{"history":[],"article":{"id":"fe821285-5e06-4d32-81d2-14b181c93935","title":"Integer","slug":"integer","content":":::summary\n:::\n\n# \nAn **integer** is the [number](https://bliptext.com/articles/number) zero ([0](https://bliptext.com/articles/0)), a positive natural [number](https://bliptext.com/articles/number) (1, 2, 3, . . .), or the negation of a positive natural [number](https://bliptext.com/articles/number) ([−1](https://bliptext.com/articles/1), −2, −3, . . .).\nThe negations or [additive inverses](https://bliptext.com/articles/additive-inverse) of the positive natural numbers are referred to as **negative integers**.\nThe [set](https://bliptext.com/articles/set-mathematics) of all integers is often denoted by the [boldface](https://bliptext.com/articles/boldface)\nZ\nor [blackboard bold](https://bliptext.com/articles/blackboard-bold) \\mathbb{Z}.\nThe set of natural numbers \\mathbb{N} is a [subset](https://bliptext.com/articles/subset) of \\mathbb{Z}, which in turn is a [subset](https://bliptext.com/articles/subset) of the set of all [rational numbers](https://bliptext.com/articles/rational-number) \\mathbb{Q}, itself a [subset](https://bliptext.com/articles/subset) of the [real numbers](https://bliptext.com/articles/real-number) \\mathbb{R}.\nLike the set of natural numbers, the set of integers \\mathbb{Z} is [countably infinite](https://bliptext.com/articles/countable-set).\nAn integer may be regarded as a real number that can be written without a [fractional component](https://bliptext.com/articles/fraction).\nFor example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1⁄2, 5/4, and [√2](https://bliptext.com/articles/square-root-of-2) are not.\nThe integers form the smallest [group](https://bliptext.com/articles/group-mathematics) and the smallest [ring](https://bliptext.com/articles/ring-mathematics) containing the [natural numbers](https://bliptext.com/articles/natural-number).\nIn [algebraic number theory](https://bliptext.com/articles/algebraic-number-theory), the integers are sometimes qualified as **rational integers** to distinguish them from the more general [algebraic integers](https://bliptext.com/articles/algebraic-integer).\nIn fact, (rational) integers are algebraic integers that are also [rational numbers](https://bliptext.com/articles/rational-number).\n# [The integers arranged on a number line | upload.wikimedia.org/wikipedia/commons/a/ae/NumberLineIntegers.svg]\n\n# History\nThe word integer comes from the [Latin](https://bliptext.com/articles/latin) integer meaning \"whole\" or (literally) \"untouched\", from in (\"not\") plus tangere (\"to touch\").\n\"[Entire](https://bliptext.com/articles/entire)\" derives from the same origin via the [French](https://bliptext.com/articles/french-language) word [entier](https://bliptext.com/articles/entier), which means both entire and integer.\nHistorically the term was used for a [number](https://bliptext.com/articles/number) that was a multiple of 1, or to the whole part of a mixed [number](https://bliptext.com/articles/number).\nOnly positive integers were considered, making the term synonymous with the [natural numbers](https://bliptext.com/articles/natural-number).\nThe definition of integer expanded over time to include [negative numbers](https://bliptext.com/articles/negative-number) as their usefulness was recognized.\nFor example [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) in his 1765 [Elements of Algebra](https://bliptext.com/articles/elements-of-algebra) defined integers to include both positive and negative numbers.\nThe phrase the set of the integers was not used before the end of the 19th century, when [Georg Cantor](https://bliptext.com/articles/georg-cantor) introduced the concept of [infinite sets](https://bliptext.com/articles/infinite-set) and [set theory](https://bliptext.com/articles/set-theory).\nThe use of the letter Z to denote the set of integers comes from the [German](https://bliptext.com/articles/german-language) word [Zahlen](https://bliptext.com/articles/zahlen) (\"numbers\") and has been attributed to [David Hilbert](https://bliptext.com/articles/david-hilbert).\nThe earliest known use of the notation in a textbook occurs in [Algèbre](https://bliptext.com/articles/l-ments-de-math-matique) written by the collective [Nicolas Bourbaki](https://bliptext.com/articles/nicolas-bourbaki), dating to 1947.\nThe notation was not adopted immediately.\nFor example, another textbook used the letter J, and a 1960 paper used Z to denote the non-negative integers.\nBut by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.\nThe symbol \\mathbb{Z} is often annotated to denote various sets, with varying usage amongst different authors:, , or for the positive integers, or for non-negative integers, and for non-zero integers.