Parity of zero
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In mathematics,
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
is an even number. In other words, its parity—the quality of an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
being even or odd—is even. This can be easily verified based on the definition of "even": it is an integer multiple of 2, specifically . As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal integer has the same parity as its last digit—so, since 10 is even, 0 will be even, and if is even then has the same parity as —and always have the same parity. Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as , require 0 to be even. Zero is the additive
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of even integers, and it is the starting case from which other even
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 n ...
s are recursively defined. Applications of this recursion from
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
to
computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...
rely on zero being even. Not only is 0 divisible by 2, it is divisible by every
power of 2 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
, which is relevant to the
binary numeral system A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notatio ...
used by computers. In this sense, 0 is the "most even" number of all. "By the same test zero surpasses all numbers in 'evenness.'"; "Thus, the integer is the most 'even.' Among the general public, the parity of zero can be a source of confusion. In
reaction time Mental chronometry is the scientific study of processing speed or reaction time on cognitive tasks to infer the content, duration, and temporal sequencing of mental operations. Reaction time (RT; sometimes referred to as "response time") is meas ...
experiments, most people are slower to identify 0 as even than 2, 4, 6, or 8. Some teachers —and some children in mathematics classes—think that zero is odd, or both even and odd, or neither. Researchers in
mathematics education In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although rese ...
propose that these misconceptions can become learning opportunities. Studying equalities like can address students' doubts about calling 0 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 c ...
and using it in
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
. Class discussions can lead students to appreciate the basic principles of mathematical reasoning, such as the importance of definitions. Evaluating the parity of this exceptional number is an early example of a pervasive theme in mathematics: the
abstraction Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstr ...
of a familiar concept to an unfamiliar setting.


Why zero is even

The standard definition of "even number" can be used to directly prove that zero is even. A number is called "even" if it is an integer multiple of 2. As an example, the reason that 10 is even is that it equals . In the same way, zero is an integer multiple of 2, namely so zero is even. It is also possible to explain why zero is even without referring to formal definitions. The following explanations make sense of the idea that zero is even in terms of fundamental number concepts. From this foundation, one can provide a rationale for the definition itself—and its applicability to zero.


Basic explanations

Given a set of objects, one uses a number to describe how many objects are in the set. Zero is the count of ''no objects''; in more formal terms, it is the number of objects in the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
. The concept of parity is used for making groups of two objects. If the objects in a set can be marked off into groups of two, with none left over, then the number of objects is even. If an object is left over, then the number of objects is odd. The empty set contains zero groups of two, and no object is left over from this grouping, so zero is even. These ideas can be illustrated by drawing objects in pairs. It is difficult to depict zero groups of two, or to emphasize the nonexistence of a leftover object, so it helps to draw other groupings and to compare them with zero. For example, in the group of five objects, there are two pairs. More importantly, there is a leftover object, so 5 is odd. In the group of four objects, there is no leftover object, so 4 is even. In the group of just one object, there are no pairs, and there is a leftover object, so 1 is odd. In the group of zero objects, there is no leftover object, so 0 is even. There is another concrete definition of evenness: if the objects in a set can be placed into two groups of equal size, then the number of objects is even. This definition is equivalent to the first one. Again, zero is even because the empty set can be divided into two groups of zero items each. Numbers can also be visualized as points on a
number line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poi ...
. When even and odd numbers are distinguished from each other, their pattern becomes obvious, especially if negative numbers are included: The even and odd numbers alternate. Starting at any even number,
counting Counting is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every ele ...
up or down by twos reaches the other even numbers, and there is no reason to skip over zero. With the introduction of
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 ...
, parity can be approached in a more formal way using arithmetic expressions. Every integer is either of the form or the former numbers are even and the latter are odd. For example, 1 is odd because and 0 is even because Making a table of these facts then reinforces the number line picture above.


