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# Parity of zero

Zero is an even number. [1]

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

## Abstraction (mathematics)

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

Addition (often signified by the plus symbol "+") is one of the four elementary, mathematical operations of arithmetic, with the others being subtraction, multiplication and division.

In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

In mathematics, the additive inverse of a number is the number that, when added to, yields zero.

Adrien-Marie Legendre (18 September 1752 – 10 January 1833) was a French mathematician.

## Algebraic structure

In mathematics, and more specifically in abstract algebra, the term algebraic structure generally refers to a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a some list of axioms.

## Algorithm

In mathematics and computer science, an algorithm is a self-contained step-by-step set of operations to be performed.

## All Things Considered

All Things Considered (ATC) is the flagship news program on the American network National Public Radio (NPR).

## Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set.

## Arithmetic

Arithmetic or arithmetics (from the Greek ἀριθμός arithmos, "number") is the oldest and most elementary branch of mathematics.

## Arthur Cayley

Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician.

## Associative property

In mathematics, the associative property is a property of some binary operations.

## École Normale Supérieure

The École normale supérieure (also known as Normale sup’, ENS Ulm, ENS Paris and most often just as ENS) is a French grande école (higher education establishment outside the framework of the public university system).

## Binary number

In mathematics and digital electronics, a binary number is a number expressed in the binary numeral system, or base-2 numeral system, which represents numeric values using two different symbols: typically 0 (zero) and 1 (one).

## Binary relation

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.

## Bipartite graph

In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets U and V (that is, U and V are each independent sets) such that every edge connects a vertex in U to one in V. Vertex set U and V are often denoted as partite sets.

## Bit-reversal permutation

In applied mathematics, a bit-reversal permutation is a permutation of a sequence of n items, where n.

## Calendar date

A calendar date is a reference to a particular day represented within a calendar system.

## Casino

In modern English, a casino is a facility which houses and accommodates certain types of gambling activities.

## Charleston Gazette-Mail

The Charleston Gazette-Mail is the only daily morning newspaper in Charleston, West Virginia.

## Christian Goldbach

Christian Goldbach (March 18, 1690 &ndash; November 20, 1764) was a German mathematician who also studied law.

## Closure (mathematics)

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.

## Complex question

A complex question, trick question, multiple question or plurium interrogationum (Latin, "of many questions") is a question that has a presupposition that is complex.

## Computational geometry

Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry.

## Computer

A computer is a general-purpose device that can be programmed to carry out a set of arithmetic or logical operations automatically.

## Computer monitor

A monitor or a display is an electronic visual display for computers.

## Concept image and concept definition

In mathematics education, concept image and concept definition are two ways of understanding a mathematical concept.

## Connectivity (graph theory)

In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to disconnect the remaining nodes from each other.

## Convention (norm)

A convention is a set of agreed, stipulated, or generally accepted standards, norms, social norms, or criteria, often taking the form of a custom.

## Cooley–Tukey FFT algorithm

The Cooley–Tukey algorithm, named after J.W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm.

## Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.

## Counterexample

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule or law.

## Counting

Counting is the action of finding the number of elements of a finite set of objects.

## Cycle (graph theory)

In graph theory, there are several different types of object called cycles, principally a closed walk and a simple cycle; also, e.g., an element of the cycle space of the graph.

## Cyclic permutation

In mathematics, and in particular in group theory, a cyclic permutation is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion, while fixing (i.e., mapping to themselves) all other elements of X. For example, the permutation of that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a cycle, while the permutation that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not (it separately permutes the pairs and). A cycle in a permutation is a subset of the elements that are permuted in this way.

## Deborah Loewenberg Ball

Deborah Loewenberg Ball is an educational researcher noted for her work in mathematics instruction and the mathematical preparation of teachers.

## Definition

A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols).

## Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class.

## Degree (graph theory)

In graph theory, the degree (or valency) of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice.

## Deseret News

The Deseret News is a newspaper published in Salt Lake City, Utah, United States.

## Distance (graph theory)

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them.

## Divisor

In mathematics a divisor of an integer n, also called a factor of n, is an integer that can be multiplied by some other integer to produce n.

## Education in England

Education in England is overseen by the United Kingdom's Department for Education and Department for Business, Innovation and Skills.

## Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors.

## Empty set

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Equivalence class

In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of elements that are related to one another, forming what are called equivalence classes.

## Equivalence relation

In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of the set is a member of one and only one cell of the partition.

## Even and odd ordinals

In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers.

## Existential quantification

In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some".

## Fast Fourier transform

A fast Fourier transform (FFT) algorithm computes the discrete Fourier transform (DFT) of a sequence, or its inverse.

Fourth grade, also called Grade 4, is a term used to refer to a year of elementary education in some countries.

## 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 greater than 1Using the empty product rule one need not exclude the number 1, and the theorem can be stated as: every positive integer has unique prime factorization.