\nSome authors use for non-zero integers, while others use it for non-negative integers, or for {–1,1} (the [group of units](https://bliptext.com/articles/group-of-units) of \\mathbb{Z}).\nAdditionally, is used to denote either the set of [[integers modulo n|integers modulo\np\n]] (i.e., the set of [congruence classes](https://bliptext.com/articles/congruence-relation) of integers), or the set of [[p-adic integer|\np\n-adic integers]].\nThe whole numbers were synonymous with the integers up until the early 1950s.\nIn the late 1950s, as part of the [New Math](https://bliptext.com/articles/new-math) movement, American elementary school teachers began teaching that whole numbers referred to the [natural numbers](https://bliptext.com/articles/natural-number), excluding negative numbers, while integer included the negative numbers.\nThe whole numbers remain ambiguous to the present day.\n\n# Algebraic properties\nLike the [natural numbers](https://bliptext.com/articles/natural-numbers), \\mathbb{Z} is [closed](https://bliptext.com/articles/closure-mathematics) under the [operations](https://bliptext.com/articles/binary-operation) of addition and [multiplication](https://bliptext.com/articles/multiplication), that is, the sum and product of any two integers is an integer.\nHowever, with the inclusion of the negative natural numbers (and importantly, ), \\mathbb{Z}, unlike the natural numbers, is also closed under [subtraction](https://bliptext.com/articles/subtraction).\nThe integers form a [ring](https://bliptext.com/articles/ring-mathematics) which is the most basic one, in the following sense: for any [ring](https://bliptext.com/articles/ring-mathematics), there is a unique [ring](https://bliptext.com/articles/ring-mathematics) homomorphism from the integers into this [ring](https://bliptext.com/articles/ring-mathematics).\nThis [universal property](https://bliptext.com/articles/universal-property), namely to be an [initial object](https://bliptext.com/articles/initial-object) in the [category of rings](https://bliptext.com/articles/category-of-rings), characterizes the ring \\mathbb{Z}.\n\\mathbb{Z} is not closed under [division](https://bliptext.com/articles/division-mathematics), since the quotient of two integers (e.g., 1 divided by 2) need not be an integer.\nAlthough the natural numbers are closed under [exponentiation](https://bliptext.com/articles/exponentiation), the integers are not (since the result can be a fraction when the exponent is negative).\nThe following table lists some of the basic properties of addition and multiplication for any integers\na\n,\nb\n, and\nc\nThe first five properties listed above for addition say that \\mathbb{Z}, under addition, is an [abelian group](https://bliptext.com/articles/abelian-group).\nIt is also a [cyclic group](https://bliptext.com/articles/cyclic-group), since every non-zero integer can be written as a finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1).\nIn fact, \\mathbb{Z} under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is [isomorphic](https://bliptext.com/articles/group-isomorphism) to \\mathbb{Z}.\nThe first four properties listed above for multiplication say that \\mathbb{Z} under multiplication is a [commutative monoid](https://bliptext.com/articles/commutative-monoid).\nHowever, not every integer has a multiplicative inverse (as is the case of the number 2), which means that \\mathbb{Z} under multiplication is not a group.\nAll the rules from the above property table (except for the last), when taken together, say that \\mathbb{Z} together with addition and multiplication is a [commutative ring](https://bliptext.com/articles/commutative-ring) with [unity](https://bliptext.com/articles/multiplicative-identity).\nIt is the prototype of all objects of such [algebraic structure](https://bliptext.com/articles/algebraic-structure).\nOnly those [equalities](https://bliptext.com/articles/equality-mathematics) of [expressions](https://bliptext.com/articles/algebraic-expression) are true in \\mathbb{Z} [for all](https://bliptext.com/articles/for-all) values of variables, which are true in any unital commutative ring.\nCertain non-[zero](https://bliptext.com/articles/additive-identity) integers map to [zero](https://bliptext.com/articles/additive-identity) in certain rings.\nThe lack of [zero divisors](https://bliptext.com/articles/zero-divisor) in the integers (last property in the table) means that the commutative ring \\mathbb{Z} is an [integral domain](https://bliptext.com/articles/integral-domain).\nThe lack of multiplicative inverses, which is equivalent to the fact that \\mathbb{Z} is not closed under division, means that \\mathbb{Z} is not a [field](https://bliptext.com/articles/field-mathematics).