Defining parity

The precise
definition A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...
of a mathematical term, such as "even" meaning "integer multiple of two", is ultimately a convention. Unlike "even", some mathematical terms are purposefully constructed to exclude
trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
or
degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
cases.
Prime number 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 ...
s are a famous example. Before the 20th century, definitions of primality were inconsistent, and significant mathematicians such as Goldbach, Lambert, Legendre, Cayley, and Kronecker wrote that 1 was prime. The modern definition of "prime number" is "positive integer with exactly 2
factor Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, suc ...
s", so 1 is not prime. This definition can be rationalized by observing that it more naturally suits mathematical theorems that concern the primes. For example, 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 ord ...
is easier to state when 1 is not considered prime. It would be possible to similarly redefine the term "even" in a way that no longer includes zero. However, in this case, the new definition would make it more difficult to state theorems concerning the even numbers. Already the effect can be seen in the algebraic rules governing even and odd numbers. The most relevant rules concern
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
,
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 ...
, 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 ...
: :even ± even = even :odd ± odd = even :even × integer = even Inserting appropriate values into the left sides of these rules, one can produce 0 on the right sides: :2 − 2 = 0 :−3 + 3 = 0 :4 × 0 = 0 The above rules would therefore be incorrect if zero were not even. At best they would have to be modified. For example, one test study guide asserts that even numbers are characterized as integer multiples of two, but zero is "neither even nor odd". Accordingly, the guide's rules for even and odd numbers contain exceptions: :even ± even = even (or zero) :odd ± odd = even (or zero) :even × nonzero integer = even These rules are given, but they are not quoted verbatim. Making an exception for zero in the definition of evenness forces one to make such exceptions in the rules for even numbers. From another perspective, taking the rules obeyed by positive even numbers and requiring that they continue to hold for integers forces the usual definition and the evenness of zero.


Mathematical contexts

Countless results in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
invoke the fundamental theorem of arithmetic and the algebraic properties of even numbers, so the above choices have far-reaching consequences. For example, the fact that positive numbers have unique
factorization In mathematics, factorization (or factorisation, see American and British English spelling differences#-ise, -ize (-isation, -ization), English spelling differences) or factoring consists of writing a number or another mathematical object as a p ...
s means that one can determine whether a number has an even or odd number of distinct prime factors. Since 1 is not prime, nor does it have prime factors, it is a product of 0 distinct primes; since 0 is an even number, 1 has an even number of distinct prime factors. This implies that the
Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
takes the value , which is necessary for it to be a
multiplicative function In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime. An arithmetic function ''f''(''n'') is ...
and for the
Möbius inversion formula In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large gener ...
to work.


Not being odd

A number is odd if there is an integer such that . One way to prove that zero is not odd is by contradiction: if then , which is not an integer. Since zero is not odd, if an unknown number is proven to be odd, then it cannot be zero. This apparently trivial observation can provide a convenient and revealing proof explaining why an odd number is nonzero. A classic result of
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
states that a
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 ...
of odd order (having an odd number of vertices) always has at least one vertex of even degree. (The statement itself requires zero to be even: the
empty 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"). Order-zero graph The order-zero graph, , is th ...
has an even order, and an
isolated vertex In discrete 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 ve ...
has an even degree.) For isolated vertices see p. 149; for groups see p. 311. In order to prove the statement, it is actually easier to prove a stronger result: any odd-order graph has an ''odd number'' of even degree vertices. The appearance of this odd number is explained by a still more general result, known as the
handshaking lemma In graph theory, a branch of mathematics, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. In more colloquial terms, in a party of people some of whom ...
: any graph has an even number of vertices of odd degree. Finally, the even number of odd vertices is naturally explained by the
degree sum formula In graph theory, a branch of mathematics, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. In more colloquial terms, in a party of people some of whom ...
.
Sperner's lemma In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an ...
is a more advanced application of the same strategy. The lemma states that a certain kind of coloring on a
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
of a
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
has a subsimplex that contains every color. Rather than directly construct such a subsimplex, it is more convenient to prove that there exists an odd number of such subsimplices through an
induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
argument. A stronger statement of the lemma then explains why this number is odd: it naturally breaks down as when one considers the two possible
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
s of a simplex.


