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Noun

integers
  1. In the context of "math}} Plural of integer
  2. context math

Extensive Definition

confused Natural number
This article discusses the concept of integers in mathematics. For the term in computer science see Integer (computer science).
The integers (from the Latin integer'', which means with untouched integrity, whole, entire) are the set of numbers consisting of the natural numbers including 0 (0, 1, 2, 3, ...) and their negatives (0, −1, −2, −3, ...). They are numbers that can be written without a fractional or decimal component, and fall within the set . For example, 65, 7, and −756 are integers; 1.6 and 1½ are not integers. In other terms, integers are the numbers one can count with items such as apples or fingers, and their negatives, including 0.
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 countably infinite set. The set of all integers is often denoted by a boldface Z (or blackboard bold \mathbb, Unicode U+2124 ℤ), which stands for Zahlen (German for numbers).
In algebraic number theory, these commonly understood integers, embedded in the field of rational numbers, are referred to as rational integers to distinguish them from the more broadly defined algebraic integers.

Algebraic properties

Like the natural numbers, Z is closed under the 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, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of 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. 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 (−1) + (−1) + ... + (−1). In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is 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 2x = 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 rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative 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. The smallest field containing the integers is the field of rational numbers. 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 ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ 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 divisors.
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 primes in an essentially unique way. This is the fundamental theorem of arithmetic.

Order-theoretic properties

Z is a totally ordered set without upper or lower bound. The ordering of Z is given by
... < −2 < −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:
  1. if a < b and c < d, then a + c < b + d
  2. if a < b and 0 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 classes of pairs of natural numbers N×N under an equivalence relation, "~", where
(a,b) \sim (c,d) \,\!
precisely when
a+d = b+c. \,\!
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:
[(a,b)]+[(c,d)] := [(a+c,b+d)].\,
[(a,b)]\cdot[(c,d)] := [(ac+bd,ad+bc)].\,
It is easily verified that the result is independent of the choice of representatives of the equivalence classes.
Typically, [(a,b)] is denoted by
\begin n, & \mbox a \ge b \\ -n, & \mbox a
where
n = |a-b|.\,
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 representation of the integers as .
Some examples are:
\begin
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

Integers in computing

An integer (sometimes known as an "int", from the name of a datatype in the C programming language) is often a primitive datatype in computer languages. 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 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 bignums, 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 computers, such as Turing machines, typically do not have infinite (but only unbounded finite) capacity.

Cardinality

The cardinality of the set of integers is equal to \aleph_0. This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from \mathbb to \mathbb. Consider the function
\begin 2x+1, & \mbox x \ge 0 \\ 2|x|, & \mbox x.
If the domain is restricted to \mathbb then each and every member of \mathbb has one and only one corresponding member of \mathbb and by the definition of cardinal equality the two sets have equal cardinality.

Notes

References

integers in Afrikaans: Heelgetal
integers in Arabic: عدد صحيح
integers in Bengali: পূর্ণ সংখ্যা
integers in Min Nan: Chéng-sò͘
integers in Belarusian: Цэлы лік
integers in Bosnian: Cijeli broj
integers in Bulgarian: Цяло число
integers in Catalan: Nombre enter
integers in Chuvash: Тулли хисеп
integers in Czech: Celé číslo
integers in Danish: Heltal
integers in German: Ganze Zahl
integers in Estonian: Täisarv
integers in Modern Greek (1453-): Ακέραιος αριθμός
integers in Spanish: Número entero
integers in Esperanto: Entjero
integers in Basque: Zenbaki oso
integers in Persian: اعداد صحیح
integers in Faroese: Heiltal
integers in French: Entier relatif
integers in Gan Chinese: 整數
integers in Galician: Número enteiro
integers in Classical Chinese: 整數
integers in Korean: 정수
integers in Hindi: पूर्ण संख्या
integers in Croatian: Cijeli broj
integers in Ido: Integro
integers in Indonesian: Bilangan bulat
integers in Interlingua (International Auxiliary Language Association): Numero integre
integers in Icelandic: Heiltölur
integers in Italian: Numero intero
integers in Hebrew: מספר שלם
integers in Lao: ຈຳນວນຖ້ວນ
integers in Latin: Numerus integer
integers in Lithuanian: Sveikasis skaičius
integers in Lombard: Nümar intreegh
integers in Hungarian: Egész számok
integers in Macedonian: Цел број
integers in Marathi: पूर्ण संख्या
integers in Dutch: Geheel getal
integers in Japanese: 整数
integers in Norwegian: Heltall
integers in Norwegian Nynorsk: Heiltal
integers in Low German: Hele Tall
integers in Polish: Liczby całkowite
integers in Portuguese: Número inteiro
integers in Romanian: Număr întreg
integers in Russian: Целое число
integers in Albanian: Numrat e plotë
integers in Sicilian: Nùmmuru rilativu
integers in Simple English: Integer
integers in Slovak: Celé číslo
integers in Slovenian: Celo število
integers in Serbian: Цео број
integers in Serbo-Croatian: Cijeli broj
integers in Finnish: Kokonaisluku
integers in Swedish: Heltal
integers in Tamil: முழு எண்
integers in Thai: จำนวนเต็ม
integers in Vietnamese: Số nguyên
integers in Turkish: Tam sayılar
integers in Ukrainian: Цілі числа
integers in Urdu: صحیح عدد
integers in Võro: Terveharv
integers in Vlaams: Gehêel getal
integers in Yiddish: גאנצע צאל
integers in Yoruba: Nọ́mbà odidi
integers in Contenese: 整數
integers in Chinese: 整数
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