# Dictionary Definition

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

# User Contributed Dictionary

### Pronunciation

### Noun

- 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

#### Derived terms

### Adjective

- Of a number, whole.

#### Translations

whole

### See also

## Dutch

### Alternative spellings

### Adjective

integer ( meer integer, meest integer)- honest, trustworthy, having integrity

## Latin

### 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 < ...

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 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) \,\!

- 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)].\,

Typically, [(a,b)] is denoted by

- \begin n, & \mbox a \ge b \\ -n, & \mbox a

- n = |a-b|.\,

This notation recovers the familiar representation
of the integers as .

Some examples are:

- \begin

## 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.

## Notes

## References

- Bell, E. T., Men of Mathematics. New York: Simon and Schuster, 1986. (Hardcover; ISBN 0-671-46400-0)/(Paperback; ISBN 0-671-62818-6)
- 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

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