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, −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 less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
s are the
additive inverses of the corresponding positive numbers. In the
language of mathematics, the
set of integers is often denoted by the
boldface or
blackboard bold .
The set of natural numbers
is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
of
, which in turn is a subset of the set of all
rational numbers
, itself a subset of the
real numbers
. Like the natural numbers,
is
countably infinite. 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 integers. In fact, (rational) integers are algebraic integers that are also
rational numbers.
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 less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
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 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
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally in ...
, 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 this ...
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 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 n ...
,
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 additi ...
, 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 which is the most basic one, in the following sense: for any unital ring, there is a unique
ring homomorphism from the integers into this ring. This
universal property, namely to be an
initial object in the
category of rings, characterizes the ring
.
is not closed under
division, 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, 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. It is also a
cyclic group, 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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to
.
The first four properties listed above for multiplication say that
under multiplication is a
commutative monoid. 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. It is the prototype of all objects of such
algebraic structure. 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 values of variables, which are true in any unital commutative ring. Certain non-zero integers map to
zero in certain rings.
The lack of
zero divisors in the integers (last property in the table) means that the commutative ring
is an
integral domain.
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 wh ...
is the field of
rational numbers. The process of constructing the rationals from the integers can be mimicked to form the
field of fractions of any integral domain. And back, starting from an
algebraic number field (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 deno ...
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 wh ...
.
Although ordinary division is not defined on
, the division "with remainder" is defined on them. It is called
Euclidean division, 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'' 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 divisors 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 principal, ...
, and any positive integer can be written as the products of
primes in an
essentially unique way. This is the
fundamental theorem of arithmetic.
Order-theoretic properties
is a
totally ordered set without
upper or lower bound. The ordering of
is given by:
An integer is ''positive'' if it is greater than
zero, 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.
The integers are the only nontrivial
totally ordered abelian group whose positive elements are
well-ordered. This is equivalent to the statement that any
Noetherian valuation ring is either a
field—or a
discrete valuation ring.
Construction
Traditional development
In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers,
zero, 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, 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 con ...
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 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 false ...
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 a ...
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 con ...
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 relation ...
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:
:
The standard ordering on the integers is given by:
: