Integer

This commodity is about the algebraic concept. For integers in computer science, see Accumulation (computer science

).

The integers (from the Latin integer, actually "untouched", appropriately "whole": the chat absolute comes from the aforementioned origin, but via French1) are formed by the accustomed numbers (including 0) (0, 1, 2, 3, ...) calm with the negatives of the non-zero accustomed numbers (−1, −2, −3, ...).They are accepted as Positive and Negative Integers respectively. Viewed as a subset of the absolute numbers, they are numbers that can be accounting after a apportioned or decimal component, and

fall aural the set {..., −2, −1, 0, 1, 2, ...}. For example, 21, 4, and −2048 are integers; 9.75, 5½, and 14% are not integers.

The set of all integers is generally denoted by a boldface Z (or blackboard adventurous \mathbb{Z}, Unicode U+2124 ℤ), which stands for Zahlen (German for numbers, arresting ˈtsaːlən).2 The set \mathbb{Z}_n is the bound set of integers modulo n (for example, \mathbb{Z}_2=\{0,1\}).citation needed



The integers (with accession as operation) anatomy the aboriginal accumulation absolute the accretion monoid of the accustomed numbers. Like the accustomed numbers, the integers anatomy a countably absolute set.

In algebraic cardinal theory, these frequently accepted integers, anchored in the acreage of rational numbers, are referred to as rational integers to analyze them from the added broadly authentic algebraic integers (but with "rational" acceptation "quotient of integers", this attack at attention suffers from circularity

).

Algebraic properties

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, zero, Z (unlike the accustomed numbers) is additionally bankrupt beneath subtraction. Z is not bankrupt beneath division, back 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 back the backer is negative

).

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The afterward lists some of the basal backdrop of accession and multiplication for any integers a, b and c.citation needed

In the accent of abstruse algebra, the aboriginal bristles backdrop listed aloft for accession say that Z beneath accession is an abelian group. As a accumulation beneath addition, Z is a circadian group, back every nonzero accumulation can be accounting as a bound sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1). In fact, Z beneath accession is the alone absolute circadian group, in the faculty that any absolute circadian accumulation is isomorphic to Z.citation needed



The aboriginal four backdrop listed aloft for multiplication say that Z beneath multiplication is a capricious monoid. However not every accumulation has a multiplicative inverse; e.g. there is no accumulation x such that 2x = 1, because the larboard duke ancillary is even, while the appropriate duke ancillary is odd. This agency that Z beneath multiplication is not a group.citation needed



All the rules from the aloft acreage table, except for the last, taken calm say that Z calm with accession and multiplication is a capricious arena with unity. Adding the aftermost acreage says that Z is an basic domain. In fact, Z provides the action for defining such a structure.citation needed

Order-theoretic properties

Z is a absolutely ordered set after high or lower bound. The acclimation of Z is accustomed by:citation needed



... −3 < −2 < −1 < 0 < 1 < 2 < 3 <

...

An accumulation is absolute if it is greater than aught 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

.

It follows that Z calm with the aloft acclimation is an ordered ring.citation needed

The integers are the alone basic area whose absolute elements are well-ordered, and in which adjustment is preserved by addition.

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Construction

The integers can be formally complete as the adequation classes of ordered pairs of accustomed numbers (a, b).3

The intuition is that (a, b) stands for the aftereffect of adding b from a.3 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) \sim (c,d) \,\!

precisely when

a+d = b+c. \,\!

Addition and multiplication of integers can be authentic in agreement of the agnate operations on the accustomed numbers3; cogent by (a,b) the adequation chic accepting (a,b) as a member, one has:

(a,b)+(c,d) := (a+c,b+d).\,

(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).\,

Hence accession can be authentic as the accession of

the

additive inverse:

(a,b)-(c,d) := (a+d,b+c).\,

The accepted acclimation on the integers is accustomed by

:

(a,b)<(c,d)\, iff

a+d < b+c.\,

It is calmly absolute that these definitions are absolute of the best of assembly of the adequation classes

.

Every adequation chic has a different affiliate that is of the anatomy (n,0) or (0,n) (or both at once). The accustomed cardinal n is articular with the chic (n,0) (in added words the accustomed numbers are anchored into the integers by map sending n to (n,0)), and the chic (0,n) is denoted −n (this covers all actual classes, and gives the chic (0,0) a additional time back −0 = 0.citation needed



Thus, (a,b) is denoted bycitation needed



\begin{cases} a-b, & \mbox{if } a \ge b \\ -(b-a), & \mbox{if } a < b. \end{cases}

If the accustomed numbers are articular with the agnate integers (using the embedding mentioned above), this assemblage creates no ambiguity.citation needed



This characters recovers the accustomed representation of the integers as {... −3,−2,−1, 0, 1, 2, 3, ...}.

Some examples are

:

\begin{align} 0 &= (0,0) &= (1,1) &= \cdots & &= (k,k) \\ 1 &= (1,0) &= (2,1) &= \cdots & &= (k+1,k) \\ -1 &= (0,1) &= (1,2) &= \cdots & &= (k,k+1) \\ 2 &= (2,0) &= (3,1) &= \cdots & &= (k+2,k) \\ -2 &= (0,2) &= (1,3) &= \cdots & &= (k,k+2). \end{align}

Integers in computing

An accumulation is generally a archaic datatype in computer languages. However, accumulation datatypes can alone represent a subset of all integers, back applied computers are of bound capacity. Also, in the accepted two's accompaniment representation, the inherent analogue of assurance distinguishes amid "negative" and "non-negative" rather than "negative, positive, and 0". (It is, however, absolutely accessible for a computer to actuate whether an accumulation amount is absolutely positive.) Anchored breadth accumulation approximation datatypes (or subsets) are denoted int or Accumulation in several programming languages (such as Algol68, C, Java, Delphi, etc.).citation needed



Variable-length representations of

integers, such as bignums, can abundance any accumulation that fits in the computer's memory. Other accumulation datatypes are implemented with a anchored size, usually a cardinal of $.25 which is a ability of 2 (4, 8, 16, etc.) or a memorable cardinal of decimal digits (e.g., 9 or 10).citation needed

Cardinality

The cardinality of the set of integers is according to \aleph_0 (aleph-null). This is readily approved by the architecture of a bijection, that is, a action that is injective and surjective from Z to N.citation needed

If N = {0, 1, 2, ...} again accede the function:

f(x) = \begin{cases} 2|x|, & \mbox{if } x < 0 \\ 0, & \mbox{if } x = 0 \\ 2x-1, & \mbox{if } x > 0. \end{cases}

{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,0) (1,1) (2,3) (3,5) ... }

If N = {1,2,3,...} again accede the function:

g(x) = \begin{cases} 2|x|, & \mbox{if } x < 0 \\ 2x+1, & \mbox{if } x \ge 0. \end{cases}

{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,1) (1,3) (2,5) (3,7) ... }

If the area is belted to Z again anniversary and every affiliate of Z has one and alone one agnate affiliate of N and by the analogue of basal adequation the two sets accept according cardinality.citation neede