An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.).
The are the of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold $\backslash mathbb\{Z\}$.The set of natural numbers $\backslash mathbb\{N\}$ is a subset of $\backslash mathbb\{Z\}$, which in turn is a subset of the set of all $\backslash mathbb\{Q\}$, itself a subset of the $\backslash mathbb\{R\}$. Like the natural numbers, $\backslash mathbb\{Z\}$ is Countable set. An integer may be regarded as a real number that can be written without a fraction. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not.
The integers form the smallest group and the smallest ring containing the . In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also .
The use of the letter Z to denote the set of integers comes from the German language word ("number")
and has been attributed to David Hilbert. The earliest known use of the notation in a textbook occurs in Algébre written by the collective Nicolas Bourbaki, dating to 1947. The notation was not adopted immediately, for example another textbook used the letter J and a 1960 paper used Z to denote the nonnegative integers. But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.The symbol $\backslash mathbb\{Z\}$ is often annotated to denote various sets, with varying usage amongst different authors: $\backslash mathbb\{Z\}^+$,$\backslash mathbb\{Z\}\_+$ or $\backslash mathbb\{Z\}^\{>\}$ for the positive integers, $\backslash mathbb\{Z\}^\{0+\}$ or $\backslash mathbb\{Z\}^\{\backslash geq\}$ for nonnegative integers, and $\backslash mathbb\{Z\}^\{\backslash neq\}$ for nonzero integers. Some authors use $\backslash mathbb\{Z\}^\{*\}$ for nonzero integers, while others use it for nonnegative integers, or for } (the group of units of $\backslash mathbb\{Z\}$). Additionally, $\backslash mathbb\{Z\}\_\{p\}$ is used to denote either the set of integers modulo (i.e., the set of congruence classes of integers), or the set of padic integer.Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008LK Turner, FJ BUdden, D Knighton, "Advanced Mathematics", Book 2, Longman 1975.
The whole numbers were synonymous with the integers up until the early 1950s. In the late 1950s, as part of the New Math movement, American elementary school teachers began teaching that "whole numbers" referred to the , excluding negative numbers, while "integer" included the negative numbers. "Whole number" remains ambiguous to the present day.
The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of rings, characterizes the ring $\backslash mathbb\{Z\}$.
$\backslash mathbb\{Z\}$ is not closed under 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 table lists some of the basic properties of addition and multiplication for any integers , and :
+Properties of addition and multiplication on integers ! !scope="col"  Addition !scope="col"  Multiplication 
The first five properties listed above for addition say that $\backslash mathbb\{Z\}$, under addition, is an abelian group. It is also a cyclic group, since every nonzero integer can be written as a finite sum or . In fact, $\backslash mathbb\{Z\}$ under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to $\backslash mathbb\{Z\}$.
The first four properties listed above for multiplication say that $\backslash mathbb\{Z\}$ under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that $\backslash mathbb\{Z\}$ under multiplication is not a group.
All the rules from the above property table (except for the last), when taken together, say that $\backslash mathbb\{Z\}$ together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in $\backslash mathbb\{Z\}$ for all values of variables, which are true in any unital commutative ring. Certain nonzero integers map to zero in certain rings.
The lack of in the integers (last property in the table) means that the commutative ring $\backslash mathbb\{Z\}$ is an integral domain.
The lack of multiplicative inverses, which is equivalent to the fact that $\backslash mathbb\{Z\}$ is not closed under division, means that $\backslash mathbb\{Z\}$ is not a field. The smallest field containing the integers as a subring is the field of . The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes $\backslash mathbb\{Z\}$ as its subring.
Although ordinary division is not defined on $\backslash mathbb\{Z\}$, the division "with remainder" is defined on them. It is called Euclidean division, and possesses the following important property: given two integers and with , there exist unique integers and such that and , where denotes the absolute value of . The integer is called the quotient and is called the remainder of the division of by . The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions.
The above says that $\backslash mathbb\{Z\}$ is a Euclidean domain. This implies that $\backslash mathbb\{Z\}$ is a principal ideal domain, and any positive integer can be written as the products of prime number in an essentially unique way.
This is the fundamental theorem of arithmetic.
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:
Thus it follows that $\backslash mathbb\{Z\}$ together with the above ordering is an ordered ring.
The integers are the only nontrivial totally ordered abelian group whose positive elements are wellordered.
. This is equivalent to the statement that any Noetherian ring valuation ring is either a field—or a discrete valuation ring.
The traditional arithmetic operations can then be defined on the integers in a piecewise fashion, for each of positive numbers, negative numbers, and zero. For example negation is defined as follows: $$x\; =\; \backslash begin\{cases\}$$
\psi(x), & \text{if } x \in P \\ \psi^{1}(x), & \text{if } x \in P^ \\ 0, & \text{if } x = 0\end{cases}
The traditional style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that integers obey the various laws of arithmetic.
.
In modern settheoretic mathematics, a more abstract constructionIvorra Castillo: Álgebra allowing one to define arithmetical operations without any case distinction is often used instead.
. The integers can thus be formally constructed as the equivalence classes of of .The intuition is that stands for the result of subtracting from . To confirm our expectation that and denote the same number, we define an equivalence relation on these pairs with the following rule:
Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using to denote the equivalence class having as a member, one has:
The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:
Hence subtraction can be defined as the addition of the additive inverse:
The standard ordering on the integers is given by:
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 or (or both at once). The natural number is identified with the class (i.e., the natural numbers are embedding into the integers by map sending to ), and the class is denoted (this covers all remaining classes, and gives the class a second time since
Thus, is denoted by
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 .
Some examples are:
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}
There exist at least ten such constructions of signed integers. These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2) and the types of arguments accepted by these operations; the presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms.
The technique for the construction of integers presented in the previous section corresponds to the particular case where there is a single basic operation pair$(x,y)$ that takes as arguments two natural numbers $x$ and $y$, and returns an integer (equal to $xy$). This operation is not free since the integer 0 can be written pair(0,0), or pair(1,1), or pair(2,2), etc. This technique of construction is used by the proof assistant Isabelle; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers.
Variablelength representations of integers, such as , can store any integer that fits in the computer's memory. Other integer data types 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).

