# User Contributed Dictionary

### Noun

integers- In the context of "math}} Plural of integer
- 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 < ...

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

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