\nThe smallest field containing the integers as a [subring](https://bliptext.com/articles/subring) is the field of [rational numbers](https://bliptext.com/articles/rational-number).\nThe process of constructing the rationals from the integers can be mimicked to form the [field of fractions](https://bliptext.com/articles/field-of-fractions) of any integral domain.\nAnd back, starting from an [algebraic number field](https://bliptext.com/articles/algebraic-number-field) (an extension of rational numbers), its [ring of integers](https://bliptext.com/articles/ring-of-integers) can be extracted, which includes \\mathbb{Z} as its [subring](https://bliptext.com/articles/subring).\nAlthough ordinary division is not defined on \\mathbb{Z}, the division \"with remainder\" is defined on them.\nIt is called [Euclidean division](https://bliptext.com/articles/euclidean-division), and possesses the following important property: given two integers\na\nand\nb\nwith\nb ≠ 0\n, there exist unique integers\nq\nand\nr\nsuch that\nand\n0 ≤ r <\n, where\ndenotes the [absolute value](https://bliptext.com/articles/absolute-value) of\nb\n. The integer\nq\nis called the quotient and\nr\nis called the [remainder](https://bliptext.com/articles/remainder) of the division of\na\nby\nb\n. The [Euclidean algorithm](https://bliptext.com/articles/euclidean-algorithm) for computing [greatest common divisors](https://bliptext.com/articles/greatest-common-divisor) works by a sequence of Euclidean divisions.\nThe above says that \\mathbb{Z} is a [Euclidean domain](https://bliptext.com/articles/euclidean-domain).\nThis implies that \\mathbb{Z} is a [principal ideal domain](https://bliptext.com/articles/principal-ideal-domain), and any positive integer can be written as the products of [primes](https://bliptext.com/articles/prime-number) in an [essentially unique](https://bliptext.com/articles/essentially-unique) way.\nThis is the [fundamental theorem of arithmetic](https://bliptext.com/articles/fundamental-theorem-of-arithmetic).\n# [Integers can be thought of as discrete, equally spaced points on an infinitely long number line. In the above, non-negative integers are shown in blue and negative integers in red. | upload.wikimedia.org/wikipedia/commons/9/93/Number-line.svg]\n\n# Order-theoretic properties\n\\mathbb{Z} is a [totally ordered set](https://bliptext.com/articles/total-order) without [upper or lower bound](https://bliptext.com/articles/upper-and-lower-bounds).\nThe ordering of \\mathbb{Z} is given by:\n. An integer is positive if it is greater than [0|zero](https://bliptext.com/articles/0-zero), and negative if it is less than zero.\nZero is defined as neither negative nor positive.\nThe ordering of integers is compatible with the algebraic operations in the following way:\na < b\nand\nc < d\n, then\na + c < b + d\na < b\nand\n0 < c\n, then\nac < bc\nThus it follows that \\mathbb{Z} together with the above ordering is an [ordered ring](https://bliptext.com/articles/ordered-ring).\nThe integers are the only nontrivial [totally ordered](https://bliptext.com/articles/totally-ordered) [abelian group](https://bliptext.com/articles/abelian-group) whose positive elements are [well-ordered](https://bliptext.com/articles/well-ordered).\nThis is equivalent to the statement that any [Noetherian](https://bliptext.com/articles/noetherian-ring) [valuation ring](https://bliptext.com/articles/valuation-ring) is either a [field](https://bliptext.com/articles/field-mathematics)—or a discrete [valuation ring](https://bliptext.com/articles/valuation-ring).\n\n# Construction\n\n## Traditional development\nIn elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, [zero](https://bliptext.com/articles/zero), and the negations of the natural numbers.\nThis can be formalized as follows.\nFirst construct the set of natural numbers according to the [Peano axioms](https://bliptext.com/articles/peano-axioms), call this P. Then construct a set P^- which is [disjoint](https://bliptext.com/articles/disjoint-sets) from P and in one-to-one correspondence with P via a function \\psi.\nFor example, take P^- to be the [ordered pairs](https://bliptext.com/articles/ordered-pair) (1,n) with the mapping.\nFinally let 0 be some object not in P or P^-, for example the ordered pair (0,0).\nThen the integers are defined to be the union.\nThe traditional arithmetic operations can then be defined on the integers in a [piecewise](https://bliptext.com/articles/piecewise) fashion, for each of positive numbers, negative numbers, and zero.\nFor example [negation](https://bliptext.com/articles/negation) is defined as follows:\nThe 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.\n\n## Equivalence classes of ordered pairs\n[[File:Relative numbers representation.svg|thumb|alt=Representation of equivalence classes for the numbers −5 to 5\nIn modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead.\nThe integers can thus be formally constructed as the [equivalence classes](https://bliptext.com/articles/equivalence-class) of [ordered pairs](https://bliptext.com/articles/ordered-pair) of [natural numbers](https://bliptext.com/articles/natural-number)\n(a,b)\n.\nThe intuition is that\n(a,b)\nstands for the result of subtracting\nb\nfrom\na\n. To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an [equivalence relation](https://bliptext.com/articles/equivalence-relation)\n~\non these pairs with the following rule:\nprecisely when\nAddition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using\n[(a,b)]\nto denote the equivalence class having\n(a,b)\nas a member, one has:\nThe negation (or additive inverse) of an integer is obtained by reversing the order of the pair:\nHence subtraction can be defined as the addition of the additive inverse:\nThe standard ordering on the integers is given by:\nIt is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.\nEvery equivalence class has a unique member that is of the form\n(n,0)\nor\n(0,n)\n(or both at once).\nThe natural number\nn\nis identified with the class\n[(n,0)]\n(i.e., the natural numbers are [embedded](https://bliptext.com/articles/embedding) into the integers by map sending\nn\nto\n[(n,0)]\n), and the class\n[(0,n)]\nis denoted\n−n\n(this covers all remaining classes, and gives the class\n[(0,0)]\na second time since –0 = 0.\nThus,\n[(a,b)]\nis denoted by\nIf the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity.\nThis notation recovers the familiar [representation](https://bliptext.com/articles/group-representation) of the integers as\n{..., −2, −1, 0, 1, 2, ...}\n.\nSome examples are:\n\n## Other approaches\nIn theoretical computer science, other approaches for the construction of integers are used by [automated theorem provers](https://bliptext.com/articles/automated-theorem-proving) and [term rewrite engines](https://bliptext.com/articles/rewriting).\nIntegers are represented as [algebraic terms](https://bliptext.com/articles/term-algebra) built using a few basic operations (e.g., **zero**, **succ**, **pred**) and using [natural numbers](https://bliptext.com/articles/natural-number), which are assumed to be already constructed (using the [Peano approach](https://bliptext.com/articles/peano-axioms)).\nThere exist at least ten such constructions of signed integers.\nThese 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.\nThe 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 x-y).\nThis operation is not free since the integer 0 can be written ******pair******(0,0), or ******pair******(1,1), or ******pair******(2,2), etc..\nThis technique of construction is used by the [proof assistant](https://bliptext.com/articles/proof-assistant) [Isabelle](https://bliptext.com/articles/isabelle-proof-assistant); 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.\n\n# Computer science\nAn integer is often a primitive [data type](https://bliptext.com/articles/data-type) in [computer languages](https://bliptext.com/articles/computer-language).\nHowever, integer data types can only represent a [subset](https://bliptext.com/articles/subset) of all integers, since practical computers are of finite capacity.\nAlso, in the common [two's complement](https://bliptext.com/articles/two-s-complement) representation, the inherent definition of [sign](https://bliptext.com/articles/sign-mathematics) distinguishes between \"negative\" and \"non-negative\" rather than \"negative, positive, and 0\".\n(It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as [Algol68](https://bliptext.com/articles/algol68), [C](https://bliptext.com/articles/c-computer-language), [Java](https://bliptext.com/articles/java-programming-language), [Delphi](https://bliptext.com/articles/object-pascal), etc.).\nVariable-length representations of integers, such as [bignums](https://bliptext.com/articles/bignum), can store any integer that fits in the computer's memory.\nOther 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).\n\n# Cardinality\nThe set of integers is [countably infinite](https://bliptext.com/articles/countably-infinite), meaning it is possible to pair each integer with a unique natural number.\nAn example of such a pairing is\n(0, 1), (1, 2), (−1, 3), (2, 4), (−2, 5),\n(3, 6), . . . ,(1 − k, 2k − 1), (k, 2k ), . . .\nMore technically, the [cardinality](https://bliptext.com/articles/cardinality) of \\mathbb{Z} is said to equal\nℵ0\n([aleph-null](https://bliptext.com/articles/aleph-number)).\nThe pairing between elements of \\mathbb{Z} and \\mathbb{N} is called a [bijection](https://bliptext.com/articles/bijection).