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The integers (with addition as operation) form the smallest group containing the additive monoid of the natural numbers. Like the natural numbers, the integers form a countably infinite set.

Integers can be thought of as discrete, equally spaced points on an infinitely long number line.

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 (but with "rational" meaning "quotient of integers", this attempt at precision suffers from circularity).

Contents

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 division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative).

The following lists some of the basic properties of addition and multiplication for any integers a, b and c.

Addition Multiplication
Closure: a + b   is an integer a × b   is an integer
Associativity: a + (b + c)  =  (a + b) + c a × (b × c)  =  (a × b) × c
Commutativity: a + b  =  b + a a × b  =  b × a
Existence of an identity element: a + 0  =  a a × 1  =  a
Existence of inverse elements: a + (−a)  =  0
Distributivity: a × (b + c)  =  (a × b) + (a × c)
No zero divisors: If a × b = 0, then either a = 0 or b = 0 (or both)

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 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:

... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ...

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 < 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.

The integers are the only integral domain whose positive elements are well-ordered, and in which order is preserved by addition.[citation needed]

Construction

Représentation des classes d'équivalence pour les nombres de -5 à 5
Red Points représents ordered pairs of natural numbers. Linked red points are equivalence classes of the blue integer at the end of the line.

The integers can be formally constructed as the equivalence classes of ordered pairs of natural numbers (a, b).

The intuition is that (a, b) stands for the result of subtracting b from a. To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule:

 (a,b) \sim (c,d) \,\!

precisely when

a+d = b+c. \,\!

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; denoting by [(a,b)] the equivalence class having (a,b) as a member, one has:

[(a,b)]+[(c,d)] := [(a+c,b+d)].\,
[(a,b)]\cdot[(c,d)] := [(ac+bd,ad+bc)].\,

The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:

-[(a,b)] := [(b,a)].\,

Hence subtraction can be defined as the addition of the additive inverse:

[(a,b)]-[(c,d)] := [(a+d,b+c)].\,

The standard ordering on the integers is given by:

[(a,b)]<[(c,d)]\, iff a+d < b+c.\,

It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.

Every equivalence class has a unique member that is of the form (n,0) or (0,n) (or both at once). The natural number n is identified with the class [(n,0)] (in other words the natural numbers are embedded into the integers by map sending n to [(n,0)]), and the class [(0,n)] is denoted −n (this covers all remaining classes, and gives the class [(0,0)] a second time since −0 = 0.

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

\begin{cases} a-b, & \mbox{if } a \ge b \\ -(b-a), & \mbox{if } a < b, \end{cases} .

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 {... −3,−2,−1, 0, 1, 2, 3, ...}.

Some examples are:

\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}

Integers in computing

An integer 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.) Fixed length integer approximation datatypes (or subsets) are denoted int in several programming languages (such as Algol68, C, Java, etc).

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.[citation needed]

Cardinality

The cardinality of the set of integers is equal to \aleph_0 (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from Z to N.

If N = {0, 1, 2, ...} then consider the function:

f(x) = \begin{cases} 2|x|, & \mbox{if } x < 0 \\ 0, & \mbox{if } x = 0 \\ 2x-1, & \mbox{if } x > 0. \end{cases}

{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,0) (1,1) (2,3) (3,5) ... }

If N = {1,2,3,...} then consider the function:

g(x) = \begin{cases} 2|x|, & \mbox{if } x < 0 \\ 2x+1, & \mbox{if } x \ge 0. \end{cases}

{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,1) (1,3) (2,5) (3,7) ... }

If the domain is restricted to Z then each and every member of Z has one and only one corresponding member of N and by the definition of cardinal equality the two sets have equal cardinality.

See also

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

This article incorporates material from Integer on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.


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Integer: The whole part of a number. Often this is the name of a computer command or function which removes the fractional portion of a number, returning the whole number. Also commonly known as any whole number, and the variable type for numbers in computer programs. For instance, in 1 1/2, 1 would be the integer.


Wiktionary

Up to date as of January 15, 2010
(Redirected to integer article)

Definition from Wiktionary, a free dictionary

Contents

English

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Wikipedia

Etymology

From Latin integer (untouched, unhurt, unchanged, sound, fresh, whole, entire, pure, honest) < in + tangere (to touch); see tangere, tact.

Pronunciation

Noun

Singular
integer

Plural
integers

integer (plural integers)

  1. (arithmetic) 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.

Hypernyms

Hyponyms

Related terms

Derived terms

Translations

External links

  • integer in Webster’s Revised Unabridged Dictionary, G. & C. Merriam, 1913
  • integer in The Century Dictionary, The Century Co., New York, 1911

Anagrams


Dutch

Alternative spellings

  • integere

Adjective

integer (comparative meer integer, superlative meest integer)

  1. honest, trustworthy, having integrity

Latin

Etymology

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

Adjective

integer

  1. complete, whole, intact

Limburgish

Etymology

Latin integer

Adjective

integer (comparative integerder; superlatives: integers, predicate form 't integers)

  1. friendly
  2. complete, whole, intact

Usage notes

Unlike Dutch, not used in the meaning of honest or trustworthy.

Inflection

Masculine Feminine Neutral
Singular Plural Singular Plural Singular Plural
nominative integere integer integer integer integer integer
genitive integers integerer integerer integerer integers integerer
locative integeres integeres integeres integeres integeres integeres
vocative² integer integer integer integer integer integer
dative¹ integere integeren integerer integeren integer integeren
accusative¹ integere integeren integer integer integer integeren
  • Dative and accusative are nowadays obsolete, use nominative instead.
  • Vocative only exists for about ten words.

Simple English

Integers are the whole numbers. Integers are the numbers used for counting and for identification, for example, house numbers, together with the negative numbers.[1]

These are some of the integers:

{...-4, -3, -2, -1, 0, +1, +2, +3, +4,...}\,\!

Integers are whole numbers and their negative counterparts. 0 is also an integer, even though it has no positive or negative signs. While integers are a subset of the rational numbers and real numbers, they are a smaller set because they do not include the numbers with decimal points. A list of the integers would have no end in either direction, which means that no number is the largest and no number is the smallest.

Another type of integer is the consecutive integer, which is viewed here: consecutive integer

In Programming

In some programming languages, like C, have types called "int" or "integer."

Notes

  1. Negative numbers have a minus (−) in front of the number.








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