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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 and their changes (cal ...
, parity is the property of an
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
of whether it is even or odd. An integer's parity is even if it is
divisible 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). ...

divisible
by two with no remainders left and its parity is odd if it isn't; that is, its remainder is 1.. For example, −4, 0, 82, and 178 are even because there is no
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 ...
when dividing it by 2. By contrast, −3, 5, 7, 21 are odd numbers as they leave a remainder of 1 when divided by 2. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, zero's parity is even. Any two consecutive integers have opposite parity. A formal definition of an even number is that it is an integer of the form ''n'' = 2''k'', where ''k'' is an integer; it can then be shown that an odd number is an integer of the form (or alternately, 2''k'' − 1). It is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. The sets of even and odd numbers can be defined as following: * Even =\ * Odd =\ A number (i.e., integer) expressed in the
decimal The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
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 Numerical digit, digits or other symbols in a consistent manner. The same s ...
is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8. The same idea will work using any even base. In particular, a number expressed in the
binary numeral system In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically "0" (zero) and "1" (one). The base-2 numeral system is a positional notat ...
is odd if its last digit is 1; and it is even if its last digit is 0. In an odd base, the number is even according to the sum of its digits—it is even if and only if the sum of its digits is even.


Arithmetic on even and odd numbers

The following laws can be verified using the properties of
divisibility 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). ...
. They are a special case of rules in
modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), a ...
, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, and multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic.


Addition and subtraction

* even ± even = even; * even ± odd = odd; * odd ± odd = even;


Multiplication

* even × even = even; * even × odd = even; * odd × odd = odd; The structure (, +, ×) is in fact a field with just two elements.


Division

The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which is neither even ''nor'' odd, since the concepts of even and odd apply only to integers. But when the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ...
is an integer, it will be even
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
the
dividend A dividend is a distribution of profit Profit may refer to: Business and law * Profit (accounting), the difference between the purchase price and the costs of bringing to market * Profit (economics), normal profit and economic profit * Profit ...
has more factors of two than the divisor.


History

The ancient Greeks considered 1, the
monad Monad may refer to: Philosophy * Monad (philosophy) Monad (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeas ...
, to be neither fully odd nor fully even. Some of this sentiment survived into the 19th century:
Friedrich Wilhelm August FröbelFriedrich may refer to: Names *Friedrich (surname)Friedrich or Friedrichs is a surname. Notable people with the surname include: Friedrich * Johannes Friedrich, Bishop of the Evangelical Lutheran Church in Bavaria * Ariane Friedrich, German high ...

Friedrich Wilhelm August Fröbel
's 1826 ''The Education of Man'' instructs the teacher to drill students with the claim that 1 is neither even nor odd, to which Fröbel attaches the philosophical afterthought,


Higher mathematics


Higher dimensions and more general classes of numbers

Integer coordinates of points in
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
s of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the face-centered cubic lattice and its higher-dimensional generalizations, the ''Dn'' lattices, consist of all of the integer points whose sum of coordinates is even. This feature manifests itself in
chess Chess is a board game Board games are tabletop game Tabletop games are game with separate sliding drawer, from 1390–1353 BC, made of glazed faience, dimensions: 5.5 × 7.7 × 21 cm, in the Brooklyn Museum (New Yor ...

chess
, where the parity of a square is indicated by its color:
bishops A bishop is an ordained Ordination is the process by which individuals are Consecration, consecrated, that is, set apart and elevated from the laity class to the clergy, who are thus then authorization, authorized (usually by the religious denom ...
are constrained to squares of the same parity; knights alternate parity between moves. This form of parity was famously used to solve the
mutilated chessboard problem The mutilated chessboard problem is a tiling puzzle Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). ...
: if two opposite corner squares are removed from a chessboard, then the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other. The parity of an ordinal number may be defined to be even if the number is a limit ordinal, or a limit ordinal plus a finite even number, and odd otherwise. Let ''R'' be 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 ...
and let ''I'' be 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 considered ...
of ''R'' whose
index Index 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 megastructure in the ''Halo'' series ...
is 2. Elements of the
coset 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). ...
0+I may be called even, while elements of the coset 1+I may be called odd. As an example, let be the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
of Z at the
prime ideal In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
(2). Then an element of ''R'' is even or odd if and only if its numerator is so in Z.


Number theory

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 considered ...
in the ring of integers, but the odd numbers do not—this is clear from the fact that the
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2. All
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
s are odd, with one exception: the prime number 2. All known
perfect number 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 Frie ...
s are even; it is unknown whether any odd perfect numbers exist.
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems 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 a ...
states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

computer
calculations have shown this conjecture to be true for integers up to at least 4 × 1018, but still no general
proof Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
has been found.


Group theory

The
parity of a 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 ord ...
(as defined 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), rings, field (mathema ...
) is the parity of the number of transpositions into which the permutation can be decomposed. For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable definition. In
Rubik's Cube The Rubik's Cube is a invented in 1974 by Hungarian sculptor and professor of architecture . Originally called the Magic Cube, the puzzle was licensed by Rubik to be sold by in 1980 via businessman Tibor Laczi and Seven Towns founder . Rubi ...