\n\n# Footnotes\n\n# Sources","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:45:58.090Z","updated_at":"2025-01-25T16:45:58.090Z","search_vector":"'integ':1","search_content":" \n \n\n# \nAn **integer** is the [number](https://bliptext.com/articles/number) zero ([0](https://bliptext.com/articles/0)), a positive natural [number](https://bliptext.com/articles/number) (1, 2, 3, . . .), or the negation of a positive natural [number](https://bliptext.com/articles/number) ([−1](https://bliptext.com/articles/1), −2, −3, . . .).\nThe negations or [additive inverses](https://bliptext.com/articles/additive-inverse) of the positive natural numbers are referred to as **negative integers**.\nThe [set](https://bliptext.com/articles/set-mathematics) of all integers is often denoted by the [boldface](https://bliptext.com/articles/boldface)\nZ\nor [blackboard bold](https://bliptext.com/articles/blackboard-bold) \\mathbb{Z}.\nThe set of natural numbers \\mathbb{N} is a [subset](https://bliptext.com/articles/subset) of \\mathbb{Z}, which in turn is a [subset](https://bliptext.com/articles/subset) of the set of all [rational numbers](https://bliptext.com/articles/rational-number) \\mathbb{Q}, itself a [subset](https://bliptext.com/articles/subset) of the [real numbers](https://bliptext.com/articles/real-number) \\mathbb{R}.\nLike the set of natural numbers, the set of integers \\mathbb{Z} is [countably infinite](https://bliptext.com/articles/countable-set).\nAn integer may be regarded as a real number that can be written without a [fractional component](https://bliptext.com/articles/fraction).\nFor example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1⁄2, 5/4, and [√2](https://bliptext.com/articles/square-root-of-2) are not.\nThe integers form the smallest [group](https://bliptext.com/articles/group-mathematics) and the smallest [ring](https://bliptext.com/articles/ring-mathematics) containing the [natural numbers](https://bliptext.com/articles/natural-number).\nIn [algebraic number theory](https://bliptext.com/articles/algebraic-number-theory), the integers are sometimes qualified as **rational integers** to distinguish them from the more general [algebraic integers](https://bliptext.com/articles/algebraic-integer).\nIn fact, (rational) integers are algebraic integers that are also [rational numbers](https://bliptext.com/articles/rational-number).\n# [The integers arranged on a number line | upload.wikimedia.org/wikipedia/commons/a/ae/NumberLineIntegers.svg]\n\n# History\nThe word integer comes from the [Latin](https://bliptext.com/articles/latin) integer meaning \"whole\" or (literally) \"untouched\", from in (\"not\") plus tangere (\"to touch\").\n\"[Entire](https://bliptext.com/articles/entire)\" derives from the same origin via the [French](https://bliptext.com/articles/french-language) word [entier](https://bliptext.com/articles/entier), which means both entire and integer.\nHistorically the term was used for a [number](https://bliptext.com/articles/number) that was a multiple of 1, or to the whole part of a mixed [number](https://bliptext.com/articles/number).\nOnly positive integers were considered, making the term synonymous with the [natural numbers](https://bliptext.com/articles/natural-number).\nThe definition of integer expanded over time to include [negative numbers](https://bliptext.com/articles/negative-number) as their usefulness was recognized.\nFor example [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) in his 1765 [Elements of Algebra](https://bliptext.com/articles/elements-of-algebra) defined integers to include both positive and negative numbers.\nThe phrase the set of the integers was not used before the end of the 19th century, when [Georg Cantor](https://bliptext.com/articles/georg-cantor) introduced the concept of [infinite sets](https://bliptext.com/articles/infinite-set) and [set theory](https://bliptext.com/articles/set-theory).\nThe use of the letter Z to denote the set of integers comes from the [German](https://bliptext.com/articles/german-language) word [Zahlen](https://bliptext.com/articles/zahlen) (\"numbers\") and has been attributed to [David Hilbert](https://bliptext.com/articles/david-hilbert).\nThe earliest known use of the notation in a textbook occurs in [Algèbre](https://bliptext.com/articles/l-ments-de-math-matique) written by the collective [Nicolas Bourbaki](https://bliptext.com/articles/nicolas-bourbaki), dating to 1947.\nThe notation was not adopted immediately.\nFor example, another textbook used the letter J, and a 1960 paper used Z to denote the non-negative integers.\nBut by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.\nThe symbol \\mathbb{Z} is often annotated to denote various sets, with varying usage amongst different authors:, , or for the positive integers, or for non-negative integers, and for non-zero integers.