# Integer

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Number in {..., –2, –1, 0, 1, 2, ...}

This article is about integers in mathematics. For integers as a data type, see [Integer (computer science)](/source/Integer_(computer_science)).

The integers arranged on a [number line](/source/Number_line)

An **integer** is the [number](/source/Number) zero ([0](/source/0)), a positive [natural number](/source/Natural_number) (1, 2, 3, ...), or the negation of a positive natural number ([−1](/source/%E2%88%921), −2, −3, ...).[1] The negations or [additive inverses](/source/Additive_inverse) of the positive natural numbers are referred to as **negative integers**.[2] The [set](/source/Set_(mathematics)) of all integers is often denoted by the [boldface](/source/Boldface) **Z** or [blackboard bold](/source/Blackboard_bold) ⁠ Z {\displaystyle \mathbb {Z} } ⁠.[3][4]

The set of natural numbers ⁠ N {\displaystyle \mathbb {N} } ⁠ is a [subset](/source/Subset) of ⁠ Z {\displaystyle \mathbb {Z} } ⁠, which in turn is a subset of the set of all [rational numbers](/source/Rational_number) ⁠ Q {\displaystyle \mathbb {Q} } ⁠, itself a subset of the [real numbers](/source/Real_number) ⁠ R {\displaystyle \mathbb {R} } ⁠.[a] Like the set of natural numbers, the set of integers ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is [countably infinite](/source/Countable_set). An integer may be regarded as a real number that can be written without a [fractional component](/source/Fraction). For example, 21, 4, 0, and −2048 are integers, while 9.75, ⁠5+1/2⁠, 5/4, and the [square root of 2](/source/Square_root_of_2) are not.[7]

The integers form the smallest [group](/source/Group_(mathematics)) and the smallest [ring](/source/Ring_(mathematics)) containing the [natural numbers](/source/Natural_number). In [algebraic number theory](/source/Algebraic_number_theory), integers are sometimes called **rational integers** to distinguish them from the more general [algebraic integers](/source/Algebraic_integer). In fact, (rational) integers are algebraic integers that are also [rational numbers](/source/Rational_number).

## History

The word integer comes from the [Latin](/source/Latin) [*integer*](https://en.wiktionary.org/wiki/integer#Latin) meaning "whole" or (literally) "untouched", from *in* ("not") plus *tangere* ("to touch"). "[Entire](https://en.wiktionary.org/wiki/entire)" derives from the same origin via the [French](/source/French_language) word *[entier](https://en.wiktionary.org/wiki/entier)*, which means both *entire* and *integer*.[8] Historically the term was used for a [number](/source/Number) that was a multiple of 1,[9][10] or to the whole part of a [mixed number](/source/Mixed_number).[11][12] Only positive integers were considered, making the term synonymous with the [natural numbers](/source/Natural_number). The definition of integer expanded over time to include [negative numbers](/source/Negative_number) as their usefulness was recognized.[13] For example [Leonhard Euler](/source/Leonhard_Euler) in his 1765 *[Elements of Algebra](/source/Elements_of_Algebra)* defined integers to include both positive and negative numbers.[14]

The phrase *the set of the integers* was not used before the end of the 19th century, when [Georg Cantor](/source/Georg_Cantor) introduced the concept of [infinite sets](/source/Infinite_set) and [set theory](/source/Set_theory). The use of the letter Z to denote the set of integers comes from the [German](/source/German_language) word *[Zahlen](https://en.wiktionary.org/wiki/Zahlen)* ("numbers")[3][4] and has been attributed to [David Hilbert](/source/David_Hilbert).[15] The earliest known use of the notation in a textbook occurs in [Algèbre](/source/%C3%89l%C3%A9ments_de_math%C3%A9matique) written by the collective [Nicolas Bourbaki](/source/Nicolas_Bourbaki), dating to 1947.[3][16] The notation was not adopted immediately. For example, another textbook used the letter J,[17] and a 1960 paper used Z to denote the non-negative integers.[18] But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.[19]

The symbol ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is often annotated to denote various sets, with varying usage amongst different authors: ⁠ Z + {\displaystyle \mathbb {Z} ^{+}} ⁠, ⁠ Z + {\displaystyle \mathbb {Z} _{+}} ⁠, or ⁠ Z > {\displaystyle \mathbb {Z} ^{>}} ⁠ for the positive integers, ⁠ Z 0 + {\displaystyle \mathbb {Z} ^{0+}} ⁠ or ⁠ Z ≥ {\displaystyle \mathbb {Z} ^{\geq }} ⁠ for non-negative integers, and ⁠ Z ≠ {\displaystyle \mathbb {Z} ^{\neq }} ⁠ for non-zero integers. Some authors use ⁠ Z ∗ {\displaystyle \mathbb {Z} ^{*}} ⁠ for non-zero integers, while others use it for non-negative integers, or for {−1,1} (the [group of units](/source/Group_of_units) of ⁠ Z {\displaystyle \mathbb {Z} } ⁠). Additionally, ⁠ Z p {\displaystyle \mathbb {Z} _{p}} ⁠ is used to denote either the set of [integers modulo *p*](/source/Integers_modulo_n) (i.e., the set of [congruence classes](/source/Congruence_relation) of integers), or the set of [*p*-adic integers](/source/P-adic_integer).[20][21]

The *whole numbers* were synonymous with the integers up until the early 1950s.[22][23][24] In the late 1950s, as part of the [New Math](/source/New_Math) movement,[25] American elementary school teachers began teaching that *whole numbers* referred to the [natural numbers](/source/Natural_number), excluding negative numbers, while *integer* included the negative numbers.[26][27] The *whole numbers* remain ambiguous to the present day.[28]

## Algebraic properties

Integers can be thought of as discrete, equally spaced points on an infinitely long [number line](/source/Number_line). In the above, non-[negative](/source/Sign_(mathematics)#Terminology_for_signs) integers are shown in blue and negative integers in red.