Even-odd alternation

The fact that zero is even, together with the fact that even and odd numbers alternate, is enough to determine the parity of every other
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 n ...
. This idea can be formalized into a
recursive definition In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set ( Aczel 1977:740ff). Some examples of recursively-definable objects include facto ...
of the set of even natural numbers: *0 is even. *(''n'' + 1) is even if and only if ''n'' is not even. This definition has the conceptual advantage of relying only on the minimal foundations of the natural numbers: the existence of 0 and of successors. As such, it is useful for computer logic systems such as LF and the
Isabelle theorem prover The Isabelle automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs withou ...
. With this definition, the evenness of zero is not a theorem but an axiom. Indeed, "zero is an even number" may be interpreted as one of 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 ...
, of which the even natural numbers are a model. A similar construction extends the definition of parity to transfinite
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...
s: every
limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...
is even, including zero, and
successor Successor may refer to: * An entity that comes after another (see Succession (disambiguation)) Film and TV * ''The Successor'' (film), a 1996 film including Laura Girling * ''The Successor'' (TV program), a 2007 Israeli television program Musi ...
s of even ordinals are odd. The classic
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. It is a special case of point location problems and finds applications in areas that dea ...
test from
computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...
applies the above ideas. To determine if a point lies within a
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
, one casts a ray from infinity to the point and counts the number of times the ray crosses the edge of polygon. The crossing number is even if and only if the point is outside the polygon. This
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
works because if the ray never crosses the polygon, then its crossing number is zero, which is even, and the point is outside. Every time the ray does cross the polygon, the crossing number alternates between even and odd, and the point at its tip alternates between outside and inside. In graph theory, a
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 and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are ...
is a graph whose vertices are split into two
colors Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associa ...
, such that neighboring vertices have different colors. If a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
graph has no odd cycles, then a bipartition can be constructed by choosing a base vertex ''v'' and coloring every vertex black or white, depending on whether its
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
from ''v'' is even or odd. Since the distance between ''v'' and itself is 0, and 0 is even, the base vertex is colored differently from its neighbors, which lie at a distance of 1.


Algebraic patterns

In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, the even integers form various
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 of ...
s that require the inclusion of zero. The fact that the
additive identity In mathematics, the additive identity of a set that 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 element ...
(zero) is even, together with the evenness of sums and
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 even numbers and the
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
of addition, means that the even integers form a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. Moreover, the group of even integers under addition is a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation âˆ—, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation âˆ—. More precisely, ''H'' is a subgroup ...
of the group of all integers; this is an elementary example of the subgroup concept. The earlier observation that the rule "even − even = even" forces 0 to be even is part of a general pattern: any
nonempty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
subset of an additive group that is
closed under In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but n ...
subtraction must be a subgroup, and in particular, must contain the
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...
. Since the even integers form a subgroup of the integers, they
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
the integers into
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s. These cosets may be described as the
equivalence classes 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 ...
of the following
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 ...
: if is even. Here, the evenness of zero is directly manifested as the reflexivity of the
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
~. There are only two cosets of this subgroup—the even and odd numbers—so it has
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
2. Analogously, the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic prop ...
is a subgroup of index 2 in the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
on ''n'' letters. The elements of the alternating group, called
even permutation In mathematics, when ''X'' is a finite set with 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. If any total ...
s, are the products of even numbers of transpositions. The
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
, an
empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
of no transpositions, is an even permutation since zero is even; it is the identity element of the group. The rule "even × integer = even" means that the even numbers form an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
in the
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
of integers, and the above equivalence relation can be described as equivalence modulo this ideal. In particular, even integers are exactly those integers ''k'' where This formulation is useful for investigating integer zeroes of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s.


