- This article discusses the concept of integers in mathematics. For the term in computer science see Integer (computer science).
 Symbol often used to denote the set of integers 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 {... −2, −1, 0, 1, 2, ...}. 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. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
In computer science, the term integer is used to refer to any data type which can represent some subset of the mathematical integers. ...
For other uses, see Latins and Latin (disambiguation). ...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
Zero redirects here. ...
This article is about the number one. ...
This article does not cite any references or sources. ...
This article is about the number. ...
A negative number is a number that is less than zero, such as −3. ...
In mathematics, −1 is the integer greater than negative two (−2) and less than 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  , Unicode U+2124 ℤ), which stands for Zahlen (German for numbers).[1] In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...
In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ...
3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ...
In mathematics, a countable set is a set with the same cardinality (i. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
An example of blackboard bold letters. ...
The Unicode Standard, Version 5. ...
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. This article or section does not cite its references or sources. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ...
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. In mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set. ...
In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...
3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ...
In mathematics, multiplication is an elementary arithmetic operation. ...
Zero redirects here. ...
5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ...
In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...
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. In mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set. ...
In mathematics, associativity is a property that a binary operation can have. ...
Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation). ...
For other uses, see identity (disambiguation). ...
In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ...
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...
In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ...
Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...
In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of...
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...
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. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...
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. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...
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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the 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. The division algorithm is a theorem in mathematics which precisely expresses the outcome of the usual process of division of integers. ...
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...
In mathematics, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder —an amount left over— is also acknowledged. ...
In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ...
In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...
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. In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ...
In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ...
In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ...
In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ...
Order-theoretic properties Z is a totally ordered set without upper or lower bound. The ordering of Z is given by In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ...
- ... < −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:
- if a < b and c < d, then a + c < b + d
- 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. Definitions In abstract algebra, an ordered ring is a commutative ring with a a total order such that if and , then if and , then . ...
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 In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...
In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ...
 precisely when  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. In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ...
Addition and multiplication of integers are defined as follows:
![[(a,b)]+[(c,d)] := [(a+c,b+d)].,](http://upload.wikimedia.org/math/b/f/3/bf337cc180aaafed2c987a227b799e47.png) ![[(a,b)]cdot[(c,d)] := [(ac+bd,ad+bc)].,](http://upload.wikimedia.org/math/2/8/8/28866cf8c27afb7c817dc2d64fcbbd58.png) It is easily verified that the result is independent of the choice of representatives of the equivalence classes.
Typically, [(a,b)] is denoted by
 where  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,...}. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
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}](http://upload.wikimedia.org/math/7/7/2/772a4e9ba107a95900b90c56be31edeb.png)
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.) In computer science, the term integer is used to refer to any data type which can represent some subset of the mathematical integers. ...
C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ...
In computer science, a datatype or data type (often simply a type) is a name or label for a set of values and some operations which one can perform on that set of values. ...
Look up computer language & a Brief History of it in Wiktionary, the free dictionary. ...
Superset redirects here. ...
The twos complement of a binary number is defined as the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ...
A negative number is a number that is less than zero, such as −3. ...
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). A bignum package in a computer or program allows internal representation of very large integers, rational numbers, decimal numbers, or floating-point numbers (limitted only by available memory), and provides a set of arithmetic operations on such numbers. ...
In contrast, theoretical models of digital computers, such as Turing machines, typically do not have infinite (but only unbounded finite) capacity. ...
For the test of artificial intelligence, see Turing test. ...
Cardinality The cardinality of the set of integers is equal to  . This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from to  . Consider the function In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ...
A bijective function. ...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
 . If the domain is restricted to then each and every member of has one and only one corresponding member of and by the definition of cardinal equality the two sets have equal cardinality.
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.
For other persons named Eric Bell, see Eric Bell (disambiguation). ...
Men of Mathematics by E.T. Bell Men of Mathematics is a well-known book on the history of mathematics written by the mathematician E.T. Bell. ...
is the 171st day of the year (172nd in leap years) in the Gregorian calendar. ...
Year 1975 (MCMLXXV) was a common year starting on Wednesday (link will display full calendar) of the Gregorian calendar. ...
Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ...
Garrett Birkhoff (January 19, 1911, Princeton, New Jersey, USA - November 22, 1996, Water Mill, New York, USA) was an American mathematician. ...
External links
This article incorporates material from Integer on PlanetMath, which is licensed under the GFDL. Wikipedia does not have an article with this exact name. ...
Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ...
The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...
PlanetMath is a free, collaborative, online mathematics encyclopedia. ...
| Number systems | | Basic | Complex extensions | Other extensions | | Natural numbers
Negative numbers
Integers
Rational numbers 
Irrational numbers In mathematics, a number system is a set of numbers, or number-like objects, together with one or more operations, such as addition or multiplication. ...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
A negative number is a number that is less than zero, such as −3. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ...
| Real numbers 
Imaginary numbers 
Complex numbers 
Algebraic numbers 
Transcendental numbers In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...
In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...
In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ...
| Quaternions 
Octonions 
Sedenions 
Cayley-Dickson construction
Split-complex numbers  Graphical representation of quaternion units product as 90°-rotation in 4D-space, ij = k, ji = -k, ij = -ji This page describes the mathematical entity. ...
In mathematics, the octonions are a nonassociative extension of the quaternions. ...
The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ...
In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. ...
In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ...
| Bicomplex numbers
Biquaternions
Coquaternions
Tessarines
Hypercomplex numbers A bicomplex number is a number written in the form, a + bi1 + ci2 + dj, where i1, i2 and j are imaginary units. ...
In mathematics, a biquaternion (or complex quaternion) is an element of the (unique) quaternion algebra over the complex numbers. ...
In abstract algebra, a coquaternion is an idea put forward by James Cockle in 1849. ...
The tessarines are a mathematical idea introduced by James Cockle in 1848. ...
The term hypercomplex number has been used in mathematics for the elements of algebras that extend or go beyond complex number arithmetic. ...
| Musean hypernumbers
Superreal numbers
Hyperreal numbers
Surreal numbers
Dual numbers
Transfinite numbers Musean Hypernumbers are a concept envisioned by Charles A. Musès (1919–2000) to form a complete, integrated, connected, and natural number system [1][2][3][4][5]. Musès sketched certain fundamental types of hypernumbers and arranged them in ten levels, each with its own associated arithmetic and geometry. ...
The superreal numbers compose a more inclusive category than hyperreal number. ...
The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ...
In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ...
A variety of dualities in mathematics are listed at duality (mathematics). ...
Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ...
| | Other | | Cardinal numbers · Computable numbers · Constructible numbers · ∞ (infinity) · Integer sequences · Large numbers · Mathematical constants · Nominal numbers · Ordinal numbers · p-adic numbers · Prime numbers · Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ...
In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. ...
A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. ...
In set theory, an infinite set is a set that is not a finite set. ...
In mathematics, an integer sequence is a sequence (i. ...
Big numbers redirects here. ...
A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...
Nominal numbers are numbers used for identification only. ...
In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...
In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ...
In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ...
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