Algebraic structure → Group theory Group theory Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic Discrete groups Lattices Integers ( Z {\displaystyle \mathbb {Z} } ) Free group Modular groups PSL(2, Z {\displaystyle \mathbb {Z} } ) SL(2, Z {\displaystyle \mathbb {Z} } ) Arithmetic group Lattice Hyperbolic group Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve v t e

Algebraic structure → Ring theory Ring theory Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring Z {\displaystyle \mathbb {Z} } • Terminal ring 0 = Z / 1 Z {\displaystyle 0=\mathbb {Z} /1\mathbb {Z} } Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Integers modulo n • Ring of integers • p-adic integers Z p {\displaystyle \mathbb {Z} _{p}} • p-adic numbers Q p {\displaystyle \mathbb {Q} _{p}} • Prüfer p-ring Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })} Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra v t e

Like the [natural numbers](/source/Natural_numbers), ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is [closed](/source/Closure_(mathematics)) under the [operations](/source/Binary_operation) of addition and [multiplication](/source/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, [0](/source/0_(number))), ⁠ Z {\displaystyle \mathbb {Z} } ⁠, unlike the natural numbers, is also closed under [subtraction](/source/Subtraction).[29]

The integers form a [ring](/source/Ring_(mathematics)) which is the most basic one, in the following sense: for any ring, there is a unique [ring homomorphism](/source/Ring_homomorphism) from the integers into this ring. This [universal property](/source/Universal_property), namely to be an [initial object](/source/Initial_object) in the [category of rings](/source/Category_of_rings), characterizes the ring ⁠ Z {\displaystyle \mathbb {Z} } ⁠. This unique homomorphism is [injective](/source/Injective) if and only if the [characteristic](/source/Characteristic_(algebra)) of the ring is zero. It follows that every ring of characteristic zero contains a subring isomorphic to ⁠ Z {\displaystyle \mathbb {Z} } ⁠, which is its smallest subring.

⁠ Z {\displaystyle \mathbb {Z} } ⁠ is not closed under [division](/source/Division_(mathematics)), 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](/source/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 *a*, *b*, and *c*:

Properties of addition and multiplication on integers 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 The only invertible integers (called units) are −1 and 1. Distributivity: a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c) No zero divisors: If a × b = 0, then a = 0 or b = 0 (or both)

The first five properties listed above for addition say that ⁠ Z {\displaystyle \mathbb {Z} } ⁠, under addition, is an [abelian group](/source/Abelian_group). It is also a [cyclic group](/source/Cyclic_group), since every non-zero integer can be written as a finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1). In fact, ⁠ Z {\displaystyle \mathbb {Z} } ⁠ under addition is the *only* infinite cyclic group—in the sense that any infinite cyclic group is [isomorphic](/source/Group_isomorphism) to ⁠ Z {\displaystyle \mathbb {Z} } ⁠.

The first four properties listed above for multiplication say that ⁠ Z {\displaystyle \mathbb {Z} } ⁠ under multiplication is a [commutative monoid](/source/Commutative_monoid). However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that ⁠ Z {\displaystyle \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 ⁠ Z {\displaystyle \mathbb {Z} } ⁠ together with addition and multiplication is a [commutative ring](/source/Commutative_ring) with [unity](/source/Multiplicative_identity). It is the prototype of all objects of such [algebraic structure](/source/Algebraic_structure). Only those [equalities](/source/Equality_(mathematics)) of [expressions](/source/Algebraic_expression) are true in ⁠ Z {\displaystyle \mathbb {Z} } ⁠ [for all](/source/For_all) values of variables, which are true in any unital commutative ring. Certain non-zero integers map to [zero](/source/Additive_identity) in certain rings.

The lack of [zero divisors](/source/Zero_divisor) in the integers (last property in the table) means that the commutative ring ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is an [integral domain](/source/Integral_domain).

The lack of multiplicative inverses, which is equivalent to the fact that ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is not closed under division, means that ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is *not* a [field](/source/Field_(mathematics)). The smallest field containing the integers as a [subring](/source/Subring) is the field of [rational numbers](/source/Rational_number). The process of constructing the rationals from the integers can be mimicked to form the [field of fractions](/source/Field_of_fractions) of any integral domain. And back, starting from an [algebraic number field](/source/Algebraic_number_field) (an extension of rational numbers), its [ring of integers](/source/Ring_of_integers) can be extracted, which includes ⁠ Z {\displaystyle \mathbb {Z} } ⁠ as its [subring](/source/Subring).