2-adic order

There is a sense in which some multiples of 2 are "more even" than others. Multiples of 4 are called
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. The former names are traditional ones, derived from ancient Gree ...
, since they can be divided by 2 twice. Not only is zero divisible by 4, zero has the unique property of being divisible by every
power of 2 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
, so it surpasses all other numbers in "evenness". One consequence of this fact appears in the bit-reversed ordering of integer data types used by some computer algorithms, such as the Cooley–Tukey fast Fourier transform. This ordering has the property that the farther to the left the first 1 occurs in a number's
binary expansion A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notatio ...
, or the more times it is divisible by 2, the sooner it appears. Zero's bit reversal is still zero; it can be divided by 2 any number of times, and its binary expansion does not contain any 1s, so it always comes first. Although 0 is divisible by 2 more times than any other number, it is not straightforward to quantify exactly how many times that is. For any nonzero integer ''n'', one may define the 2-adic order of ''n'' to be the number of times ''n'' is divisible by 2. This description does not work for 0; no matter how many times it is divided by 2, it can always be divided by 2 again. Rather, the usual convention is to set the 2-order of 0 to be
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
as a special case. This convention is not peculiar to the 2-order; it is one of the axioms of an additive valuation in higher algebra. The powers of two—1, 2, 4, 8, ...—form a simple
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of numbers of increasing 2-order. In the 2-adic numbers, such sequences actually
converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines *CONVERGE CFD s ...
to zero.


Education

The subject of the parity of zero is often treated within the first two or three years of
primary education Primary education or elementary education is typically the first stage of formal education, coming after preschool/kindergarten and before secondary school. Primary education takes place in ''primary schools'', ''elementary schools'', or first ...
, as the concept of even and odd numbers is introduced and developed.


Students' knowledge

The chart on the right depicts children's beliefs about the parity of zero, as they progress from Year 1 to
Year 6 Year 6 is an educational year group in schools in many countries including the United Kingdom, Australia and New Zealand. It is usually the sixth year of compulsory education and incorporates students aged between ten and eleven however some chil ...
of the English education system. The data is from Len Frobisher, who conducted a pair of surveys of English schoolchildren. Frobisher was interested in how knowledge of single-digit parity translates to knowledge of multiple-digit parity, and zero figures prominently in the results. In a preliminary survey of nearly 400 seven-year-olds, 45% chose ''even'' over ''odd'' when asked the parity of zero. A follow-up investigation offered more choices: ''neither'', ''both'', and ''don't know''. This time the number of children in the same age range identifying zero as even dropped to 32%. Success in deciding that zero is even initially shoots up and then levels off at around 50% in Years 3 to 6. For comparison, the easiest task, identifying the parity of a single digit, levels off at about 85% success. In interviews, Frobisher elicited the students' reasoning. One fifth-year decided that 0 was even because it was found on the 2
times 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. The decimal multiplication table was traditionally taught as an essenti ...
. A couple of fourth-years realized that zero can be split into equal parts. Another fourth-year reasoned "1 is odd and if I go down it's even." The interviews also revealed the misconceptions behind incorrect responses. A second-year was "quite convinced" that zero was odd, on the basis that "it is the first number you count". A fourth-year referred to 0 as "none" and thought that it was neither odd nor even, since "it's not a number". In another study, Annie Keith observed a class of 15 second-graders who convinced each other that zero was an even number based on even-odd alternation and on the possibility of splitting a group of zero things in two equal groups. More in-depth investigations were conducted by Esther Levenson, Pessia Tsamir, and Dina Tirosh, who interviewed a pair of sixth-grade students in the USA who were performing highly in their mathematics class. One student preferred deductive explanations of mathematical claims, while the other preferred practical examples. Both students initially thought that 0 was neither even nor odd, for different reasons. Levenson et al. demonstrated how the students' reasoning reflected their concepts of zero and division.
Deborah Loewenberg Ball Deborah Loewenberg Ball is an educational researcher noted for her work in mathematics instruction and the mathematical preparation of teachers. From 2017 to 2018 she serves as president of the American Educational Research Association. She served ...
analyzed US third grade students' ideas about even and odd numbers and zero, which they had just been discussing with a group of fourth-graders. The students discussed the parity of zero, the rules for even numbers, and how mathematics is done. The claims about zero took many forms, as seen in the list on the right., Figure 1.5 "Mathematical claims about zero." Ball and her coauthors argued that the episode demonstrated how students can "do mathematics in school", as opposed to the usual reduction of the discipline to the mechanical solution of exercises. One of the themes in the research literature is the tension between students' concept images of parity and their concept definitions. Levenson et al.'s sixth-graders both defined even numbers as multiples of 2 or numbers divisible by 2, but they were initially unable to apply this definition to zero, because they were unsure how to multiply or divide zero by 2. The interviewer eventually led them to conclude that zero was even; the students took different routes to this conclusion, drawing on a combination of images, definitions, practical explanations, and abstract explanations. In another study, David Dickerson and Damien Pitman examined the use of definitions by five advanced
undergraduate Undergraduate education is education conducted after secondary education and before postgraduate education. It typically includes all postsecondary programs up to the level of a bachelor's degree. For example, in the United States, an entry-lev ...
mathematics
major Major (commandant in certain jurisdictions) is a military rank of commissioned officer status, with corresponding ranks existing in many military forces throughout the world. When used unhyphenated and in conjunction with no other indicators ...
s. They found that the undergraduates were largely able to apply the definition of "even" to zero, but they were still not convinced by this reasoning, since it conflicted with their concept images.


