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--[[User:71.245.69.130|71.245.69.130]] 00:17, 12 September 2007 (UTC)#REDIRECT [[#REDIRECT [[Insert text]]<br /><sup><sub>Superscript text</sub><small><blockquote>
[[Image:Latex_integers.svg|thumb|100px|The integers are often denoted by the above symbol.]]
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An '''integer''' is a whole number (one that can be written without a fractional part, or a fractional part which is zero&mdash;for example: <tt>7, 1, 0, &minus;234, 5.00</tt>).

In [[mathematics]], the ''integers'' ([[Latin]], ''integer'', literally, untouched, whole, entire, i.e., a [[whole number]]) are defined in the abstract; they include the positive [[natural number]]s ([[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], &hellip;), their [[negative and non-negative numbers|negative]]s ([[&minus;1]], &minus;2, &minus;3, ...), and the number [[zero]].

More formally, the integers are the only [[integral domain]] whose positive elements are [[well-ordered]], and in which order is preserved by [[addition]]. Like the natural numbers, the integers form a [[Countable set|countably infinite]] set. The [[set]] of all integers is often denoted by a boldface '''Z''' (or [[blackboard bold]] <math>\mathbb{Z}</math>, [[Unicode]] U+2124), which stands for ''Zahlen'' ([[German language|German]] for ''numbers'').<ref>[http://members.aol.com/jeff570/nth.html "Earliest Uses of Symbols of Number Theory"]</ref>

In [[algebraic number theory]], these commonly understood integers, embedded in the [[field (mathematics)|field]] of [[rational number]]s, are referred to as '''rational integers''' to distinguish them from the more broadly defined [[algebraic integer]]s.

== Algebraic properties ==
Like the natural numbers, '''Z''' is [[closure (mathematics)|closed]] under the [[binary operation|operations]] of [[addition]] and [[multiplication]], that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, [[0 (number)|zero]], '''Z''' (unlike the natural numbers) is also closed under [[subtraction]]. '''Z''' is not closed under the operation of [[division (mathematics)|division]], since the quotient of two integers (''e.g.'', 1 divided by 2), need not be an integer.

The following lists some of the basic properties of addition and multiplication for any integers ''a'', ''b'' and ''c''.
{| class="wikitable"
| || addition || multiplication
|-
|-
<blockquote>
| [[Closure (mathematics)|closure]]: || ''a''&nbsp;+&nbsp;''b''&nbsp;&nbsp; is an integer || ''a''&nbsp;&times;&nbsp;''b''&nbsp;&nbsp; is an integer
! header 1
! header 2
! header 3
|-
|-
| row 1, cell 1
| [[associativity]]: || ''a''&nbsp;+&nbsp;(''b''&nbsp;+&nbsp;''c'')&nbsp;&nbsp;=&nbsp;&nbsp;(''a''&nbsp;+&nbsp;''b'')&nbsp;+&nbsp;''c'' || ''a''&nbsp;&times;&nbsp;(''b''&nbsp;&times;&nbsp;''c'')&nbsp;&nbsp;=&nbsp;&nbsp;(''a''&nbsp;&times;&nbsp;''b'')&nbsp;&times;&nbsp;''c''
| row 1, cell 2
| row 1, cell 3
|-
|-
| row 2, cell 1
| [[commutativity]]: || ''a''&nbsp;+&nbsp;''b''&nbsp;&nbsp;=&nbsp;&nbsp;''b''&nbsp;+&nbsp;''a'' || ''a''&nbsp;&times;&nbsp;''b''&nbsp;&nbsp;=&nbsp;&nbsp;''b''&nbsp;&times;&nbsp;''a''
| row 2, cell 2
| row 2, cell 3
</blockquote><blockquote>
<blockquote>
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</blockquote>{| class="wikitable"
|-
|-
<!-- ! header 1
| existence of an [[identity element]]: || ''a''&nbsp;+&nbsp;0&nbsp;&nbsp;=&nbsp;&nbsp;''a'' || ''a''&nbsp;&times;&nbsp;1&nbsp;&nbsp;=&nbsp;&nbsp;''a''
! header 2
! header 3
|-
|-
| row 1, cell 1
| existence of [[inverse element]]s: || ''a''&nbsp;+&nbsp;(&minus;''a'')&nbsp;&nbsp;=&nbsp;&nbsp;0 ||
| row 1, cell 2
| row 1, cell 3
|-
|-
| row 2, cell 1
| [[distributivity]]: || colspan=2 align=center| ''a''&nbsp;&times;&nbsp;(''b''&nbsp;+&nbsp;''c'')&nbsp;&nbsp;=&nbsp;&nbsp;(''a''&nbsp;&times;&nbsp;''b'')&nbsp;+&nbsp;(''a''&nbsp;&times;&nbsp;''c'')
| row 2, cell 2
|-
| row 2, cell 3 -->[''http://www.example.com link title'''''[Bold text][[Image:Example.jpg]]''']
| No [[zero divisors]]: || || if ''ab'' = 0, then either ''a'' = 0 or ''b'' = 0 (or both)
|}
|}
</blockquote>

