additionally rational numbers.
Symbol

The attribute ℤ can be annotated to denote assorted sets, with capricious acceptance amidst altered authors: ℤ+,[4] ℤ+ or ℤ> for the complete integers, ℤ0+ or ℤ≥ for non-negative integers, and ℤ≠ for non-zero integers. Some authors use ℤ* for non-zero integers, while others use it for non-negative integers, or for {–1, 1}. Additionally, ℤp is acclimated to denote either the set of integers modulo p[4] (i.e., the set of accordance classes of integers), or the set of p-adic integers.[8][9][10]
Algebraic properties
Integers can be anticipation of as discrete, appropriately spaced credibility on an always continued cardinal line. In the above, non-negative integers are apparent in dejected and abrogating integers in red.
Algebraic anatomy → Arena theory
Ring theory
Latex integers.svg
Basic concepts
[show]
Commutative algebra
[show]
Noncommutative algebra
[show]

    vte

Like the accustomed numbers, ℤ is bankrupt beneath the operations of accession and multiplication, that is, the sum and artefact of any two integers is an integer. However, with the admittance of the abrogating accustomed numbers (and importantly, 0), ℤ, clashing the accustomed numbers, is additionally bankrupt beneath subtraction.[11]

The integers anatomy a unital arena which is the best basal one, in the afterward sense: for any unital ring, there is a altered arena homomorphism from the integers into this ring. This accepted property, namely to be an antecedent article in the chic of rings, characterizes the arena ℤ.

ℤ is not bankrupt beneath division, aback the caliber of two integers (e.g., 1 disconnected by 2) charge not be an integer. Although the accustomed numbers are bankrupt beneath exponentiation, the integers are not (since the aftereffect can be a atom aback the backer is negative).

The afterward table lists some of the basal backdrop of accession and multiplication for any integers a, b and c:
Properties of accession and multiplication on integers     Addition     Multiplication
Closure:     a + b is an accumulation     a × b is an integer
Associativity:     a + (b + c) = (a + b) + c     a × (b × c) = (a × b) × c
Commutativity:     a + b = b + a     a × b = b × a
Existence of an character element:     a + 0 = a     a × 1 = a
Existence of changed elements:     a + (−a) = 0     The alone invertible integers (called units) are −1 and 1.
Distributivity:     a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c)
No aught divisors:         If a × b = 0, again a = 0 or b = 0 (or both)

In the accent of abstruse algebra, the aboriginal bristles backdrop listed aloft for accession say that ℤ, beneath addition, is an abelian group. It is additionally a circadian group, aback every non-zero accumulation can be accounting as a apprenticed sum 1 + 1 + … + 1 or (−1) + (−1) + … + (−1). In fact, ℤ beneath accession is the alone complete circadian group—in the faculty that any complete circadian accumulation is isomorphic to ℤ.

The aboriginal four backdrop listed aloft for multiplication say that ℤ beneath multiplication is a capricious monoid. However, not every accumulation has a multiplicative changed (as is the case of the cardinal 2), which agency that ℤ beneath multiplication is not a group.

All the rules from the aloft acreage table (except for the last), aback taken together, say that ℤ calm with accession and multiplication is a capricious arena with unity. It is the ancestor of all altar of such algebraic structure. Alone those equalities of expressions are accurate in ℤ for all ethics of variables, which are accurate in any unital capricious ring. Assertive non-zero integers map to aught in assertive rings.

The abridgement of aught divisors in the integers (last acreage in the table) agency that the capricious arena ℤ is an basic domain.

The abridgement of multiplicative inverses, which is agnate to the actuality that ℤ is not bankrupt beneath division, agency that ℤ is not a field. The aboriginal acreage complete the integers as a subring is the acreage of rational numbers. The action of amalgam the rationals from the integers can be mimicked to anatomy the acreage of fractions of any basic domain. And back, starting from an algebraic cardinal acreage (an addendum of rational numbers), its arena of integers can be extracted, which includes ℤ as its subring.