Although ordinary division is not defined on ⁠ Z {\displaystyle \mathbb {Z} } ⁠, the division "with remainder" is defined on them. It is called [Euclidean division](/source/Euclidean_division), and possesses the following important property: 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](/source/Absolute_value) of *b*. The integer *q* is called the *quotient* and *r* is called the *[remainder](/source/Remainder)* of the division of *a* by *b*. The [Euclidean algorithm](/source/Euclidean_algorithm) for computing [greatest common divisors](/source/Greatest_common_divisor) works by a sequence of Euclidean divisions.

The above says that ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is a [Euclidean domain](/source/Euclidean_domain). This implies that ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is a [principal ideal domain](/source/Principal_ideal_domain), and any positive integer can be written as the products of [primes](/source/Prime_number) in an [essentially unique](/source/Essentially_unique) way.[30] This is the [fundamental theorem of arithmetic](/source/Fundamental_theorem_of_arithmetic).

## Order-theoretic properties

⁠ Z {\displaystyle \mathbb {Z} } ⁠ is a [totally ordered set](/source/Total_order) without [upper or lower bound](/source/Upper_and_lower_bounds). The ordering of ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is given by:

- ⋯ < − 3 < − 2 < − 1 < 0 < 1 < 2 < 3 < ⋯ . {\displaystyle \cdots <-3<-2<-1<0<1<2<3<\cdots .}

An integer is *positive* if it is greater than [zero](/source/0), 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*

1. If *a* < *b* and 0 < *c*, then *ac* < *bc*

Thus it follows that ⁠ Z {\displaystyle \mathbb {Z} } ⁠ together with the above ordering is an [ordered ring](/source/Ordered_ring).

The integers are the only nontrivial [totally ordered](/source/Totally_ordered) [abelian group](/source/Abelian_group) whose positive elements are [well-ordered](/source/Well-ordered).[31] This is equivalent to the statement that any [Noetherian](/source/Noetherian_ring) [valuation ring](/source/Valuation_ring) is either a [field](/source/Field_(mathematics))—or a [discrete valuation ring](/source/Discrete_valuation_ring).

## Construction

### Traditional development

In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, [zero](/source/Zero), and the negations of the natural numbers. This can be formalized as follows.[32] First construct the set of natural numbers according to the [Peano axioms](/source/Peano_axioms), call this ⁠ P {\displaystyle P} ⁠. Then construct a set ⁠ P − {\displaystyle P^{-}} ⁠ which is [disjoint](/source/Disjoint_sets) from ⁠ P {\displaystyle P} ⁠ and in one-to-one correspondence with ⁠ P {\displaystyle P} ⁠ via a function ⁠ ψ {\displaystyle \psi } ⁠. For example, take ⁠ P − {\displaystyle P^{-}} ⁠ to be the [ordered pairs](/source/Ordered_pair) ⁠ ( 1 , n ) {\displaystyle (1,n)} ⁠ with the mapping ⁠ ψ = n ↦ ( 1 , n ) {\displaystyle \psi =n\mapsto (1,n)} ⁠. Finally let 0 be some object not in ⁠ P {\displaystyle P} ⁠ or ⁠ P − {\displaystyle P^{-}} ⁠, for example the ordered pair (0,0). Then the integers are defined to be the union ⁠ P ∪ P − ∪ { 0 } {\displaystyle P\cup P^{-}\cup \{0\}} ⁠.

The traditional arithmetic operations can then be defined on the integers in a [piecewise](/source/Piecewise) fashion, for each of positive numbers, negative numbers, and zero. For example [negation](/source/Negation) is defined as follows:

− x = { ψ ( x ) , if x ∈ P ψ − 1 ( x ) , if x ∈ P − 0 , if x = 0 {\displaystyle -x={\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.[33]

### Equivalence classes of ordered pairs

Red points represent ordered pairs of [natural numbers](/source/Natural_number). Linked red points are equivalence classes representing the blue integers at the end of the line.

In modern set-theoretic mathematics, a more abstract construction[34][35] allowing one to define arithmetical operations without any case distinction is often used instead.[36] The integers can thus be formally constructed as the [equivalence classes](/source/Equivalence_class) of [ordered pairs](/source/Ordered_pair) of [natural numbers](/source/Natural_number) (*a*,*b*).[37]

The intuition is that (*a*,*b*) stands for the result of subtracting *b* from *a*.[37] To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an [equivalence relation](/source/Equivalence_relation) ~ on these pairs with the following rule:

- ( a , b ) ∼ ( c , d ) {\displaystyle (a,b)\sim (c,d)}

precisely when

- a + d = b + c {\displaystyle a+d=b+c} .

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;[37] by using [(*a*,*b*)] to denote the equivalence class having (*a*,*b*) as a member, one has:

- [ ( a , b ) ] + [ ( c , d ) ] := [ ( a + c , b + d ) ] {\displaystyle [(a,b)]+[(c,d)]:=[(a+c,b+d)]} .

- [ ( a , b ) ] ⋅ [ ( c , d ) ] := [ ( a c + b d , a d + b c ) ] {\displaystyle [(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)]} .