\nSome authors use for non-zero integers, while others use it for non-negative integers, or for {–1,1} (the [group of units](https://bliptext.com/articles/group-of-units) of \\mathbb{Z}).\nAdditionally, is used to denote either the set of [[integers modulo n|integers modulo\np\n]] (i.e., the set of [congruence classes](https://bliptext.com/articles/congruence-relation) of integers), or the set of [[p-adic integer|\np\n-adic integers]].\nThe whole numbers were synonymous with the integers up until the early 1950s.\nIn the late 1950s, as part of the [New Math](https://bliptext.com/articles/new-math) movement, American elementary school teachers began teaching that whole numbers referred to the [natural numbers](https://bliptext.com/articles/natural-number), excluding negative numbers, while integer included the negative numbers.\nThe whole numbers remain ambiguous to the present day.\n\n# Algebraic properties\nLike the [natural numbers](https://bliptext.com/articles/natural-numbers), \\mathbb{Z} is [closed](https://bliptext.com/articles/closure-mathematics) under the [operations](https://bliptext.com/articles/binary-operation) of addition and [multiplication](https://bliptext.com/articles/multiplication), that is, the sum and product of any two integers is an integer.\nHowever, with the inclusion of the negative natural numbers (and importantly, ), \\mathbb{Z}, unlike the natural numbers, is also closed under [subtraction](https://bliptext.com/articles/subtraction).\nThe integers form a [ring](https://bliptext.com/articles/ring-mathematics) which is the most basic one, in the following sense: for any [ring](https://bliptext.com/articles/ring-mathematics), there is a unique [ring](https://bliptext.com/articles/ring-mathematics) homomorphism from the integers into this [ring](https://bliptext.com/articles/ring-mathematics).\nThis [universal property](https://bliptext.com/articles/universal-property), namely to be an [initial object](https://bliptext.com/articles/initial-object) in the [category of rings](https://bliptext.com/articles/category-of-rings), characterizes the ring \\mathbb{Z}.\n\\mathbb{Z} is not closed under [division](https://bliptext.com/articles/division-mathematics), since the quotient of two integers (e.g., 1 divided by 2) need not be an integer.\nAlthough the natural numbers are closed under [exponentiation](https://bliptext.com/articles/exponentiation), the integers are not (since the result can be a fraction when the exponent is negative).\nThe following table lists some of the basic properties of addition and multiplication for any integers\na\n,\nb\n, and\nc\nThe first five properties listed above for addition say that \\mathbb{Z}, under addition, is an [abelian group](https://bliptext.com/articles/abelian-group).\nIt is also a [cyclic group](https://bliptext.com/articles/cyclic-group), since every non-zero integer can be written as a finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1).\nIn fact, \\mathbb{Z} under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is [isomorphic](https://bliptext.com/articles/group-isomorphism) to \\mathbb{Z}.\nThe first four properties listed above for multiplication say that \\mathbb{Z} under multiplication is a [commutative monoid](https://bliptext.com/articles/commutative-monoid).\nHowever, not every integer has a multiplicative inverse (as is the case of the number 2), which means that \\mathbb{Z} under multiplication is not a group.\nAll the rules from the above property table (except for the last), when taken together, say that \\mathbb{Z} together with addition and multiplication is a [commutative ring](https://bliptext.com/articles/commutative-ring) with [unity](https://bliptext.com/articles/multiplicative-identity).\nIt is the prototype of all objects of such [algebraic structure](https://bliptext.com/articles/algebraic-structure).\nOnly those [equalities](https://bliptext.com/articles/equality-mathematics) of [expressions](https://bliptext.com/articles/algebraic-expression) are true in \\mathbb{Z} [for all](https://bliptext.com/articles/for-all) values of variables, which are true in any unital commutative ring.\nCertain non-[zero](https://bliptext.com/articles/additive-identity) integers map to [zero](https://bliptext.com/articles/additive-identity) in certain rings.\nThe lack of [zero divisors](https://bliptext.com/articles/zero-divisor) in the integers (last property in the table) means that the commutative ring \\mathbb{Z} is an [integral domain](https://bliptext.com/articles/integral-domain).\nThe lack of multiplicative inverses, which is equivalent to the fact that \\mathbb{Z} is not closed under division, means that \\mathbb{Z} is not a [field](https://bliptext.com/articles/field-mathematics).