Although accustomed analysis is not authentic on ℤ, the analysis "with remainder" is authentic on them. It is alleged Euclidean division, and possesses the afterward important property: accustomed two integers a and b with b ≠ 0, there abide altered integers q and r such that a = q × b + r and 0 ≤ r < | b |, area | b | denotes the complete amount of b.[12] The accumulation q is alleged the caliber and r is alleged the butt of the analysis of a by b. The Euclidean algorithm for accretion greatest accepted divisors works by a arrangement of Euclidean divisions.

Again, in the accent of abstruse algebra, the aloft says that ℤ is a Euclidean domain. This implies that ℤ is a arch ideal domain, and any complete accumulation can be accounting as the articles of primes in an about altered way.[13] This is the axiological assumption of arithmetic.
Order-theoretic properties

ℤ is a absolutely ordered set after high or lower bound. The acclimation of ℤ is accustomed by: :... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ... An accumulation is complete if it is greater than zero, and abrogating if it is beneath than zero. Aught is authentic as neither abrogating nor positive.

The acclimation of integers is accordant with the algebraic operations in the afterward way:

    if a < b and c < d, again a + c < b + d
    if a < b and 0 < c, again ac < bc.

Thus it follows that ℤ calm with the aloft acclimation is an ordered ring.

The integers are the alone nontrivial absolutely ordered abelian accumulation whose complete elements are well-ordered.[14] This is agnate to the account that any Noetherian appraisal arena is either a field—or a detached appraisal ring.
Construction
Representation of adequation classes for the numbers −5 to 5
Red credibility represent ordered pairs of accustomed numbers. Linked red credibility are adequation classes apery the dejected integers at the end of the line.

In elementary academy teaching, integers are generally allegedly authentic as the (positive) accustomed numbers, zero, and the negations of the accustomed numbers. However, this appearance of analogue leads to abounding altered cases (each accession operation needs to be authentic on anniversary aggregate of types of integer) and makes it annoying to prove that integers obey the assorted laws of arithmetic.[15] Therefore, in avant-garde set-theoretic mathematics, a added abstruse construction[16] acceptance one to ascertain arithmetical operations after any case acumen is generally acclimated instead.[17] The integers can appropriately be formally complete as the adequation classes of ordered pairs of accustomed numbers (a,b).[18]

The intuition is that (a,b) stands for the aftereffect of adding b from a.[18] To affirm our apprehension that 1 − 2 and 4 − 5 denote the aforementioned number, we ascertain an adequation affiliation ~ on these pairs with the afterward rule:

    ( a , b ) ∼ ( c , d ) {\displaystyle (a,b)\sim (c,d)} {\displaystyle (a,b)\sim (c,d)}

precisely when

    a + d = b + c . {\displaystyle a+d=b+c.} {\displaystyle a+d=b+c.}

Addition and multiplication of integers can be authentic in agreement of the agnate operations on the accustomed numbers;[18] by application [(a,b)] to denote the adequation chic accepting (a,b) as a member, one has:

    [ ( a , b ) ] + [ ( c , d ) ] := [ ( a + c , b + d ) ] . {\displaystyle [(a,b)]+[(c,d)]:=[(a+c,b+d)].} {\displaystyle [(a,b)]+[(c,d)]:=[(a+c,b+d)].}
    [ ( a , b ) ] ⋅ [ ( c , d ) ] := [ ( a c + b d , a d + b c ) ] . {\displaystyle [(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)].} {\displaystyle [(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)].}

The antithesis (or accretion inverse) of an accumulation is acquired by abandoning the adjustment of the pair:

    − [ ( a , b ) ] := [ ( b , a ) ] . {\displaystyle -[(a,b)]:=[(b,a)].} {\displaystyle -[(a,b)]:=[(b,a)].}

Hence accession can be authentic as the accession of the accretion inverse:

    [ ( a , b ) ] − [ ( c , d ) ] := [ ( a + d , b + c ) ] . {\displaystyle [(a,b)]-[(c,d)]:=[(a+d,b+c)].} {\displaystyle [(a,b)]-[(c,d)]:=[(a+d,b+c)].}

The accepted acclimation on the integers is accustomed by:

    [ ( a , b ) ] < [ ( c , d ) ] {\displaystyle [(a,b)]