# integer

The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …). In non-mathematical terms, 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.

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), which stands for Zahlen (German for numbers).[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.

## 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.
 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 ab = 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, 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 properties from the above 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 < ...
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.

## 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.

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 {…,−3,−2,−1,0,1,2,3,…}.

Some examples are:

## 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 . This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from to . 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.

## References

• 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.

Latin}}}
Official status
Official language of: Vatican City
Used for official purposes, but not spoken in everyday speech
Regulated by: Opus Fundatum Latinitas
Roman Catholic Church
Language codes
ISO 639-1: la
ISO 639-2: lat
In mathematics, a natural number can mean either an element of the set (i.e the positive integers or the counting numbers) or an element of the set (i.e. the non-negative integers).
0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.

0 1 2 3 4 5 6 7 8 9

2 (two) is a number, numeral, and glyph. It is the natural number following 1 and preceding 3.

## In mathematics

Two has many properties in mathematics.[1] An integer is called even if it is divisible by 2.
3 (three) is a number, numeral, and glyph. It is the natural number following 2 and preceding 4.

## In mathematics

Three is the first odd prime number, and the second smallest positive prime.
A negative number is a number that is less than zero, such as −3. A positive number is a number that is greater than zero, such as 3. Zero itself is neither positive nor negative.
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 that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero (the zero-product property); that is, there
In mathematics, a well-order relation (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. Equivalently, a well-ordering is a well-founded total order.
Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection.
countable set is a set with the same cardinality (i.e., number of elements) as some subset of the set of natural numbers. The term was originated by Georg Cantor; it stems from the fact that the natural numbers are often called counting numbers.
SET may stand for:
• Sanlih Entertainment Television, a television channel in Taiwan
• Secure electronic transaction, a protocol used for credit card processing,

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, or near-vertical lines) are doubled. The symbols usually describe number sets.
Unicode is an industry standard allowing computers to consistently represent and manipulate text expressed in any of the world's writing systems. Developed in tandem with the Universal Character Set standard and published in book form as The Unicode Standard
German language (Deutsch, ] ) is a West Germanic language and one of the world's major languages.
Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic numbers which are mathematic roots of polynomials with rational number coefficients.
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. Non-integer rational numbers (commonly called fractions) are usually written as the vulgar fraction , where b is not zero.
In number theory, an algebraic integer is a complex number which is a root of some monic polynomial (leading coefficient 1) with integer coefficients. The set of all algebraic integers is closed under addition and multiplication so it forms a subring of complex numbers denoted by
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. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 7 are both natural numbers, but the result of 3
In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator.
Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection.
Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:

0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.
Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. Subtraction is denoted by a minus sign in infix notation.

The traditional names for the parts of the formula
cb = a
are
In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.

Specifically, if c times b equals a, written:
where b is not zero, then a