|}</small></sup>]]
In the language of [[abstract algebra]], the first five properties listed above for addition say that '''Z''' under addition is an [[abelian group]]. As a group under addition, '''Z''' is a [[cyclic group]], since every nonzero integer can be written as a finite sum 1 + 1 + ... 1 or (&minus;1) + (&minus;1) + ... + (&minus;1). In fact, '''Z''' under addition is the ''only'' infinite cyclic group, in the sense that any infinite cyclic group is [[group isomorphism|isomorphic]] to '''Z'''.

The first four properties listed above for multiplication say that '''Z''' under multiplication is a [[commutative monoid]]. However, note that not every integer has a multiplicative inverse; e.g. there is no integer ''x'' such that 2''x'' = 1, because the left hand side is even, while the right hand side is odd. This means that '''Z''' under multiplication is not a group.

All the properties from the above table, except for the last, taken together say that '''Z''' together with addition and multiplication is a commutative [[ring (mathematics)|ring]] with unity. Adding the last property says that '''Z''' is an [[integral domain]]. In fact, '''Z''' provides the motivation for defining such a structure.

The lack of multiplicative inverses, which is equivalent to the fact that '''Z''' is not closed under division, means that '''Z''' is ''not'' a [[field (mathematics)|field]]. The smallest field containing the integers is the field of [[rational number]]s. This process can be mimicked to form the [[field of fractions]] of any integral domain.

Although ordinary division is not defined on '''Z''', it does possess an important property called the [[division algorithm]]: that is, given two integers ''a'' and ''b'' with ''b''&nbsp;&ne;&nbsp;0, there exist unique integers ''q'' and ''r'' such that ''a''&nbsp;=&nbsp;''q'' &times; ''b''&nbsp;+&nbsp;''r'' and 0 &le; ''r'' < |''b''|, where |''b''| denotes the [[absolute value]] of ''b''. The integer ''q'' is called the ''quotient'' and ''r'' is called the ''[[remainder]]'', resulting from division of ''a'' by ''b''. This is the basis for the [[Euclidean algorithm]] for computing [[greatest common divisor]]s.

Again, in the language of abstract algebra, the above says that '''Z''' is a [[Euclidean domain]]. This implies that '''Z''' is a [[principal ideal domain]] and any positive integer can be written as the products of [[prime number|primes]] in an essentially unique way. This is the [[fundamental theorem of arithmetic]].

==Order-theoretic properties ==
'''Z''' is a [[total order|totally ordered set]] without upper or lower bound. The ordering of '''Z''' is given by
: ... < &minus;2 < &minus;1 < 0 < 1 < 2 < ...
An integer is ''positive'' if it is greater than zero and ''negative'' if it is less than zero. Zero is defined as neither negative nor positive.

The ordering of integers is compatible with the algebraic operations in the following way:
# if ''a'' < ''b'' and ''c'' < ''d'', then ''a'' + ''c'' < ''b'' + ''d''
# if ''a'' < ''b'' and 0 < ''c'', then ''ac'' < ''bc''. (From this fact, one can show that if ''c'' < 0, then ''ac'' > ''bc''.)

It follows that '''Z''' together with the above ordering is an [[ordered ring]].

==Construction==
The integers can be constructed from the natural numbers by defining [[equivalence class]]es of pairs of natural numbers '''N'''×'''N''' under an [[equivalence relation]], "~", where
:<math> (a,b) \sim (c,d) \,\! </math>
precisely when
:<math>a+d = b+c. \,\!</math>

Taking 0 to be a natural number, the natural numbers may be considered to be integers by the [[embedding]] that maps ''n'' to [(''n'',0)], where [(''a'',''b'')] denotes the equivalence class having (''a'',''b'') as a member.

Addition and multiplication of integers are defined as follows:
:<math>[(a,b)]+[(c,d)] := [(a+c,b+d)].\,</math>
:<math>[(a,b)]\cdot[(c,d)] := [(ac+bd,ad+bc)].\,</math>
It is easily verified that the result is independent of the choice of representatives of the equivalence classes.

