An integer (from the

natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s, which are assumed to be already constructed (using, say, the Peano axioms, Peano approach).
There exist at least ten such constructions of signed integers. These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2) and the types of arguments accepted by these operations; the presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms.
The technique for the construction of integers presented above in this 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 $x-y$). This operation is not free since the integer 0 can be written pair(0,0), or pair(1,1), or pair(2,2), etc. This technique of construction is used by the proof assistant Isabelle (proof assistant), Isabelle; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers.

The Positive Integers – divisor tables and numeral representation tools

On-Line Encyclopedia of Integer Sequences

cf OEIS * {{Authority control Elementary mathematics Abelian group theory Ring theory Integers, Elementary number theory Algebraic number theory

Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

meaning "whole") is colloquially defined as a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not.
The set of integers consists of zero (), the positive natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s (, , , ...), also called ''whole numbers'' or ''counting numbers'', and their additive inverse
In mathematics, the additive inverse of a is the number that, when to , yields . This number is also known as the opposite (number), sign change, and negation. For a , it reverses its : the additive inverse (opposite number) of a is negative, ...

s (the negative integers, i.e., −1
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, −2, −3, ...). The set of integers is often denoted by the boldface
emphasis using the technique of changing fonts
In typography, emphasis is the strengthening of words in a text with a font in a different style from the rest of the text, to highlight them. It is the equivalent of Stress (linguistics)#Prosodic ...

() or blackboard bold
Image:Blackboard bold.svg, 250px, An example of blackboard bold letters
Blackboard bold is a typeface style that is often used for certain symbols in mathematics, mathematical texts, in which certain lines of the symbol (usually vertical or near-v ...

$(\backslash mathbb)$ letter "Z"—standing originally for the German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of Germany, see also German nationality law
* German language
The German la ...

word ''Zahlen
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

'' ("numbers").
$\backslash mathbb$ is a subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the set of all rational
Rationality is the quality or state of being rational – that is, being based on or agreeable to reason
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογι ...

numbers $\backslash mathbb$, which in turn is a subset of the real
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

numbers $\backslash mathbb$. Like the natural numbers, $\backslash mathbb$ is countably infinite
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
The integers form the smallest group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

and the smallest ring containing the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s. In algebraic number theory
Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory.
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, ...

, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integer
In algebraic number theory
Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory.
Algebraic number theory is a branch of number theory that uses the techniques of abstrac ...

s. In fact, (rational) integers are algebraic integers that are also rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s.
Symbol

The symbol $\backslash mathbb$ can be annotated to denote various sets, with varying usage amongst different authors: $\backslash mathbb^+$,$\backslash mathbb\_+$ or $\backslash mathbb^$ for the positive integers, $\backslash mathbb^$ or $\backslash mathbb^$ for non-negative integers, and $\backslash mathbb^$ for non-zero integers. Some authors use $\backslash mathbb^$ for non-zero integers, while others use it for non-negative integers, or for . Additionally, $\backslash mathbb\_$ is used to denote either the set of integers modulo (i.e., the set of congruence classes of integers), or the set of -adic integers.Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008Algebraic properties

Like thenatural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

, $\backslash mathbb$ is closed under the operations
Operation or Operations may refer to:
Science and technology
* Surgical operation
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical or dental specialty that ...

of addition and multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, ), $\backslash mathbb$, unlike the natural numbers, is also closed under subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

.
The integers form a unital ring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...

from the integers into this ring. This universal property
In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

, namely to be an initial object
In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...

in the category of rings
In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categor ...

, characterizes the ring $\backslash mathbb$.
$\backslash mathbb$ is not closed under division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting o ...

, 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
Exponentiation is a mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

, the integers are not (since the result can be a fraction when the exponent is negative).
The following table lists some of the basic properties of addition and multiplication for any integers , and :
The first five properties listed above for addition say that $\backslash mathbb$, under addition, is an abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. It is also a cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

, since every non-zero integer can be written as a finite sum or . In fact, $\backslash mathbb$ under addition is the ''only'' infinite cyclic group—in the sense that any infinite cyclic group is isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to $\backslash mathbb$.
The first four properties listed above for multiplication say that $\backslash mathbb$ under multiplication is a commutative monoid
In abstract algebra, a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...

. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that $\backslash mathbb$ under multiplication is not a group.
All the rules from the above property table (except for the last), when taken together, say that $\backslash mathbb$ together with addition and multiplication is a commutative ring
In ring theory
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...

with unity
Unity may refer to:
Buildings
* Unity Building
The Unity Building, in Oregon, Illinois, is a historic building in that city's Oregon Commercial Historic District. As part of the district the Oregon Unity Building has been listed on the National R ...

. It is the prototype of all objects of such algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. Only those equalities of expressions
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphor#Common types, Metaphorical expression, a parti ...

are true in $\backslash mathbb$ for all
In mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...

values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

in certain rings.
The lack of zero divisor
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s in the integers (last property in the table) means that the commutative ring $\backslash mathbb$ is an integral domain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

.
The lack of multiplicative inverses, which is equivalent to the fact that $\backslash mathbb$ is not closed under division, means that $\backslash mathbb$ is ''not'' a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

. The smallest field containing the integers as a subring
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is the field of rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s. The process of constructing the rationals from the integers can be mimicked to form the field of fractions
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...

of any integral domain. And back, starting from an algebraic number field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

(an extension of rational numbers), its ring of integersIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

can be extracted, which includes $\backslash mathbb$ as its subring
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
Although ordinary division is not defined on $\backslash mathbb$, the division "with remainder" is defined on them. It is called Euclidean division
In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...

, and possesses the following important property: given two integers and with , there exist unique integers and such that and , where denotes the absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of . The integer is called the ''quotient'' and is called the ''remainder
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

'' of the division of by . The Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...

for computing greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

s works by a sequence of Euclidean divisions.
The above says that $\backslash mathbb$ is a Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a #Definition, Euclidean function which allows a suitable generalization of the Euclidean division of ...

. This implies that $\backslash mathbb$ is a principal ideal domain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

, and any positive integer can be written as the products of primes
A prime number (or a prime) is a natural number
File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...)
In mathematics, the natural numbers are those numbers used for counting (as in "the ...

in an essentially uniqueIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

way. This is the fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...

.
Order-theoretic properties

$\backslash mathbb$ is atotally ordered set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

without upper or lower bound. The ordering of $\backslash mathbb$ is given by:
An integer is ''positive'' if it is greater than zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

, 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:
# if and , then
# if and , then .
Thus it follows that $\backslash mathbb$ together with the above ordering is an ordered ring 350px, The real numbers are an ordered ring which is also an ordered field. The integers">ordered_field.html" ;"title="real numbers are an ordered ring which is also an ordered field">real numbers are an ordered ring which is also an ordered field. ...

.
The integers are the only nontrivial totally ordered
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

whose positive elements are well-ordered
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. This is equivalent to the statement that any NoetherianIn mathematics, the adjective
In linguistics, an adjective (list of glossing abbreviations, abbreviated ) is a word that grammatical modifier, modifies a noun or noun phrase or describes its referent. Its Semantics, semantic role is to change inf ...

valuation ringIn abstract algebra, a valuation ring is an integral domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometr ...

is either a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

—or a discrete valuation ringIn abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra ...

.
Construction

In elementary school teaching, integers are often intuitively defined as the (positive) natural numbers,zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

, and the negations of the natural numbers. However, this 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. Therefore, in modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead. The integers can thus be formally constructed as the equivalence class
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

es of ordered pair
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s of natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s .
The intuition is that stands for the result of subtracting from . To confirm our expectation that and denote the same number, we define an equivalence relation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

on these pairs with the following rule:
:$(a,b)\; \backslash sim\; (c,d)$
precisely when
:$a\; +\; d\; =\; b\; +\; c.$
Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using to denote the equivalence class having as a member, one has:
:$;\; href="/html/ALL/s/a,b).html"\; ;"title="a,b)">a,b)$
:$;\; href="/html/ALL/s/a,b).html"\; ;"title="a,b)">a,b)$
The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:
:$-;\; href="/html/ALL/s/a,b).html"\; ;"title="a,b)">a,b)$
Hence subtraction can be defined as the addition of the additive inverse:
:$;\; href="/html/ALL/s/a,b).html"\; ;"title="a,b)">a,b)$
The standard ordering on the integers is given by:
:$;\; href="/html/ALL/s/a,b).html"\; ;"title="a,b)">a,b)$if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