The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:

- − [ ( a , b ) ] := [ ( b , a ) ] {\displaystyle -[(a,b)]:=[(b,a)]} .

Hence subtraction can be defined as the addition of the additive inverse:

- [ ( a , b ) ] − [ ( c , d ) ] := [ ( a + d , b + c ) ] {\displaystyle [(a,b)]-[(c,d)]:=[(a+d,b+c)]} .

The standard ordering on the integers is given by:

- [ ( a , b ) ] < [ ( c , d ) ] {\displaystyle [(a,b)]<[(c,d)]} [if and only if](/source/If_and_only_if) a + d < b + c {\displaystyle a+d<b+c} .

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 (*n*,0) or (0,*n*) (or both at once). The natural number *n* is identified with the class [(*n*,0)] (i.e., the natural numbers are [embedded](/source/Embedding) into the integers by map sending *n* to [(*n*,0)]), and the class [(0,*n*)] is denoted −*n* (this covers all remaining classes, and gives the class [(0,0)] a second time since −0 = 0.

Thus, [(*a*,*b*)] is denoted by

- { a − b , if a ≥ b − ( b − a ) , if a < b {\displaystyle {\begin{cases}a-b,&{\mbox{if }}a\geq b\\-(b-a),&{\mbox{if }}a<b\end{cases}}}

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](/source/Group_representation) of the integers as {..., −2, −1, 0, 1, 2, ...} .

Some examples are:

- 0 = [ ( 0 , 0 ) ] = [ ( 1 , 1 ) ] = ⋯ = [ ( k , k ) ] , 1 = [ ( 1 , 0 ) ] = [ ( 2 , 1 ) ] = ⋯ = [ ( k + 1 , k ) ] , − 1 = [ ( 0 , 1 ) ] = [ ( 1 , 2 ) ] = ⋯ = [ ( k , k + 1 ) ] , 2 = [ ( 2 , 0 ) ] = [ ( 3 , 1 ) ] = ⋯ = [ ( k + 2 , k ) ] , − 2 = [ ( 0 , 2 ) ] = [ ( 1 , 3 ) ] = ⋯ = [ ( k , k + 2 ) ] . {\displaystyle {\begin{alignedat}{3}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{alignedat}}}

### Other approaches

In theoretical computer science, other approaches for the construction of integers are used by [automated theorem provers](/source/Automated_theorem_proving) and [term rewrite engines](/source/Rewriting). Integers are represented as [algebraic terms](/source/Term_algebra) built using a few basic operations (e.g., **zero**, **succ**, **pred**) and using [natural numbers](/source/Natural_number), which are assumed to be already constructed (using the [Peano approach](/source/Peano_axioms)).

There exist at least ten such constructions of signed integers.[38] 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 ) {\displaystyle (x,y)} that takes as arguments two natural numbers ⁠ x {\displaystyle x} ⁠ and ⁠ y {\displaystyle y} ⁠, and returns an integer (equal to ⁠ x − y {\displaystyle x-y} ⁠). 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](/source/Proof_assistant) [Isabelle](/source/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.

## Computer science

Main article: [Integer (computer science)](/source/Integer_(computer_science))

An integer is often a primitive [data type](/source/Data_type) in [computer languages](/source/Computer_language). However, integer data types can only represent a [subset](/source/Subset) of all integers, since practical computers are of finite capacity. Also, in the common [two's complement](/source/Two's_complement) representation, the inherent definition of [sign](/source/Sign_(mathematics)) 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.) Fixed length integer approximation data types (or subsets) are denoted *int* or Integer in several programming languages (such as [Algol68](/source/Algol68), [C](/source/C_(computer_language)), [Java](/source/Java_(programming_language)), [Delphi](/source/Object_Pascal), etc.).

Variable-length representations of integers, such as [bignums](/source/Bignum), 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).

## Cardinality

The set of integers is [countably infinite](/source/Countably_infinite), meaning it is possible to pair each integer with a unique natural number. An example of such a pairing is

- (0, 1), (1, 2), (−1, 3), (2, 4), (−2, 5), (3, 6), . . . ,(1 − *k*, 2*k* − 1), (*k*, 2*k* ), . . .

More technically, the [cardinality](/source/Cardinality) of ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is said to equal ℵ0 ([aleph-null](/source/Aleph_number)). The pairing between elements of ⁠ Z {\displaystyle \mathbb {Z} } ⁠ and ⁠ N {\displaystyle \mathbb {N} } ⁠ is called a [bijection](/source/Bijection).

