AskDefine | Define integer

Dictionary Definition

integer n : any of the natural numbers (positive or negative) or zero [syn: whole number]

User Contributed Dictionary

English

Etymology

From integer (which see for more information)

Pronunciation

Noun

  1. An element of the infinite and numerable set [...,-3,-2,-1,0,1,2,3,...].

Synonyms

  • whole number, when understood to include negative numbers and zero.

Translations

integer
  • Catalan: enter
  • Croatian: cijeli broj
  • Czech: celé číslo
  • Danish: heltal
  • Dutch: geheel getal
  • Finnish: kokonaisluku
  • French: entier
  • German: ganze Zahl
  • Greek: ακέραιος αριθμός (akéraios arithmós)
  • Hungarian: egész szám
  • Italian: intero
  • Japanese: 整数
  • Norwegian: heltall
  • Polish: liczba całkowita
  • Portuguese: inteiro
  • Romanian: întreg
  • Spanish: entero
  • Swedish: heltal

Adjective

  1. Of a number, whole.

Translations

whole
  • Danish: heltallig
  • Finnish: kokonais-
  • French: entier, entière
  • Hungarian: egész
  • Romanian: întreg, întreagă

Dutch

Alternative spellings

Adjective

integer ( meer integer, meest integer)

Latin

Etymology

in-, a negating prefix, + tangere, to touch.

Adjective

integer

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

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

Synonyms, Antonyms and Related Words

Gaussian integer, algebraic number, article, cardinal, cardinal number, cipher, collectivity, complex, complex number, defective number, digit, embodiment, entirety, entity, even number, figure, finite number, fraction, imaginary number, impair, individual, infinity, integration, integrity, irrational, irrational number, item, mixed number, module, numeral, oneness, ordinal, organic unity, pair, person, persona, point, polygonal number, prime number, pure imaginary, rational, rational number, real, real number, rectangular number, round number, serial number, single, singleton, soul, surd, totality, transcendental number, transfinite number, unit, unity, whole, whole number
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