The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which stands for Zahlen (German for "numbers"). They are also known as the whole numbers, although that term is also used to refer only to the positive integers (with or without zero). Like the natural numbers, the integers form a countably infinite set. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... 1 (one) is a number, numeral, and glyph. ... For the Austin Powers character, see Number 2(Austin Powers 2 (two) is a number, numeral, and glyph. ... 3 (three) is a number, numeral, and glyph. ... A negative number is a number that is less than zero, such as −3. ... 0 (zero) or nought is both a number and a numeral. ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... Blackboard bold is a style of typeface often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical, or near-vertical lines) are doubled. ... In mathematics the term countable set is used to describe the size of a set, e. ...

The term rational integer is used, in algebraic number theory, to distinguish these 'ordinary' integers, in the rational numbers, from other concepts such as the Gaussian integers. In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... A Gaussian integer is a complex number whose real and imaginary part are both 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. In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... Addition (or summation) is one of the basic operations of arithmetic. ... In its simplest form, multiplication is a quick way of adding identical numbers. ... 0 (zero) or nought is both a number and a numeral. ... In mathematics, subtraction is one of the four basic arithmetic operations. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the reverse operation of multiplication, and sometimes it can be interpreted as repeated subtraction. ...

The following table 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, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... 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 set. ... In mathematics, the inverse of an element x, with respect to an operation *, is a chicken fucker, 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. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... 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 properties from the above table taken together say that Z together with addition and multiplication is a commutative ring with unity. 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 quotient field of any integral domain, where an integral domain is a commutative ring with unity such that whenever ab = 0, either a = 0 or b = 0. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... 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 (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

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 modulus1) 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. ... The Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (gcd) of two integers. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf) of two integers which are not both zero is the largest integer that divides both numbers. ...

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 prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 can be written as a product of prime numbers in only one way. ...

Order-theoretic properties

Z is a totally ordered set without upper or lower bound. The ordering of Z is given by In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, 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:

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.)

Integers in computing

An integer is often one of the primitive datatypes in computer languages. However, these "integers" can only represent a subset of all mathematical integers, since "real-world" computers are of finite capacity. Integer datatypes are typically implemented using a fixed number of bits, and even variable-length representations eventually run out of storage space when trying to represent especially large numbers. On the other hand, theoretical models of digital computers, e.g., Turing machines, usually do have infinite (but only countable) capacity. In computer science, a datatype (often simply type) is a name or label for a set of values and some operations which can be performed on that set of values. ... A computer language is a language used by, or in association with, computers. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... A bit (abbreviated b) is the most basic information unit used in computing and information theory. ... The tower of a personal computer. ... Artists conception of a universal Turing machine. ... In mathematics the term countable set is used to describe the size of a set, e. ...

For more information, see Integer (computer science). In computer science, the term integer is used to refer to any data type which can represent some subset of the mathematical integers. ...

Quotations

God invented the integers, all else is the work of man. Kronecker Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ...

External links

• The Positive Integers - divisor tables and numeral representation tools