## See also

- [Mathematics portal](https://en.wikipedia.org/wiki/Portal:Mathematics)

- [Canonical factorization of a positive integer](/source/Canonical_representation_of_a_positive_integer)

- [Complex integer](/source/Complex_integer)

- [Hyperinteger](/source/Hyperinteger)

- [Integer complexity](/source/Integer_complexity)

- [Integer lattice](/source/Integer_lattice)

- [Integer part](/source/Integer_part)

- [Integer sequence](/source/Integer_sequence)

- [Integer-valued function](/source/Integer-valued_function)

- [Mathematical symbols](/source/Mathematical_symbols)

- [Parity (mathematics)](/source/Parity_(mathematics))

- [Profinite integer](/source/Profinite_integer)

[Set inclusions](/source/Set_inclusion) between the [natural numbers](/source/Natural_number) (

          N

    {\displaystyle \mathbb {N} }

), the integers (

          Z

    {\displaystyle \mathbb {Z} }

), the [rational numbers](/source/Rational_number) (

          Q

    {\displaystyle \mathbb {Q} }

), the [real numbers](/source/Real_number) (

          R

    {\displaystyle \mathbb {R} }

), and the [complex numbers](/source/Complex_number) (

          C

    {\displaystyle \mathbb {C} }

)

## Footnotes

1. **[^](#cite_ref-7)** More precisely, each system is [embedded](/source/Embedding) in the next, isomorphically mapped to a subset.[5] The commonly-assumed set-theoretic containment may be obtained by constructing the reals, discarding any earlier constructions, and defining the other sets as subsets of the reals.[6]

## References

1. **[^](#cite_ref-1)** [*Science and Technology Encyclopedia*](https://books.google.com/books?id=PZIdcYCCf2kC&dq=integer&pg=PA280). University of Chicago Press. September 2000. p. 280. [ISBN](/source/ISBN_(identifier)) [978-0-226-74267-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-226-74267-0).

1. **[^](#cite_ref-2)** Hillman, Abraham P.; Alexanderson, Gerald L. (1963). [*Algebra and trigonometry;*](https://archive.org/details/algebratrigonome0000hill/page/42/mode/2up). Boston: Allyn and Bacon.

1. ^ [***a***](#cite_ref-earliest_3-0) [***b***](#cite_ref-earliest_3-1) [***c***](#cite_ref-earliest_3-2) Miller, Jeff (29 August 2010). ["Earliest Uses of Symbols of Number Theory"](https://web.archive.org/web/20100131022510/http://jeff560.tripod.com/nth.html). Archived from [the original](http://jeff560.tripod.com/nth.html) on 31 January 2010. Retrieved 20 September 2010.

1. ^ [***a***](#cite_ref-Cameron1998_4-0) [***b***](#cite_ref-Cameron1998_4-1) Peter Jephson Cameron (1998). [*Introduction to Algebra*](https://books.google.com/books?id=syYYl-NVM5IC&pg=PA4). Oxford University Press. p. 4. [ISBN](/source/ISBN_(identifier)) [978-0-19-850195-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-850195-4). [Archived](https://web.archive.org/web/20161208142220/https://books.google.com/books?id=syYYl-NVM5IC&pg=PA4) from the original on 8 December 2016. Retrieved 15 February 2016.

1. **[^](#cite_ref-5)** Partee, Barbara H.; Meulen, Alice ter; Wall, Robert E. (30 April 1990). [*Mathematical Methods in Linguistics*](https://books.google.com/books?id=qV7TUuaYcUIC&pg=PA80). Springer Science & Business Media. pp. 78–82. [ISBN](/source/ISBN_(identifier)) [978-90-277-2245-4](https://en.wikipedia.org/wiki/Special:BookSources/978-90-277-2245-4). The natural numbers are not themselves a subset of this set-theoretic representation of the integers. Rather, the set of all integers contains a subset consisting of the positive integers and zero which is isomorphic to the set of natural numbers.

1. **[^](#cite_ref-6)** Wohlgemuth, Andrew (10 June 2014). [*Introduction to Proof in Abstract Mathematics*](https://books.google.com/books?id=PEP_AwAAQBAJ&pg=PA237). Courier Corporation. p. 237. [ISBN](/source/ISBN_(identifier)) [978-0-486-14168-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-14168-8).

1. **[^](#cite_ref-8)** Prep, Kaplan Test (4 June 2019). [*GMAT Complete 2020: The Ultimate in Comprehensive Self-Study for GMAT*](https://books.google.com/books?id=6l_sDwAAQBAJ&pg=PA708). Simon and Schuster. [ISBN](/source/ISBN_(identifier)) [978-1-5062-4844-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-5062-4844-8).

1. **[^](#cite_ref-9)** Evans, Nick (1995). "A-Quantifiers and Scope". In Bach, Emmon W. (ed.). [*Quantification in Natural Languages*](https://books.google.com/books?id=NlQL97qBSZkC). Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. p. 262. [ISBN](/source/ISBN_(identifier)) [978-0-7923-3352-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7923-3352-4).

1. **[^](#cite_ref-10)** Smedley, Edward; Rose, Hugh James; Rose, Henry John (1845). [*Encyclopædia Metropolitana*](https://books.google.com/books?id=ZVI_AQAAMAAJ&pg=PA537). B. Fellowes. p. 537. An integer is a multiple of unity

1. **[^](#cite_ref-11)** [Encyclopaedia Britannica 1771](#CITEREFEncyclopaedia_Britannica1771), p. [367](https://books.google.com/books?id=d50qAQAAMAAJ&pg=PA367)

1. **[^](#cite_ref-12)** [Pisano, Leonardo](/source/Fibonacci); Boncompagni, Baldassarre (transliteration) (1202). [*Incipit liber Abbaci compositus to Lionardo filio Bonaccii Pisano in year Mccij*](https://bibdig.museogalileo.it/tecanew/opera?bid=1072400&seq=30) [*The Book of Calculation*] (Manuscript) (in Latin). Translated by Sigler, Laurence E. Museo Galileo. p. 30. Nam rupti uel fracti semper ponendi sunt post integra, quamuis prius integra quam rupti pronuntiari debeant. [And the fractions are always put after the whole, thus first the integer is written, and then the fraction]

1. **[^](#cite_ref-13)** [Encyclopaedia Britannica 1771](#CITEREFEncyclopaedia_Britannica1771), p. [83](https://books.google.com/books?id=d50qAQAAMAAJ&pg=PA83)

1. **[^](#cite_ref-negmath_14-0)** Martinez, Alberto (2014). *Negative Math*. Princeton University Press. pp. 80–109.