Typically, [(''a'',''b'')] is denoted by
:<math>\begin{cases} n, & \mbox{if } a \ge b \\ -n, & \mbox{if } a < b, \end{cases} </math>
where
:<math>n = |a-b|.\,</math>
If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity.

This notation recovers the familiar [[group representation|representation]] of the integers as {…,−3,−2,−1,0,1,2,3,…}.

Some examples are:
:<math>\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}</math>

==Integers in computing==
{{Main|Integer (computer science)}}

An integer (sometimes known as an "<tt>int</tt>", from the name of a datatype in the [[C (programming language)|C programming language]]) is often a primitive [[datatype]] in [[computer language]]s. However, integer datatypes can only represent a [[subset]] of all integers, since practical computers are of finite capacity. Also, in the common [[two's complement]] representation, the inherent definition of [[sign (mathematics)]] distinguishes between "negative" and "non-negative" rather than "negative, positive, and 0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.)

Variable-length representations of integers, such as [[bignum]]s, can store any integer that fits in the computer's memory. Other integer datatypes are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, ''etc.'') or a memorable number of decimal digits (''e.g.'', 9 or 10).

In contrast, theoretical models of [[digital computer]]s, such as [[Turing machine]]s, typically do not have infinite (but only ''unbounded finite'') capacity.

==Notes==
{{reflist}}

==References==
* Herstein, I. N., ''Topics in Algebra'', Wiley; 2 edition (June 20, 1975), ISBN 0-471-01090-1.
* Mac Lane, Saunders, and Garrett Birkhoff; ''Algebra'', American Mathematical Society; 3rd edition (April 1999). ISBN 0-8218-1646-2.

==External links==
{{Wiktionarypar|integer}}
* [http://www.positiveintegers.org The Positive Integers - divisor tables and numeral representation tools]
* [http://www.research.att.com/~njas/sequences/ On-Line Encyclopedia of Integer Sequences] cf [[OEIS]]

----
{{planetmath|id=403|title=Integer}}

[[Category:Elementary mathematics]]
[[Category:Abelian group theory]]
[[Category:Ring theory]]
[[Category:Integers| ]]
[[Category:Elementary number theory]]
[[Category:Algebraic number theory]]

[[af:Heelgetal]]
[[ar:عدد صحيح]]
[[zh-min-nan:Chéng-sò͘]]
[[bs:Cijeli broj]]
[[bg:Цяло число]]
[[ca:Nombre enter]]
[[cs:Celé číslo]]
[[da:Heltal]]
[[de:Ganze Zahl]]
[[et:Täisarv]]
[[es:Número entero]]
[[eo:Entjero]]
[[eu:Zenbaki oso]]
[[fa:اعداد صحیح]]
[[fo:Heiltal]]
[[fr:Entier relatif]]
[[gl:Número enteiro]]
[[ko:정수]]
[[hi:पूर्ण संख्या]]
[[hr:Cijeli broj]]
[[io:Integro]]
[[id:Bilangan bulat]]
[[ia:Numero integre]]
[[is:Heiltölur]]
[[it:Numero intero]]
[[he:מספר שלם]]
[[lt:Sveikasis skaičius]]
[[lmo:Nümar intreegh]]
[[hu:Egész számok]]
[[mk:Цел број]]
[[nl:Geheel getal]]
[[ja:整数]]
[[no:Heltall]]
[[nn:Heiltal]]
[[nds:Hele Tall]]
[[pl:Liczby całkowite]]
[[pt:Número inteiro]]
[[ro:Număr întreg]]
[[ru:Целое число]]
[[sq:Numrat e plotë]]
[[scn:Nùmmuru rilativu]]
[[simple:Integer]]
[[sk:Celé číslo]]
[[sl:Celo število]]
[[sr:Цео број]]
[[sh:Cijeli broj]]
[[fi:Kokonaisluku]]
[[sv:Heltal]]
[[ta:முழு எண்]]
[[th:จำนวนเต็ม]]
[[vi:Số nguyên]]
[[tr:Tam sayılar]]
[[ur:صحیح عدد]]
[[yi:גאנצע צאל]]
[[yo:Nọ́mbà odidi]]
[[zh-yue:整數]]
[[zh:整数]]

Revision as of 00:17, 12 September 2007

--71.245.69.130 00:17, 12 September 2007 (UTC)#REDIRECT [[#REDIRECT Insert text

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