
An integer is the number zero (), a positive
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
(, , , etc.) or a negative integer with a minus sign (
−1
In mathematics, −1 (also known as negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less ...
, −2, −3, etc.). The
negative number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is inequality (mathematics), less than 0 (number), zero. Negative numbers are often used to represent the magnitude of a loss ...
s are the
additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
s of the corresponding positive numbers. In the
language of mathematics, the
set of integers is often denoted by the
boldface
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 prosody stress in speech.
Methods and use
The most common methods in W ...
or
blackboard bold
Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of p ...
.
The set of natural numbers is a subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of , which in turn is a subset of the set of all rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s , itself a subset of the real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s . Like the natural numbers, is countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not.
The integers form the smallest group and the smallest ring containing the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integer
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficien ...
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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s.
History
The word integer comes from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...
''integer'' meaning "whole" or (literally) "untouched", from ''in'' ("not") plus ''tangere'' ("to touch"). " Entire" derives from the same origin via the French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
** French language, which originated in France, and its various dialects and accents
** French people, a nation and ethnic group identified with Franc ...
word '' entier'', which means both ''entire'' and ''integer''. Historically the term was used for a number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ...
that was a multiple of 1, or to the whole part of a mixed number. Only positive integers were considered, making the term synonymous with the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s. The definition of integer expanded over time to include negative number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is inequality (mathematics), less than 0 (number), zero. Negative numbers are often used to represent the magnitude of a loss ...
s as their usefulness was recognized.[ For example ]Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in his 1765 '' Elements of Algebra'' defined integers to include both positive and negative numbers. However, European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century.
The use of the letter Z to denote the set of integers comes from the German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
word '' Zahlen'' ("number") and has been attributed to David Hilbert. The earliest known use of the notation in a textbook occurs in Algébre written by the collective Nicolas Bourbaki, dating to 1947.[ The notation was not adopted immediately, for example another textbook used the letter J and a 1960 paper used Z to denote the non-negative integers. But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.
The symbol 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. Some authors use for non-zero integers, while others use it for non-negative integers, or for (the ]group of units
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
vu = uv = 1,
where is the multiplicative identity; the element is unique for th ...
of ). Additionally, 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 2008]
The whole numbers were synonymous with the integers up until the early 1950s. In the late 1950s, as part of the New Math
New Mathematics or New Math was a dramatic but temporary change in the mathematics education, way mathematics was taught in American grade schools, and to a lesser extent in European countries and elsewhere, during the 1950s1970s. Curriculum top ...
movement, American elementary school teachers began teaching that "whole numbers" referred to the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, excluding negative numbers, while "integer" included the negative numbers. "Whole number" remains ambiguous to the present day.
Algebraic properties
Like the natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
, is closed under the operations
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
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 mathematical operations of arithmetic, with the other ones being ad ...
, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, ), , 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, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying prop ...
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 between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition prese ...
from the integers into this ring. This universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...
, namely to be an initial object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element) ...
in the category of rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings ...
, characterizes the ring .
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 ...
, 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 operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
, 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 , under addition, is an abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
. It is also a cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
, since every non-zero integer can be written as a finite sum or . In fact, under addition is the ''only'' infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to .
The first four properties listed above for multiplication say that under multiplication is a commutative monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids a ...
. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that under multiplication is not a group.
All the rules from the above property table (except for the last), when taken together, say that together with addition and multiplication is a commutative ring with unity
Unity may refer to:
Buildings
* Unity Building, Oregon, Illinois, US; a historic building
* Unity Building (Chicago), Illinois, US; a skyscraper
* Unity Buildings, Liverpool, UK; two buildings in England
* Unity Chapel, Wyoming, Wisconsin, US; a ...
. It is the prototype of all objects of such algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
. Only those equalities
In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality b ...
of expressions are true in for all
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...
values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
in certain rings.
The lack of zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s in the integers (last property in the table) means that the commutative ring is an integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
.
The lack of multiplicative inverses, which is equivalent to the fact that is not closed under division, means that is ''not'' a field. The smallest field containing the integers as a subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
is the field of rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s. The process of constructing the rationals from the integers can be mimicked to form the field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of any integral domain. And back, starting from an algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
(an extension of rational numbers), its ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often d ...
can be extracted, which includes as its subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
.
Although ordinary division is not defined on , 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 dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
, and possesses the following important property: given two integers and with , there exist unique integers and such that and , where denotes the absolute value of . The integer is called the ''quotient'' and is called the ''remainder
In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algeb ...
'' 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 ...
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'' i ...
s works by a sequence of Euclidean divisions.
The above says that 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 Euclidean function which allows a suitable generalization of the Euclidean division of integers. ...
. This implies that is a principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princip ...
, and any positive integer can be written as the products of primes
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
in an essentially unique In mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the property are equivalent to each other. The notion of es ...
way. This is the fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the o ...
.
Order-theoretic properties
is a totally ordered set
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexiv ...
without upper or lower bound. The ordering of is given by:
An integer is ''positive'' if it is greater than zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
, 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 together with the above ordering is an ordered ring
In abstract algebra, an ordered ring is a (usually commutative) ring ''R'' with a total order ≤ such that for all ''a'', ''b'', and ''c'' in ''R'':
* if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''.