1. **[^](#cite_ref-15)** Euler, Leonhard (1771). [*Vollstandige Anleitung Zur Algebra*](https://archive.org/details/1770LEULERVollstandigeAnleitungZurAlgebraVol1/page/n31/mode/2up) [*Complete Introduction to Algebra*] (in German). Vol. 1. p. 10. Alle diese Zahlen, so wohl positive als negative, führen den bekannten Nahmen der gantzen Zahlen, welche also entweder größer oder kleiner sind als nichts. Man nennt dieselbe gantze Zahlen, um sie von den gebrochenen, und noch vielerley andern Zahlen, wovon unten gehandelt werden wird, zu unterscheiden. [All these numbers, both positive and negative, are called whole numbers, which are either greater or lesser than nothing. We call them whole numbers, to distinguish them from fractions, and from several other kinds of numbers of which we shall hereafter speak.]

1. **[^](#cite_ref-16)** [*The University of Leeds Review*](https://books.google.com/books?id=Z-7kAAAAMAAJ). Vol. 31–32. University of Leeds. 1989. p. 46. Incidentally, Z comes from "Zahl": the notation was created by Hilbert.

1. **[^](#cite_ref-17)** Bourbaki, Nicolas (1951). [*Algèbre, Chapter 1*](https://archive.org/details/algebrebour00bour/page/26/mode/2up) (in French) (2nd ed.). Paris: Hermann. p. 27. Le symétrisé de **N** se note **Z**; ses éléments sont appelés entiers rationnels. [The group of differences of **N** is denoted by **Z**; its elements are called the rational integers.]

1. **[^](#cite_ref-18)** Birkhoff, Garrett (1948). [*Lattice Theory*](https://archive.org/details/in.ernet.dli.2015.166886/page/n63/mode/2up) (Revised ed.). American Mathematical Society. p. 63. the set *J* of all integers

1. **[^](#cite_ref-19)** Society, Canadian Mathematical (1960). [*Canadian Journal of Mathematics*](https://books.google.com/books?id=uMAXOmLTCGsC&dq=integer%20set%20Z&pg=PA374). Canadian Mathematical Society. p. 374. Consider the set *Z* of non-negative integers

1. **[^](#cite_ref-20)** Bezuszka, Stanley (1961). [*Contemporary Progress in Mathematics: Teacher Supplement \[to\] Part 1 and Part 2*](https://books.google.com/books?id=dhJPAQAAMAAJ&q=integer+set+Z). Boston College. p. 69. Modern Algebra texts generally designate the set of integers by the capital letter Z.

1. **[^](#cite_ref-edexcelc1_21-0)** Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008

1. **[^](#cite_ref-22)** LK Turner, FJ BUdden, D Knighton, "Advanced Mathematics", Book 2, Longman 1975.

1. **[^](#cite_ref-23)** Mathews, George Ballard (1892). [*Theory of Numbers*](https://books.google.com/books?id=iQ_vAAAAMAAJ&pg=PA2). Deighton, Bell and Company. p. 2.

1. **[^](#cite_ref-24)** Betz, William (1934). [*Junior Mathematics for Today*](https://books.google.com/books?id=RzNCAAAAIAAJ). Ginn. The whole numbers, or integers, when arranged in their natural order, such as 1, 2, 3, are called consecutive integers.

1. **[^](#cite_ref-25)** Peck, Lyman C. (1950). [*Elements of Algebra*](https://books.google.com/books?id=tclXAAAAYAAJ&q=integers+whole+numbers). McGraw-Hill. p. 3. The numbers which so arise are called positive whole numbers, or positive integers.

1. **[^](#cite_ref-26)** Hayden, Robert (1981). [*A history of the "new math" movement in the United States*](https://web.archive.org/web/20251008043152/https://dr.lib.iastate.edu/handle/20.500.12876/80303) (PhD). Iowa State University. p. 145. [doi](/source/Doi_(identifier)):[10.31274/rtd-180813-5631](https://doi.org/10.31274%2Frtd-180813-5631). Archived from [the original](https://dr.lib.iastate.edu/handle/20.500.12876/80303) on 8 October 2025. A much more influential force in bringing news of the "new math" to high school teachers and administrators was the National Council of Teachers of Mathematics (NCTM).

