This article discusses the concept of integers in mathematics. For the term in computer science see Integer (computer science). In computer science the term integer is used to refer to a Data type which represents some finite subset of the mathematical Integers These are also known as

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, . The musical instrument is spelled Cymbal. A symbol is something --- such as an object, Picture, written word a sound a piece Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In mathematics Two has many properties in Mathematics. An Integer is called Even if it is divisible by 2 ---- In mathematics Three is the first odd Prime number, and the second smallest prime . . ) and their negatives (0, −1, −2, −3, . A negative number is a Number that is less than zero, such as −2 Why is &minus1 times &minus1 equal to 1? Why is &minus1 multiplied by &minus1 equal to 1? More generally why is a negative times a negative a positive? There are two ways . . ). 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.

More formally, the integers are the only integral domain whose positive elements are well-ordered, and in which order is preserved by addition. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every Addition is the mathematical process of putting things together 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). Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical In Computing, Unicode is an Industry standard allowing Computers to consistently represent and manipulate text expressed in most of the world's The German language (de ''Deutsch'') is a West Germanic language and one of the world's major languages. ^[1]

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. In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions This article deals with the ring of complex numbers integral over Z.

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. 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 Operands, in other words an operation whose Arity is two Addition is the mathematical process of putting things together However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract Z is not closed under the operation of division, since the quotient of two integers (e. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. 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. 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 In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0 Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the As a group under addition, Z is a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 + . 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 . . 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 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

The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation 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. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Adding the last property says that Z is an integral domain. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such 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. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division The smallest field containing the integers is the field of rational numbers. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions This process can be mimicked to form the field of fractions of any integral domain. In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients

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 division algorithm is a Theorem in Mathematics which precisely expresses the outcome of the usual process of division of Integers The name In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. The integer q is called the quotient and r is called the remainder, resulting from division of a by b. In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left This is the basis for the Euclidean algorithm for computing greatest common divisors. In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero

Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. In Abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies 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. In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 This is the fundamental theorem of arithmetic. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written

Order-theoretic properties

Z is a totally ordered set without upper or lower bound. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation 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 < 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. In Abstract algebra, an ordered ring is a Commutative ring R with a Total order \leq such that if a\leq

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

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, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group

Addition and multiplication of integers are defined as follows:

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 {. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of . . ,−3,−2,−1,0,1,2,3,. . . }.

Some examples are:

Integers in computing

Main article: Integer (computer science)

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. In computer science the term integer is used to refer to a Data type which represents some finite subset of the mathematical Integers These are also known as tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured A data type in Programming languages is an attribute of a datum which tells the computer (and the programmer something about the kind of datum it is 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". The two's complement of a Binary number is defined as the value obtained by subtracting the number from a large power of two (specifically from 2 N for A negative number is a Number that is less than zero, such as −2 (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. On a Computer, arbitrary-precision arithmetic, also called bignum arithmetic is a technique whereby Computer programs perform Calculations on 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. A computer is a Machine that manipulates data according to a list of instructions. Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm

Cardinality

The cardinality of the set of integers is equal to . In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from to . In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every Consider the function

.

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. Eric Temple Bell ( February 7 Men of Mathematics is a well-known 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. Events 451 - Battle of Chalons: Flavius Aetius ' defeats Attila the Hun. Year 1975 ( MCMLXXV) was a Common year starting on Wednesday (link will display full calendar of the Gregorian calendar.
• Mac Lane, Saunders, and Garrett Birkhoff; Algebra, American Mathematical Society; 3rd edition (April 1999). Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Garrett Birkhoff ( January 19, 1911, Princeton, New Jersey, USA – November ISBN 0-8218-1646-2.

External links

• The Positive Integers - divisor tables and numeral representation tools
• On-Line Encyclopedia of Integer Sequences cf OEIS
• "Integer Explorer" by Chris Boucher, The Wolfram Demonstrations Project. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

This article incorporates material from Integer on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.