* if 0 ≤ ''a'' and 0 ≤ ''b'' the ...
.
The integers are the only nontrivial totally ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexiv ...
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
whose positive elements are well-ordered
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-o ...
. This is equivalent to the statement that any Noetherian valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''.
Given a field ''F'', if ''D'' is a subring of ''F'' such ...
is either a field—or a discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R' ...
.
Construction
Traditional development
In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
, and the negations of the natural numbers. This can be formalized as follows. First construct the set of natural numbers according to the Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
, call this . Then construct a set which is disjoint from and in one-to-one correspondence with via a function . For example, take to be the ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...
s with the mapping . Finally let 0 be some object not in or , for example the ordered pair . Then the integers are defined to be the union .
The traditional arithmetic operations can then be defined on the integers in a piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
fashion, for each of positive numbers, negative numbers, and zero. For example negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fals ...
is defined as follows:
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.
Equivalence classes of ordered pairs
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, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...
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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
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, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on these pairs with the following rule:
:
precisely when
:
Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;[ by using to denote the equivalence class having as a member, one has:
:
:
The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:
:
Hence subtraction can be defined as the addition of the additive inverse:
:]a+d,b+c)
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes'' ...
The standard ordering on the integers is given by:
:if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
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
Embedded or embedding (alternatively imbedded or imbedding) may refer to:
Science
* Embedding, in mathematics, one instance of some mathematical object contained within another instance
** Graph embedding
* Embedded generation, a distributed ge ...
into the integers by map sending to ), and the class is denoted (this covers all remaining classes, and gives the class a second time since
Thus, is denoted by
:
If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity.
This notation recovers the familiar representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of the integers as .
Some examples are:
:0,0)
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
&= 1,1)&= \cdots & &= k,k)\\
1 &= 1,0)&= 2,1)&= \cdots & &= k+1,k)\\
-1 &= 0,1)&= 1,2)
Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "On ...
&= \cdots & &= k,k+1)\\
2 &= 2,0)
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
&= 3,1)&= \cdots & &= k+2,k)\\
-2 &= 0,2)
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
&= 1,3)&= \cdots & &= k,k+2)
\end
Other approaches
In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a maj ...
and term rewrite engines.
Integers are represented as algebraic terms built using a few basic operations (e.g., zero, succ, pred) and, possibly, using natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, which are assumed to be already constructed (using, say, the 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 in the previous section corresponds to the particular case where there is a single basic operation pair that takes as arguments two natural numbers and , and returns an integer (equal to ). 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
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor ...
Isabelle
Isabel is a female name of Spanish origin. Isabelle is a name that is similar, but it is of French origin. It originates as the medieval Spanish form of ''Elisabeth'' (ultimately Hebrew '' Elisheva''), Arising in the 12th century, it became popul ...
; 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
An integer is often a primitive data type
In computer science and computer programming, a data type (or simply type) is a set of possible values and a set of allowed operations on it. A data type tells the compiler or interpreter how the programmer intends to use the data. Most progra ...
in computer language
A computer language is a formal language used to communicate with a computer. Types of computer languages include:
* Construction language – all forms of communication by which a human can specify an executable problem solution to a compu ...
s. However, integer data types can only represent a subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of all integers, since practical computers are of finite capacity. Also, in the common two's complement
Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the binary digit with the greatest place value (the leftmost bit in big- endian ...
representation, the inherent definition of sign
A sign is an Physical object, object, quality (philosophy), quality, event, or Non-physical entity, entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to ...
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
ALGOL 68 (short for ''Algorithmic Language 1968'') is an imperative programming language that was conceived as a successor to the ALGOL 60 programming language, designed with the goal of a much wider scope of application and more rigorously de ...
, C, Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mo ...
, Delphi
Delphi (; ), in legend previously called Pytho (Πυθώ), in ancient times was a sacred precinct that served as the seat of Pythia, the major oracle who was consulted about important decisions throughout the ancient classical world. The oracl ...
, etc.).
Variable-length representations of integers, such as bignum
In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are l ...
s, 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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of the set of integers is equal to (aleph-null
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named ...
). This is readily demonstrated by the construction of a bijection, that is, a function that is injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
and surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
from to Such a function may be defined as
:
with graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
(set of the pairs is
:.
Its inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon ...
is defined by
:
with graph
:.
See also
* Canonical factorization of a positive integer
* Hyperinteger
* Integer complexity
* Integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or gri ...
* Integer part
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least in ...
* Integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.
An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. Fo ...
* Integer-valued function
* Mathematical symbols
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula ...
* Parity (mathematics)
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because
\begin
-2 \cdot 2 &= -4 \\
0 \cdot 2 &= 0 \\
41 ...
* Profinite integer In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)
:\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p
where
:\varprojlim \mathbb/n\mathbb
indicates the profinite completion of \mat ...
Footnotes
References
Sources
* )
*
*
*
External links
*
The Positive Integers – divisor tables and numeral representation tools
On-Line Encyclopedia of Integer Sequences
cf OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...
*
{{Authority control
Elementary mathematics
Abelian group theory
Ring theory
Elementary number theory
Algebraic number theory
Sets of real numbers