## Gasoline

Gasoline, also known as petrol outside of North America, is a transparent, petroleum-derived liquid that is used primarily as a fuel in internal combustion engines.

The Graduate Management Admission Test (GMAT ()) is a computer adaptive test (CAT) intended to assess certain analytical, writing, quantitative, verbal, and reading skills in written English for use in admission to a graduate management program, such as an MBA.

## Graduate Record Examinations

The Graduate Record Examination (GRE) is a standardized test that is an admissions requirement for most graduate schools in the United States.

## Graph (mathematics)

In mathematics, and more specifically in graph theory, a graph is a representation of a set of objects where some pairs of objects are connected by links.

## Graph coloring

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints.

## Graph theory

In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element.

## Handshaking lemma

In graph theory, a branch of mathematics, the handshaking lemma is the statement that every finite undirected graph has an even number of vertices with odd degree (the number of edges touching the vertex).

## Heute

heute (German for 'today') is a television news program on the German channel ZDF.

## Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

## Identity element

In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set.

## Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.

## Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).

## Infinity

Infinity (symbol) is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first, literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Integer (computer science)

In computer science, an integer is a datum of integral data type, a data type which represents some finite subset of the mathematical integers.

## Integer factorization

In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers.

## Internet

The Internet is the global system of interconnected computer networks that use the Internet protocol suite (TCP/IP) to link billions of devices worldwide.

## Isabelle (proof assistant)

The Isabelle theorem prover is an interactive theorem prover, a Higher Order Logic (HOL) theorem prover.

## Jewish World Review

Jewish World Review is a free, online magazine updated Monday through Friday (except for legal holidays and holy days), which seeks to appeal to "people of faith and those interested in learning more about contemporary Judaism from Jews who take their religion seriously." It carries informational articles related to Judaism, dozens of syndicated columns written mostly by politically conservative writers, both Jewish and Gentile, advice columns on a number of issues, and cartoons.

## Johann Heinrich Lambert

Johann Heinrich Lambert (Jean-Henri Lambert in French; 26 August 1728 – 25 September 1777) was a Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections.

## Lemma (mathematics)

In mathematics, a "helping theorem" or lemma (plural lemmata or lemmas) from the Ancient Greek λῆμμα (lemma, "anything which is received, such as a gift, profit, or a bribe”) is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself.

## Leopold Kronecker

Leopold Kronecker (7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory and algebra.

## Limit of a sequence

As the positive integer n becomes larger and larger, the value n sin(1/n) becomes arbitrarily close to 1.

## Limit ordinal

In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal.

## Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

## Logical framework

In logic, a logical framework provides a means to define (or present) a logic as a signature in a higher-order type theory in such a way that provability of a formula in the original logic reduces to a type inhabitation problem in the framework type theory.

In the United States and Canada, an academic major (informally major) is the academic discipline to which an undergraduate student formally commits.

## Maryland

Maryland is a state located in the Mid-Atlantic region of the United States, bordering Virginia, West Virginia, and Washington, D.C. to its south and west; Pennsylvania to its north; and Delaware to its east.

## Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers.

## Mathematical proof

In mathematics, a proof is a deductive argument for a mathematical statement.

## Mathematics education

In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research.

## Möbius function

The classical Möbius function μ(n) is an important multiplicative function in number theory and combinatorics.

## Möbius inversion formula

In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius.

## Mental chronometry

Mental chronometry is the use of response time in perceptual-motor tasks to infer the content, duration, and temporal sequencing of cognitive operations.

## Millisecond

A millisecond (from milli- and second; symbol: ms) is a thousandth (0.001 or 10−3 or 1/1,000) of a second.

## Milwaukee Journal Sentinel

The Milwaukee Journal Sentinel is a daily morning broadsheet printed in Milwaukee, Wisconsin.

## Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus.

## Multiple (mathematics)

In mathematics, a multiple is the product of any quantity and an integer.

## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "·" or by the absence of symbol) is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division.

## Multiplication table

In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.

## Multiplicative function

In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1).

## Natural number

In mathematics, the natural numbers (sometimes called the whole numbers): "whole number An integer, though sometimes it is taken to mean only non-negative integers, or just the positive integers." give definitions of "whole number" under several headwords: INTEGER … Syn. whole number.

## New South Wales

New South Wales (abbreviated as NSW) is a state on the east coast of:Australia.

## NPR

National Public Radio (NPR) is a privately and publicly funded non-profit membership media organization that serves as a national syndicator to a network of 900 public radio stations in the United States.

## Null graph

In the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph").

## Number

A number is a mathematical object used to count, measure and label.

## Number line

In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point.

## Number theory

Number theory (or arithmeticEspecially in older sources; see two following notes.) is a branch of pure mathematics devoted primarily to the study of the integers.

## Numeral (linguistics)

In linguistics, a numeral is a member of a word class (or sometimes even a part of speech) designating numbers, such as the English word 'two' and the compound 'seventy-seven'.

## Numeral system

A numeral system (or system of numeration) is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

## Numerical cognition

Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics.