Teachers' knowledge

Researchers of
mathematics education In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although rese ...
at the
University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...
have included the true-or-false prompt "0 is an even number" in a database of over 250 questions designed to measure teachers' content knowledge. For them, the question exemplifies "common knowledge ... that any well-educated adult should have", and it is "ideologically neutral" in that the answer does not vary between
traditional A tradition is a belief or behavior (folk custom) passed down within a group or society with symbolic meaning or special significance with origins in the past. A component of cultural expressions and folklore, common examples include holidays or ...
and
reform mathematics Reform mathematics is an approach to mathematics education, particularly in North America. It is based on principles explained in 1989 by the National Council of Teachers of Mathematics (NCTM). The NCTM document ''Curriculum and Evaluation Standa ...
. In a 2000–2004 study of 700 primary teachers in the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
, overall performance on these questions significantly predicted improvements in students'
standardized test A standardized test is a test that is administered and scored in a consistent, or "standard", manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predete ...
scores after taking the teachers' classes. In a more in-depth 2008 study, the researchers found a school where all of the teachers thought that zero was neither odd nor even, including one teacher who was exemplary by all other measures. The misconception had been spread by a math coach in their building. It is uncertain how many teachers harbor misconceptions about zero. The Michigan studies did not publish data for individual questions. Betty Lichtenberg, an associate professor of mathematics education at the
University of South Florida The University of South Florida (USF) is a public research university with its main campus located in Tampa, Florida, and other campuses in St. Petersburg and Sarasota. It is one of 12 members of the State University System of Florida. USF is ...
, in a 1972 study reported that when a group of prospective elementary school teachers were given a true-or-false test including the item "Zero is an even number", they found it to be a "tricky question", with about two thirds answering "False".


Implications for instruction

Mathematically, proving that zero is even is a simple matter of applying a definition, but more explanation is needed in the context of education. One issue concerns the foundations of the proof; the definition of "even" as "integer multiple of 2" is not always appropriate. A student in the first years of primary education may not yet have learned what "integer" or "multiple" means, much less how to multiply with 0. Additionally, stating a definition of parity for all integers can seem like an arbitrary conceptual shortcut if the only even numbers investigated so far have been positive. It can help to acknowledge that as the number concept is extended from positive integers to include zero and negative integers, number properties such as parity are also extended in a nontrivial way.


