TheInfoList

An integer (from the
Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

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
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() 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 ...

$\left(\mathbb\right)$ 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"). $\mathbb$ 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 $\mathbb$, 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 $\mathbb$. Like the natural numbers, $\mathbb$ 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 $\mathbb$ can be annotated to denote various sets, with varying usage amongst different authors: $\mathbb^+$,$\mathbb_+$ or $\mathbb^$ for the positive integers, $\mathbb^$ or $\mathbb^$ for non-negative integers, and $\mathbb^$ for non-zero integers. Some authors use $\mathbb^$ for non-zero integers, while others use it for non-negative integers, or for . Additionally, $\mathbb_$ 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 2008

# 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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

, $\mathbb$ 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 ...
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, ), $\mathbb$, 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 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 ...
, 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 $\mathbb$. $\mathbb$ 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 $\mathbb$, under addition, is an
abelian group 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 ...
. 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, $\mathbb$ 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 $\mathbb$. The first four properties listed above for multiplication say that $\mathbb$ 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 $\mathbb$ under multiplication is not a group. All the rules from the above property table (except for the last), when taken together, say that $\mathbb$ together with addition and multiplication is a
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
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 $\mathbb$
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 $\mathbb$ 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 $\mathbb$ is not closed under division, means that $\mathbb$ 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, a subring of ''R'' is a subset of a ring (mathematics), 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 ' ...
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 $\mathbb$ as its
subring In mathematics, a subring of ''R'' is a subset of a ring (mathematics), 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 ' ...
. Although ordinary division is not defined on $\mathbb$, 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 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 gene ...

s works by a sequence of Euclidean divisions. The above says that $\mathbb$ 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 $\mathbb$ 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), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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

$\mathbb$ is a
totally ordered set 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 ...
without upper or lower bound. The ordering of $\mathbb$ 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 $\mathbb$ 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 ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 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). ...
es of
ordered pair 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 ...

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 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). ...
on these pairs with the following rule: :$\left(a,b\right) \sim \left(c,d\right)$ 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: : : 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: :
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 :$\begin a - b, & \mbox a \ge b \\ -\left(b - a\right), & \mbox a < b. \end$ 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: : 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
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$\left(x,y\right)$ 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.

# Computer science

An integer is often a primitive data type in computer languages. However, integer data types can only represent 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 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 $\mathbb$ to $\mathbb= \.$ Such a function may be defined as :$f\left(x\right) = \begin -2x, & \mbox x \leq 0\\ 2x-1, & \mbox x > 0, \end$ with graph of a function, graph (set of the pairs $\left(x, f\left(x\right)\right)$ is :. Its inverse function is defined by :$\beging\left(2x\right) = -x\\g\left(2x-1\right)=x, \end$ with graph :.

* 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 integer

# 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). .