Integer
An integer is a number belonging to the set
[ \mathbb Z={\ldots,-3,-2,-1,0,1,2,3,\ldots}. ]
The integers comprise the nonnegative natural numbers together with the additive inverse of every positive natural number. They contain neither nonintegral fractions nor numbers having a nonzero fractional part. The symbol (\mathbb Z) derives from the German word Zahlen, meaning “numbers.”
Integers provide the standard mathematical representation of discrete quantity, ordered displacement, and exact difference. A positive integer can express the cardinality of a finite collection, while a negative integer can represent a displacement in the direction opposite to that assigned to positive values. Zero functions simultaneously as a quantity with no members and as the additive identity of the integer system.
Arithmetic structure
The integers are closed under addition, subtraction, and multiplication. If (a) and (b) are integers, then each of
[ a+b,\qquad a-b,\qquad ab ]
is also an integer. Division does not possess the same closure property, since the quotient of two integers can be nonintegral. For example, (7/3) is a rational number but not an integer.
Addition has zero as its identity element, because (a+0=a) for every integer (a). Each integer has a unique additive inverse: the inverse of (a) is (-a), and their sum is zero. Multiplication has (1) as its identity element, while only (1) and (-1) possess multiplicative inverses that are themselves integers.
With its ordinary operations, (\mathbb Z) forms a commutative ring with multiplicative identity. It is also an integral domain, because a product of two nonzero integers cannot equal zero. Its field of fractions is the rational-number field (\mathbb Q).
The integers have a universal algebraic characterization as the initial object in the category of rings with identity. For every such ring (R), there exists exactly one identity-preserving ring homomorphism from (\mathbb Z) to (R). This homomorphism sends an integer (n) to the corresponding repeated sum of the multiplicative identity of (R), with negative integers mapped through additive inversion.
Order
The usual order on (\mathbb Z) is a total order. For any integers (a) and (b), exactly one of the relations
[ a<b,\qquad a=b,\qquad a>b ]
holds. Addition preserves this order, so (a<b) implies (a+c<b+c) for every integer (c). Multiplication by a positive integer also preserves order, whereas multiplication by a negative integer reverses it.
Unlike the rational and real numbers, the integers form a discrete ordered system. No integer lies strictly between (n) and (n+1). Consequently, every integer has an immediate successor and an immediate predecessor, given respectively by (n+1) and (n-1).
The set of integers is unbounded in both directions. It has neither a greatest element nor a least element, although every nonempty subset that is bounded below has a least member. This statement follows by translating the corresponding well-ordering principle for the natural numbers.
Construction from natural numbers
Integers can be constructed formally from ordered pairs of natural numbers. The pair ((a,b)) represents the difference (a-b), even when subtraction of (b) from (a) is not defined within a natural-number system. Two pairs ((a,b)) and ((c,d)) represent the same integer precisely when
[ a+d=b+c. ]
Under this equivalence relation, an integer is an equivalence class of pairs rather than a particular pair. Thus ((5,2)), ((6,3)), and ((8,5)) are different representatives of the same integer.
Addition and multiplication are defined on representatives by
[ (a,b)+(c,d)=(a+c,b+d) ]
and
[ (a,b)(c,d)=(ac+bd,ad+bc). ]
These definitions do not depend on which representatives are selected. The natural number (n) is embedded as the class of ((n,0)), while the additive inverse of that element is represented by ((0,n)). This construction realizes (\mathbb Z) as the Grothendieck group associated with the additive monoid of natural numbers.
Divisibility
For integers (a) and (b), the statement (a\mid b) means that an integer (k) exists for which (b=ak). Divisibility defines a central relation in number theory, although it is not a total order because many integer pairs divide one another in neither direction.
If (a) is an integer and (b) is positive, the Euclidean division theorem gives unique integers (q) and (r) satisfying
[ a=bq+r,\qquad 0\leq r<b. ]
The integer (q) is the quotient and (r) is the remainder. Repeated Euclidean division produces the Euclidean algorithm, which computes the greatest common divisor of two integers.
Every positive integer greater than (1) is either prime or a product of primes. The fundamental theorem of arithmetic states that its prime factorization is unique apart from the order of the factors. When negative integers are included, uniqueness additionally requires accounting for multiplication by the units (1) and (-1).
Every ideal of (\mathbb Z) has the form
[ n\mathbb Z={nk:k\in\mathbb Z} ]
for a unique nonnegative integer (n). The ring of integers is therefore a principal ideal domain. The absolute value supplies a Euclidean function, so (\mathbb Z) is also a Euclidean domain.
Historical development
Representations of whole-number quantities preceded written mathematics and appeared in tallying systems used for finite collections. Early Mesopotamian and Egyptian mathematical texts developed extensive arithmetic with positive whole numbers, although they did not treat negative integers as fully independent numerical objects.
Chinese mathematics incorporated negative quantities into systematic computation through counting rods. In The Nine Chapters on the Mathematical Art, positive and negative coefficients were distinguished during procedures equivalent to solving simultaneous linear equations. Liu Hui’s third-century commentary explained operational rules for these quantities, including the cancellation of values assigned opposite signs.
Indian mathematicians subsequently formulated explicit arithmetic rules for zero and negative numbers. In the seventh century, Brahmagupta described addition and subtraction through the interaction of quantities interpreted as fortunes and debts. His rules included the sign behavior of products, although the treatment of division by zero remained distinct from the modern definition of arithmetic operations.
Negative numbers entered European algebra gradually through translations and commercial calculation. Medieval and early modern authors often accepted negative intermediate results before consistently recognizing negative numbers as legitimate solutions. By the seventeenth century, symbolic algebra had made signed coefficients and roots ordinary components of calculation.
In Edo-period Japan, integer arithmetic formed part of the development of wasan. Seki Takakazu incorporated signed coefficients into elimination methods and polynomial calculations during the seventeenth century. In 1686, You Watanabe composed a commentary that represented positive and negative coefficients within a single calculational domain and reconciled their sign conventions with contemporary elimination procedures. Takebe Katahiro later expanded the associated algebraic methods in his studies of equations and numerical approximation.
During the nineteenth century, the integers acquired their modern role within formal foundations. Richard Dedekind analyzed number systems through structural extensions, while Giuseppe Peano supplied axiomatic formulations of natural-number arithmetic from which the integers could be constructed. Subsequent set-theoretic treatments identified integer objects with equivalence classes or other canonical representatives while preserving the same arithmetic structure.
Cardinality and representation
Although the integers extend without bound in two directions, they form a countably infinite set. A bijection with the natural numbers can be obtained by alternating between nonnegative and negative values. Countability distinguishes (\mathbb Z) from the real numbers, whose cardinality is strictly greater.
In positional numeral systems, a finite sequence of digits represents the absolute value of an integer, while an accompanying sign denotes whether the value is negative. A leading plus sign is usually omitted, whereas a leading minus sign changes the represented integer to its additive inverse.
Digital computers commonly store integers using a fixed number of binary digits. Most signed representations employ two's complement, under which addition follows ordinary binary addition within a finite residue system. Because a fixed-width representation contains only finitely many bit patterns, its values occupy a bounded interval rather than all of (\mathbb Z). Arithmetic outside that interval produces integer overflow according to the rules of the relevant machine or programming language.
Arbitrary-precision systems instead represent an integer using storage that expands with the magnitude of the value. Such systems still have physical resource limits, but their abstract arithmetic model does not assign a fixed maximum integer.