Numerical cognition

Adults who do believe that zero is even can nevertheless be unfamiliar with thinking of it as even, enough so to measurably slow them down in a
reaction time Mental chronometry is the scientific study of processing speed or reaction time on cognitive tasks to infer the content, duration, and temporal sequencing of mental operations. Reaction time (RT; sometimes referred to as "response time") is meas ...
experiment.
Stanislas Dehaene Stanislas Dehaene (born May 12, 1965) is a French author and cognitive neuroscientist whose research centers on a number of topics, including numerical cognition, the neural basis of reading and the neural correlates of consciousness. As of 201 ...
, a pioneer in the field of
numerical cognition Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes ...
, led a series of such experiments in the early 1990s. A numeral is flashed to the subject on a
monitor Monitor or monitor may refer to: Places * Monitor, Alberta * Monitor, Indiana, town in the United States * Monitor, Kentucky * Monitor, Oregon, unincorporated community in the United States * Monitor, Washington * Monitor, Logan County, West ...
, and a
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
records the time it takes the subject to push one of two buttons to identify the number as odd or even. The results showed that 0 was slower to process than other even numbers. Some variations of the experiment found delays as long as 60
millisecond A millisecond (from '' milli-'' and second; symbol: ms) is a unit of time in the International System of Units (SI) equal to one thousandth (0.001 or 10−3 or 1/1000) of a second and to 1000 microseconds. A unit of 10 milliseconds may be called ...
s or about 10% of the average reaction time—a small difference but a significant one. Dehaene's experiments were not designed specifically to investigate 0 but to compare competing models of how parity information is processed and extracted. The most specific model, the mental calculation hypothesis, suggests that reactions to 0 should be fast; 0 is a small number, and it is easy to calculate . (Subjects are known to compute and name the result of multiplication by zero faster than multiplication of nonzero numbers, although they are slower to verify proposed results like .) The results of the experiments suggested that something quite different was happening: parity information was apparently being recalled from memory along with a cluster of related properties, such as being
prime 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 ...
or a
power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
. Both the sequence of powers of two and the sequence of positive even numbers 2, 4, 6, 8, ... are well-distinguished mental categories whose members are prototypically even. Zero belongs to neither list, hence the slower responses. Repeated experiments have shown a delay at zero for subjects with a variety of ages and national and linguistic backgrounds, confronted with number names in numeral form, spelled out, and spelled in a mirror image. Dehaene's group did find one differentiating factor: mathematical expertise. In one of their experiments, students in the
École Normale Supérieure École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...
were divided into two groups: those in literary studies and those studying mathematics, physics, or biology. The slowing at 0 was "essentially found in the iterarygroup", and in fact, "before the experiment, some L subjects were unsure whether 0 was odd or even and had to be reminded of the mathematical definition". This strong dependence on familiarity again undermines the mental calculation hypothesis. The effect also suggests that it is inappropriate to include zero in experiments where even and odd numbers are compared as a group. As one study puts it, "Most researchers seem to agree that zero is not a typical even number and should not be investigated as part of the mental number line."