\nThe smallest field containing the integers as a [subring](https://bliptext.com/articles/subring) is the field of [rational numbers](https://bliptext.com/articles/rational-number).\nThe process of constructing the rationals from the integers can be mimicked to form the [field of fractions](https://bliptext.com/articles/field-of-fractions) of any integral domain.\nAnd back, starting from an [algebraic number field](https://bliptext.com/articles/algebraic-number-field) (an extension of rational numbers), its [ring of integers](https://bliptext.com/articles/ring-of-integers) can be extracted, which includes \\mathbb{Z} as its [subring](https://bliptext.com/articles/subring).\nAlthough ordinary division is not defined on \\mathbb{Z}, the division \"with remainder\" is defined on them.\nIt is called [Euclidean division](https://bliptext.com/articles/euclidean-division), and possesses the following important property: given two integers\na\nand\nb\nwith\nb ≠ 0\n, there exist unique integers\nq\nand\nr\nsuch that\nand\n0 ≤ r <\n, where\ndenotes the [absolute value](https://bliptext.com/articles/absolute-value) of\nb\n. The integer\nq\nis called the quotient and\nr\nis called the [remainder](https://bliptext.com/articles/remainder) of the division of\na\nby\nb\n. The [Euclidean algorithm](https://bliptext.com/articles/euclidean-algorithm) for computing [greatest common divisors](https://bliptext.com/articles/greatest-common-divisor) works by a sequence of Euclidean divisions.\nThe above says that \\mathbb{Z} is a [Euclidean domain](https://bliptext.com/articles/euclidean-domain).\nThis implies that \\mathbb{Z} is a [principal ideal domain](https://bliptext.com/articles/principal-ideal-domain), and any positive integer can be written as the products of [primes](https://bliptext.com/articles/prime-number) in an [essentially unique](https://bliptext.com/articles/essentially-unique) way.\nThis is the [fundamental theorem of arithmetic](https://bliptext.com/articles/fundamental-theorem-of-arithmetic).\n# [Integers can be thought of as discrete, equally spaced points on an infinitely long number line. In the above, non-negative integers are shown in blue and negative integers in red. | upload.wikimedia.org/wikipedia/commons/9/93/Number-line.svg]\n\n# Order-theoretic properties\n\\mathbb{Z} is a [totally ordered set](https://bliptext.com/articles/total-order) without [upper or lower bound](https://bliptext.com/articles/upper-and-lower-bounds).\nThe ordering of \\mathbb{Z} is given by:\n. An integer is positive if it is greater than [0|zero](https://bliptext.com/articles/0-zero), and negative if it is less than zero.\nZero is defined as neither negative nor positive.\nThe ordering of integers is compatible with the algebraic operations in the following way:\na < b\nand\nc < d\n, then\na + c < b + d\na < b\nand\n0 < c\n, then\nac < bc\nThus it follows that \\mathbb{Z} together with the above ordering is an [ordered ring](https://bliptext.com/articles/ordered-ring).\nThe integers are the only nontrivial [totally ordered](https://bliptext.com/articles/totally-ordered) [abelian group](https://bliptext.com/articles/abelian-group) whose positive elements are [well-ordered](https://bliptext.com/articles/well-ordered).\nThis is equivalent to the statement that any [Noetherian](https://bliptext.com/articles/noetherian-ring) [valuation ring](https://bliptext.com/articles/valuation-ring) is either a [field](https://bliptext.com/articles/field-mathematics)—or a discrete [valuation ring](https://bliptext.com/articles/valuation-ring).\n\n# Construction\n\n## Traditional development\nIn elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, [zero](https://bliptext.com/articles/zero), and the negations of the natural numbers.\nThis can be formalized as follows.\nFirst construct the set of natural numbers according to the [Peano axioms](https://bliptext.com/articles/peano-axioms), call this P. Then construct a set P^- which is [disjoint](https://bliptext.com/articles/disjoint-sets) from P and in one-to-one correspondence with P via a function \\psi.\nFor example, take P^- to be the [ordered pairs](https://bliptext.com/articles/ordered-pair) (1,n) with the mapping.\nFinally let 0 be some object not in P or P^-, for example the ordered pair (0,0).\nThen the integers are defined to be the union.\nThe traditional arithmetic operations can then be defined on the integers in a [piecewise](https://bliptext.com/articles/piecewise) fashion, for each of positive numbers, negative numbers, and zero.\nFor example [negation](https://bliptext.com/articles/negation) is defined as follows:\nThe 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.\n\n## Equivalence classes of ordered pairs\n[[File:Relative numbers representation.svg|thumb|alt=Representation of equivalence classes for the numbers −5 to 5\nIn modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead.\nThe integers can thus be formally constructed as the [equivalence classes](https://bliptext.com/articles/equivalence-class) of [ordered pairs](https://bliptext.com/articles/ordered-pair) of [natural numbers](https://bliptext.com/articles/natural-number)\n(a,b)\n.\nThe intuition is that\n(a,b)\nstands for the result of subtracting\nb\nfrom\na\n. To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an [equivalence relation](https://bliptext.com/articles/equivalence-relation)\n~\non these pairs with the following rule:\nprecisely when\nAddition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using\n[(a,b)]\nto denote the equivalence class having\n(a,b)\nas a member, one has:\nThe negation (or additive inverse) of an integer is obtained by reversing the order of the pair:\nHence subtraction can be defined as the addition of the additive inverse:\nThe standard ordering on the integers is given by:\nIt is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.\nEvery equivalence class has a unique member that is of the form\n(n,0)\nor\n(0,n)\n(or both at once).\nThe natural number\nn\nis identified with the class\n[(n,0)]\n(i.e., the natural numbers are [embedded](https://bliptext.com/articles/embedding) into the integers by map sending\nn\nto\n[(n,0)]\n), and the class\n[(0,n)]\nis denoted\n−n\n(this covers all remaining classes, and gives the class\n[(0,0)]\na second time since –0 = 0.\nThus,\n[(a,b)]\nis denoted by\nIf the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity.\nThis notation recovers the familiar [representation](https://bliptext.com/articles/group-representation) of the integers as\n{..., −2, −1, 0, 1, 2, ...}\n.\nSome examples are:\n\n## Other approaches\nIn theoretical computer science, other approaches for the construction of integers are used by [automated theorem provers](https://bliptext.com/articles/automated-theorem-proving) and [term rewrite engines](https://bliptext.com/articles/rewriting).\nIntegers are represented as [algebraic terms](https://bliptext.com/articles/term-algebra) built using a few basic operations (e.g., **zero**, **succ**, **pred**) and using [natural numbers](https://bliptext.com/articles/natural-number), which are assumed to be already constructed (using the [Peano approach](https://bliptext.com/articles/peano-axioms)).\nThere exist at least ten such constructions of signed integers.\nThese 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.\nThe 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 x-y).\nThis operation is not free since the integer 0 can be written ******pair******(0,0), or ******pair******(1,1), or ******pair******(2,2), etc..\nThis technique of construction is used by the [proof assistant](https://bliptext.com/articles/proof-assistant) [Isabelle](https://bliptext.com/articles/isabelle-proof-assistant); 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.\n\n# Computer science\nAn integer is often a primitive [data type](https://bliptext.com/articles/data-type) in [computer languages](https://bliptext.com/articles/computer-language).\nHowever, integer data types can only represent a [subset](https://bliptext.com/articles/subset) of all integers, since practical computers are of finite capacity.\nAlso, in the common [two's complement](https://bliptext.com/articles/two-s-complement) representation, the inherent definition of [sign](https://bliptext.com/articles/sign-mathematics) distinguishes between \"negative\" and \"non-negative\" rather than \"negative, positive, and 0\".\n(It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as [Algol68](https://bliptext.com/articles/algol68), [C](https://bliptext.com/articles/c-computer-language), [Java](https://bliptext.com/articles/java-programming-language), [Delphi](https://bliptext.com/articles/object-pascal), etc.).\nVariable-length representations of integers, such as [bignums](https://bliptext.com/articles/bignum), can store any integer that fits in the computer's memory.\nOther 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).\n\n# Cardinality\nThe set of integers is [countably infinite](https://bliptext.com/articles/countably-infinite), meaning it is possible to pair each integer with a unique natural number.\nAn example of such a pairing is\n(0, 1), (1, 2), (−1, 3), (2, 4), (−2, 5),\n(3, 6), . . . ,(1 − k, 2k − 1), (k, 2k ), . . .\nMore technically, the [cardinality](https://bliptext.com/articles/cardinality) of \\mathbb{Z} is said to equal\nℵ0\n([aleph-null](https://bliptext.com/articles/aleph-number)).\nThe pairing between elements of \\mathbb{Z} and \\mathbb{N} is called a [bijection](https://bliptext.com/articles/bijection).\n\n# Footnotes\n\n# Sources","revisionCount":0},"hasMore":false,"nextCursor":null}