$a+d\; <\; b+c.$
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 or (or both at once). The natural number is identified with the class (i.e., the natural numbers are embedded into the integers by map sending to ), and the class is denoted (this covers all remaining classes, and gives the class a second time since
Thus, is denoted by
:$\backslash begin\; a\; -\; b,\; \&\; \backslash mbox\; a\; \backslash ge\; b\; \backslash \backslash \; -(b\; -\; a),\; \&\; \backslash mbox\; a\; <\; b.\; \backslash end$
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
Representation may refer to:
Law and politics
*Representation (politics)
Political representation is the activity of making citizens "present" in public policy making processes when political actors act in the best interest of citizens. This def ...

of the integers as .
Some examples are:
:$\backslash begin\; 0\; \&=$0,0)
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

&= 1,1)&= \cdots & &= k,k)\\
1 &= 1,0)
Onekama ( ) is a village in Manistee County
Manistee County is a County (United States), county located in the U.S. state of Michigan. As of the 2010 United States Census, 2010 census, the population was 24,733. The county seat is Manistee, Michig ...

&= 2,1)&= \cdots & &= k+1,k)\\
-1 &= 0,1)&= 1,2)
Onekama ( ) is a village in Manistee County
Manistee County is a County (United States), county located in the U.S. state of Michigan. As of the 2010 United States Census, 2010 census, the population was 24,733. The county seat is Manistee, Michig ...

&= \cdots & &= k,k+1)\\
2 &= 2,0)
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

&= 3,1)&= \cdots & &= k+2,k)\\
-2 &= 0,2)&= 1,3)&= \cdots & &= [(k,k+2)].
\end
In theoretical computer science, other approaches for the construction of integers are used by Automated theorem proving, automated theorem provers and Rewriting, term rewrite engines.
Integers are represented as Term algebra, algebraic terms built using a few basic operations (e.g., zero, succ, pred) and, possibly, using Computer science

An integer is often a primitive data type in computer languages. However, integer data types can only represent asubset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of all integers, since practical computers are of finite capacity. Also, in the common two's complement representation, the inherent definition of sign (mathematics), sign distinguishes between "negative" and "non-negative" rather than "negative, positive, and 0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted ''int'' or Integer in several programming languages (such as Algol68, C (computer language), C, Java (programming language), Java, Object Pascal, Delphi, etc.).
Variable-length representations of integers, such as bignums, 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 cardinality of the set of integers is equal to (Aleph number, aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from $\backslash mathbb$ to $\backslash mathbb=\; \backslash .$ Such a function may be defined as :$f(x)\; =\; \backslash begin\; -2x,\; \&\; \backslash mbox\; x\; \backslash leq\; 0\backslash \backslash \; 2x-1,\; \&\; \backslash mbox\; x\; >\; 0,\; \backslash end$ with graph of a function, graph (set of the pairs $(x,\; f(x))$ is :. Its inverse function is defined by :$\backslash beging(2x)\; =\; -x\backslash \backslash g(2x-1)=x,\; \backslash end$ with graph :.See also

* Canonical representation of a positive integer, Canonical factorization of a positive integer * Hyperinteger * Integer complexity * Integer lattice * Integer part * Integer sequence * Integer-valued function * Mathematical symbols * Parity (mathematics) * Profinite integerFootnotes

References

Sources

* Eric Temple Bell, Bell, E.T., ''Men of Mathematics.'' New York: Simon & Schuster, 1986. (Hardcover; )/(Paperback; ) * Herstein, I.N., ''Topics in Algebra'', Wiley; 2 edition (June 20, 1975), . * Saunders Mac Lane, Mac Lane, Saunders, and Garrett Birkhoff; ''Algebra'', American Mathematical Society; 3rd edition (1999). .External links

*The Positive Integers – divisor tables and numeral representation tools

On-Line Encyclopedia of Integer Sequences

cf OEIS * {{Authority control Elementary mathematics Abelian group theory Ring theory Integers, Elementary number theory Algebraic number theory