Everyday contexts

Some of the contexts where the parity of zero makes an appearance are purely rhetorical. The issue provides material for
Internet The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, pub ...
message boards and ask-the-expert websites. Linguist Joseph Grimes muses that asking "Is zero an even number?" to married couples is a good way to get them to disagree. People who think that zero is neither even nor odd may use the parity of zero as proof that every rule has a
counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
, or as an example of a trick question. Around the year 2000, media outlets noted a pair of unusual milestones: "1999/11/19" was the last
calendar date A calendar date is a reference to a particular day represented within a calendar system. The calendar date allows the specific day to be identified. The number of days between two dates may be calculated. For example, "25 " is ten days after " ...
composed of all odd digits that would occur for a very long time, and that "2000/02/02" was the first all-even date to occur in a very long time. Since these results make use of 0 being even, some readers disagreed with the idea. In
standardized test A standardized test is a test that is administered and scored in a consistent, or "standard", manner. Standardized tests are designed in such a way that the questions and interpretations are consistent and are administered and scored in a predete ...
s, if a question asks about the behavior of even numbers, it might be necessary to keep in mind that zero is even. Official publications relating to the
GMAT 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 ...
and
GRE The Graduate Record Examinations (GRE) is a standardized test that is an admissions requirement for many graduate schools in the United States and Canada and a few other countries. The GRE is owned and administered by Educational Testing Servi ...
tests both state that 0 is even. The parity of zero is relevant to odd–even rationing, in which cars may drive or purchase
gasoline Gasoline (; ) or petrol (; ) (see ) is a transparent, petroleum-derived flammable liquid that is used primarily as a fuel in most spark-ignited internal combustion engines (also known as petrol engines). It consists mostly of organic co ...
on alternate days, according to the parity of the last digit in their
license plate A vehicle registration plate, also known as a number plate (British English), license plate (American English), or licence plate ( Canadian English), is a metal or plastic plate attached to a motor vehicle or trailer for official identificati ...
s. Half of the numbers in a given range end in 0, 2, 4, 6, 8 and the other half in 1, 3, 5, 7, 9, so it makes sense to include 0 with the other even numbers. However, in 1977, a Paris rationing system led to confusion: on an odd-only day, the police avoided fining drivers whose plates ended in 0, because they did not know whether 0 was even. To avoid such confusion, the relevant legislation sometimes stipulates that zero is even; such laws have been passed in
New South Wales ) , nickname = , image_map = New South Wales in Australia.svg , map_caption = Location of New South Wales in AustraliaCoordinates: , subdivision_type = Country , subdivision_name = Australia , established_title = Before federation , es ...
and
Maryland Maryland ( ) is a state in the Mid-Atlantic region of the United States. It shares borders with Virginia, West Virginia, and the District of Columbia to its south and west; Pennsylvania to its north; and Delaware and the Atlantic Ocean to ...
. On U.S. Navy vessels, even-numbered compartments are found on the
port A port is a maritime facility comprising one or more wharves or loading areas, where ships load and discharge cargo and passengers. Although usually situated on a sea coast or estuary, ports can also be found far inland, such as Ham ...
side, but zero is reserved for compartments that intersect the centerline. That is, the numbers read 6-4-2-0-1-3-5 from port to starboard. In the game of
roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...
, the number 0 does not count as even or odd, giving the
casino A casino is a facility for certain types of gambling. Casinos are often built near or combined with hotels, resorts, restaurants, retail shopping, cruise ships, and other tourist attractions. Some casinos are also known for hosting live entertai ...
an advantage on such bets. Similarly, the parity of zero can affect payoffs in
prop bet In gambling, a "proposition bet" (prop bet, prop, novelty, or a side bet) is a bet made regarding the occurrence or non-occurrence during a game (usually a gambling game) of an event not directly affecting the game's final outcome. Proposition be ...
s when the outcome depends on whether some randomized number is odd or even, and it turns out to be zero. The game of " odds and evens" is also affected: if both players cast zero fingers, the total number of fingers is zero, so the even player wins. One teachers' manual suggests playing this game as a way to introduce children to the concept that 0 is divisible by 2.


References


Bibliography

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Further reading

* *


External links

*
Is Zero Even? - Numberphile
video with James Grime,
University of Nottingham The University of Nottingham is a public university, public research university in Nottingham, United Kingdom. It was founded as University College Nottingham in 1881, and was granted a royal charter in 1948. The University of Nottingham belongs t ...
{{Featured article Elementary arithmetic
Zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
0 (number)