## Odd–even rationing

Odd–even rationing is a method of rationing in which access to some resource is restricted to half the population on any given day.

## Odds and evens

Odds and evens, also known as swords, choosies, pick, odds-on poke, or bucking up, is a hand game played between two people, used to decide an issue.

## Ordinal number

In set theory, an ordinal number, or ordinal, is the order type of a well-ordered set.

## Orientation (vector space)

In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed.

## Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second-oldest, after Cambridge University Press.

In mathematics the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems.

## Parity (mathematics)

Parity is a mathematical term that describes the property of an integer's inclusion in one of two categories: even or odd.

## Parity of a permutation

In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations.

## Partition of a set

In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.

## Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.

## Point in polygon

In computational geometry, the point-in-polygon (PIP) problem asks whether a given point in the plane lies inside, outside, or on the boundary of a polygon.

## Polygon

In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

## Port and starboard

Port and starboard are nautical terms for left and right, respectively.

## Power of two

In mathematics, a power of two means a number of the form where is an integer, i.e. the result of exponentiation with number two as the base and integer as the exponent.

## Primary education

Primary education or elementary education often in primary school or elementary school is typically the first stage of compulsory education, coming between early childhood education and secondary education.

## Prime number

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

## Proof by contradiction

In logic, proof by contradiction is a form of proof, and more specifically a form of indirect proof, that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction.

## Proposition bet

In gambling, the term "proposition bet" (prop bet, prop, exotic, novelty, or a side bet in the first context below) has two definitions.

## Recursive definition

A recursive definition (or inductive definition) in mathematical logic and computer science is used to define the elements in a set in terms of other elements in the set (Aczel 1978:740ff).

## Reflexive relation

In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself.

## Reform mathematics

Reform mathematics is an approach to mathematics education, particularly in North America.

## Ring (mathematics)

In mathematics, and more specifically in algebra, a ring is an algebraic structure with operations that generalize the arithmetic operations of addition and multiplication.

## Roulette

Roulette is a casino game named after the French word meaning little wheel.

Second grade (called Grade 2 in some regions) is a year of primary education in many nations.

## Sequence

In mathematics, a sequence is an ordered collection of objects in which repetitions are allowed.

## Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

## Singly and doubly even

In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not.

## Sperner's lemma

In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which is equivalent to it.

## Standardized test

A standardized test is a test that is administered and scored in a consistent, or "standard", manner.

## Stanislas Dehaene

Stanislas Dehaene (born 12 May 1965) is a professor at the Collège de France, author, and (since 1989) director of INSERM Unit 562, "Cognitive Neuroimaging.

## Subgroup

In mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

## Subtraction

Subtraction is a mathematical operation that represents the operation of removing objects from a collection.

## Successor function

In mathematics, the successor function or successor operation is a primitive recursive function S such that S(n).

## Successor ordinal

In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α.

## Symmetric group

In abstract algebra, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutation operations that can be performed on n distinct symbols, and whose group operation is the composition of such permutation operations, which are defined as bijective functions from the set of symbols to itself.

## The Guardian

The Guardian is a British national daily newspaper.

Third grade is the term used for the third year of primary education in the US.

Traditional mathematics (sometimes classical math education) was the predominant method of mathematics education in the United States in the early-to-mid 20th century.

## Triangulation (geometry)

In geometry, a triangulation of a planar object is a subdivision into triangles, and by extension the subdivision of higher-dimension geometric objects into simplices.

## Triviality (mathematics)

In mathematics, the adjective trivial is frequently used for objects (for example, groups or topological spaces) that have a very simple structure.

Undergraduate education is the post-secondary education previous to the postgraduate education.

## United States

The United States of America (USA), commonly referred to as the United States (U.S.) or America, is a federal republic composed of 50 states, a federal district, five major territories and various possessions.

## University of Michigan

The University of Michigan (U-M, UM, UMich, or U of M), frequently referred to simply as Michigan, is a public research university located in Ann Arbor, Michigan, United States.

## University of Nottingham

The University of Nottingham is a public research university based in Nottingham, Nottinghamshire, England, United Kingdom.

## University of South Florida

The University of South Florida, also known as USF, is a member institution of the State University System of Florida and a public research university located in Tampa, Florida, USA.

## Valuation (algebra)

In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field.

## Vehicle registration plate

A vehicle registration plate is a metal or plastic plate attached to a motor vehicle or trailer for official identification purposes.

## Vertex (graph theory)

In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices).

## Year One (education)

Year One is an educational year group in schools in many countries including England, Wales, Australia and New Zealand.

## Year Six

Year 6 is an educational year group in schools in many countries including England, Wales, Australia and New Zealand.

## Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation In other words, a "zero" of a function is an input value that produces an output of zero (0).

## 0 (number)

0 (zero; BrE: or AmE) is both a number and the numerical digit used to represent that number in numerals.

## 2 (number)

2 (Two) is a number, numeral, and glyph.

## References

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