1. **[^](#cite_ref-27)** [*The Growth of Mathematical Ideas, Grades K-12: 24th Yearbook*](https://books.google.com/books?id=OO9RAQAAIAAJ&pg=PA14). National Council of Teachers of Mathematics. 1959. p. 14. [ISBN](/source/ISBN_(identifier)) [9780608166186](https://en.wikipedia.org/wiki/Special:BookSources/9780608166186). {{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date))

1. **[^](#cite_ref-28)** Deans, Edwina (1963). [*Elementary School Mathematics: New Directions*](https://books.google.com/books?id=bAUJAQAAMAAJ&pg=PA42). U.S. Department of Health, Education, and Welfare, Office of Education. p. 42.

1. **[^](#cite_ref-29)** ["entry: whole number"](https://www.ahdictionary.com/word/search.html?q=whole+number). *The American Heritage Dictionary*. HarperCollins.

1. **[^](#cite_ref-30)** ["Integer | mathematics"](https://www.britannica.com/science/integer). *Encyclopedia Britannica*. Retrieved 11 August 2020.

1. **[^](#cite_ref-31)** [Lang, Serge](/source/Serge_Lang) (1993). *Algebra* (3rd ed.). Addison-Wesley. pp. 86–87. [ISBN](/source/ISBN_(identifier)) [978-0-201-55540-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-55540-0).

1. **[^](#cite_ref-32)** Warner, Seth (2012). [*Modern Algebra*](https://books.google.com/books?id=TqHDAgAAQBAJ&pg=PA185). Dover Books on Mathematics. Courier Corporation. Theorem 20.14, p. 185. [ISBN](/source/ISBN_(identifier)) [978-0-486-13709-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-13709-4). [Archived](https://web.archive.org/web/20150906083836/https://books.google.com/books?id=TqHDAgAAQBAJ&pg=PA185) from the original on 6 September 2015. Retrieved 29 April 2015..

1. **[^](#cite_ref-33)** Mendelson, Elliott (1985). [*Number systems and the foundations of analysis*](https://archive.org/details/numbersystemsfou0000mend/page/152/mode/2up). Malabar, Fla. : R.E. Krieger Pub. Co. p. 153. [ISBN](/source/ISBN_(identifier)) [978-0-89874-818-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-89874-818-5).

1. **[^](#cite_ref-34)** Mendelson, Elliott (2008). [*Number Systems and the Foundations of Analysis*](https://books.google.com/books?id=3domViIV7HMC&pg=PA86). Dover Books on Mathematics. Courier Dover Publications. p. 86. [ISBN](/source/ISBN_(identifier)) [978-0-486-45792-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-45792-5). [Archived](https://web.archive.org/web/20161208233040/https://books.google.com/books?id=3domViIV7HMC&pg=PA86) from the original on 8 December 2016. Retrieved 15 February 2016..

1. **[^](#cite_ref-35)** Ivorra Castillo: *Álgebra*

1. **[^](#cite_ref-36)** Kramer, Jürg; von Pippich, Anna-Maria (2017). *From Natural Numbers to Quaternions* (1st ed.). Switzerland: Springer Cham. pp. 78–81. [doi](/source/Doi_(identifier)):[10.1007/978-3-319-69429-0](https://doi.org/10.1007%2F978-3-319-69429-0). [ISBN](/source/ISBN_(identifier)) [978-3-319-69427-6](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-69427-6).

1. **[^](#cite_ref-37)** Frobisher, Len (1999). [*Learning to Teach Number: A Handbook for Students and Teachers in the Primary School*](https://books.google.com/books?id=KwJQIt4jQHUC&pg=PA126). The Stanley Thornes Teaching Primary Maths Series. Nelson Thornes. p. 126. [ISBN](/source/ISBN_(identifier)) [978-0-7487-3515-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7487-3515-0). [Archived](https://web.archive.org/web/20161208121843/https://books.google.com/books?id=KwJQIt4jQHUC&pg=PA126) from the original on 8 December 2016. Retrieved 15 February 2016..

1. ^ [***a***](#cite_ref-Campbell-1970-p83_38-0) [***b***](#cite_ref-Campbell-1970-p83_38-1) [***c***](#cite_ref-Campbell-1970-p83_38-2) Campbell, Howard E. (1970). [*The structure of arithmetic*](https://archive.org/details/structureofarith00camp/page/83). Appleton-Century-Crofts. p. [83](https://archive.org/details/structureofarith00camp/page/83). [ISBN](/source/ISBN_(identifier)) [978-0-390-16895-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-390-16895-5).

1. **[^](#cite_ref-39)** Garavel, Hubert (2017). [*On the Most Suitable Axiomatization of Signed Integers*](https://hal.inria.fr/hal-01667321). Post-proceedings of the 23rd International Workshop on Algebraic Development Techniques (WADT'2016). Lecture Notes in Computer Science. Vol. 10644. Springer. pp. 120–134. [doi](/source/Doi_(identifier)):[10.1007/978-3-319-72044-9_9](https://doi.org/10.1007%2F978-3-319-72044-9_9). [ISBN](/source/ISBN_(identifier)) [978-3-319-72043-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-72043-2). [Archived](https://web.archive.org/web/20180126125528/https://hal.inria.fr/hal-01667321) from the original on 26 January 2018. Retrieved 25 January 2018.