Rubik's Cube
,
Megaminx Image:Megaminx12.jpg, A 12-color Megaminx in a star-pattern arrangement The Megaminx or Mégaminx (, ) is a dodecahedron-shaped puzzle similar to the Rubik's Cube. It has a total of 50 movable pieces to rearrange, compared to the 20 movable pieces ...

Megaminx
, and other twisting puzzles, the moves of the puzzle allow only even permutations of the puzzle pieces, so parity is important in understanding the configuration space of these puzzles. The Feit–Thompson theorem states that a
finite group 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), ...
is always solvable if its order is an odd number. This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order" is far from obvious.


Analysis

The parity of a function describes how its values change when its arguments are exchanged with their negations. An even function, such as an even power of a variable, gives the same result for any argument as for its negation. An odd function, such as an odd power of a variable, gives for any argument the negation of its result when given the negation of that argument. It is possible for a function to be neither odd nor even, and for the case ''f''(''x'') = 0, to be both odd and even. The
Taylor series In , the Taylor series of a is an of terms that are expressed in terms of the function's s at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after ...
of an even function contains only terms whose exponent is an even number, and the Taylor series of an odd function contains only terms whose exponent is an odd number.


Combinatorial game theory

In
combinatorial game theory at a combinatorial game theory workshop Combinatorial game theory (CGT) is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (al ...
, an ''evil number'' is a number that has an even number of 1's in its binary representation, and an ''odious number'' is a number that has an odd number of 1's in its binary representation; these numbers play an important role in the strategy for the game
Kayles Kayles is a simple impartial gameIn combinatorial game theory, an impartial game is a Mathematical game, game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs a ...
. The
parity functionIn Boolean algebra (logic), Boolean algebra, a parity function is a Boolean function whose value is 1 if and only if the input vector has an odd number of ones. The parity function of two inputs is also known as the exclusive or, XOR function. The ...
maps a number to the number of 1's in its binary representation, modulo 2, so its value is zero for evil numbers and one for odious numbers. The
Thue–Morse sequence frame, This graphic demonstrates the repeating and complementary makeup of the Thue–Morse sequence. In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence (an infinite sequence of 0s and 1s) obtained ...
, an infinite sequence of 0's and 1's, has a 0 in position ''i'' when ''i'' is evil, and a 1 in that position when ''i'' is odious.


Additional applications

In
information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...
, a
parity bit A parity bit, or check bit, is a bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of al ...
appended to a binary number provides the simplest form of error detecting code. If a single bit in the resulting value is changed, then it will no longer have the correct parity: changing a bit in the original number gives it a different parity than the recorded one, and changing the parity bit while not changing the number it was derived from again produces an incorrect result. In this way, all single-bit transmission errors may be reliably detected. Some more sophisticated error detecting codes are also based on the use of multiple parity bits for subsets of the bits of the original encoded value. In
wind instrument A wind instrument is a musical instrument A musical instrument is a device created or adapted to make musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose that the object ...

wind instrument
s with a cylindrical bore and in effect closed at one end, such as the
clarinet The clarinet is a family of woodwind instrument Woodwind instruments are a family of musical instruments A musical instrument is a device created or adapted to make musical sounds. In principle, any object that produces sound can be cons ...

clarinet
at the mouthpiece, the
harmonic A harmonic is any member of the harmonic series Harmonic series may refer to either of two related concepts: *Harmonic series (mathematics) *Harmonic series (music) {{Disambig .... The term is employed in various disciplines, including music ...
s produced are odd multiples of the
fundamental frequency The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is th ...
. (With cylindrical pipes open at both ends, used for example in some
organ stop An organ stop is a component of a pipe organ that admits pressurized air (known as ''wind'') to a set of organ pipes. Its name comes from the fact that stops can be used selectively by the organist; each can be "on" (admitting the passage of air ...
s such as the open diapason, the harmonics are even multiples of the same frequency for the given bore length, but this has the effect of the fundamental frequency being doubled and all multiples of this fundamental frequency being produced.) See
harmonic series (music) A harmonic series (also overtone series) is the sequence of , s, or s in which each frequency is an multiple of a . s are often based on an acoustic such as a string or a column of air, which at numerous simultaneously. At the frequencies o ...
. In some countries,
house numbering House numbering is the system of giving a unique number to each building in a street or area, with the intention of making it easier to locate a particular building. The house number is often part of a postal address. The term describes the number ...
s are chosen so that the houses on one side of a street have even numbers and the houses on the other side have odd numbers. Similarly, among
United States numbered highways The United States Numbered Highway System (often called U.S. Routes or U.S. Highways) is an integrated network of roads and highways numbered within a nationwide grid in the contiguous United States. As the designation and numbering of these high ...
, even numbers primarily indicate east–west highways while odd numbers primarily indicate north–south highways. Among airline
flight number 200px, Flight numbers on a split-flap display (Frankfurt airport) In the aviation industry, a flight number or flight designator is a code for an airline service consisting of two-character Airline codes#IATA airline designator, airline designator ...
s, even numbers typically identify eastbound or northbound flights, and odd numbers typically identify westbound or southbound flights..


See also

*
Divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

Divisor


References

{{reflist, 30em Elementary arithmetic Mathematical concepts