## Sources

- [Bell, E.T.](/source/Eric_Temple_Bell) (1986). *[Men of Mathematics](/source/Men_of_Mathematics)*. New York: Simon & Schuster. [ISBN](/source/ISBN_(identifier)) [0-671-46400-0](https://en.wikipedia.org/wiki/Special:BookSources/0-671-46400-0).)

- Herstein, I.N. (1975). *Topics in Algebra* (2nd ed.). Wiley. [ISBN](/source/ISBN_(identifier)) [0-471-01090-1](https://en.wikipedia.org/wiki/Special:BookSources/0-471-01090-1).

- [Mac Lane, Saunders](/source/Saunders_Mac_Lane); [Birkhoff, Garrett](/source/Garrett_Birkhoff) (1999). *Algebra* (3rd ed.). American Mathematical Society. [ISBN](/source/ISBN_(identifier)) [0-8218-1646-2](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-1646-2).

- A Society of Gentlemen in Scotland (1771). [*Encyclopaedia Britannica*](https://books.google.com/books?id=d50qAQAAMAAJ). Edinburgh.

## External links

Look up ***[integer](https://en.wiktionary.org/wiki/Special:Search/integer)*** in Wiktionary, the free dictionary.

- ["Integer"](https://www.encyclopediaofmath.org/index.php?title=Integer), *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*, [EMS Press](/source/European_Mathematical_Society), 2001 [1994]

- [The Positive Integers – divisor tables and numeral representation tools](https://www.positiveintegers.org)

- [On-Line Encyclopedia of Integer Sequences](https://oeis.org/) cf [OEIS](/source/OEIS)

- [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Integer"](https://mathworld.wolfram.com/Integer.html). *[MathWorld](/source/MathWorld)*.

*This article incorporates material from Integer on [PlanetMath](/source/PlanetMath), which is licensed under the [Creative Commons Attribution/Share-Alike License](https://en.wikipedia.org/wiki/Wikipedia:CC-BY-SA).*

v t e Z {\displaystyle \mathbb {Z} } Integers −2, −1 0 to 199 0 to 99 100 to 199 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 to 399 200 to 299 300 to 399 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 318 323 325 341 353 359 360 363 365 369 377 384 400 to 999 400s, 500s, and 600s 700s, 800s, and 900s 400 420 440 495 496 500 501 511 512 555 600 610 613 616 666 693 700 720 743 744 777 786 800 801 836 840 880 881 888 900 911 971 987 999 1000s and 10,000s 1000s 1000 1001 1023 1024 1089 1093 1105 1234 1289 1458 1510 1728 1729 1980 1987 2000 2520 3000 3511 4000 5000 5040 6000 6174 7000 7744 7825 8000 8128 8192 9000 9855 9999 10,000s 10,000 16,807 20,000 30,000 40,000 50,000 60,000 64,079 65,535 65,536 65,537 70,000 80,000 90,000 100,000s to 10,000,000,000,000s 100,000 142,857 144,000 1,000,000 10,000,000 43,112,609 100,000,000 1,000,000,000 2,147,483,647 4,294,967,295 10,000,000,000 100,000,000,000 1,000,000,000,000 10,000,000,000,000 Large numbers

v t e Number systems Sets of definable numbers Natural numbers ( N {\displaystyle \mathbb {N} } ) Integers ( Z {\displaystyle \mathbb {Z} } ) Rational numbers ( Q {\displaystyle \mathbb {Q} } ) Constructible numbers Algebraic numbers ( A {\displaystyle \mathbb {A} } ) Closed-form numbers Periods ( P {\displaystyle {\mathcal {P}}} ) Computable numbers Arithmetical numbers Set-theoretically definable numbers Gaussian integers Gaussian rationals Eisenstein integers Composition algebras Division algebras: Real numbers ( R {\displaystyle \mathbb {R} } ) Complex numbers ( C {\displaystyle \mathbb {C} } ) Quaternions ( H {\displaystyle \mathbb {H} } ) Octonions ( O {\displaystyle \mathbb {O} } ) Split types Over R {\displaystyle \mathbb {R} } : Split-complex numbers Split-quaternions Split-octonions Over C {\displaystyle \mathbb {C} } : Bicomplex numbers Biquaternions Bioctonions Other hypercomplex Dual numbers Dual quaternions Dual-complex numbers Hyperbolic quaternions Sedenions ( S {\displaystyle \mathbb {S} } ) Trigintaduonions ( T {\displaystyle \mathbb {T} } ) Split-biquaternions Multicomplex numbers Geometric algebra/Clifford algebra Algebra of physical space Spacetime algebra Plane-based geometric algebra Infinities and infinitesimals Cardinal numbers Extended natural numbers Extended real numbers Projective Extended complex numbers Hyperreal numbers Levi-Civita field Ordinal numbers Supernatural numbers Surreal numbers Superreal numbers Other types Irrational numbers Fuzzy numbers Transcendental numbers p-adic numbers (p-adic solenoids) Profinite integers Normal numbers Classification List

v t e Rational numbers Integer Dedekind cut Dyadic rational Half-integer Superparticular ratio

Authority control databases International GND National Japan Czech Republic Other Yale LUX

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Adapted from the Wikipedia article [Integer](https://en.wikipedia.org/wiki/Integer) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Integer?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.