even and odd numbers
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, parity is the
property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to Consumables, consume, alte ...
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 of ...
of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. 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. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing number A number is a mathematical object used to count, measure, and label. The original example ...
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 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 notation ...
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.


Definition

An even number is an integer of the form x = 2k where ''k'' is an integer; an odd number is an integer of the form x = 2k +1. An equivalent definition is that an even number is
divisible 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 ...
by 2: 2 \ , \ x and an odd number is not: 2\not, \ x The sets of even and odd numbers can be defined as following: \ \ The set of ''even'' numbers is a normal subgroup of \mathbb and create the factor group \mathbb/2\mathbb. Parity can then be defined as a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
from \mathbb to \mathbb/2\mathbb where odd numbers are 1 and even numbers are 0. The consequences of this homomorphism are covered below.


Properties

The following laws can be verified using the properties of
divisibility 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 ...
. They are a special case of rules in
modular arithmetic In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
, 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 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, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division (mathematics), division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to a ...
is an integer, it will be even
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
the
dividend A dividend is a distribution of Profit (accounting), profits by a corporation to its shareholders. When a corporation earns a profit or surplus, it is able to pay a portion of the profit as a dividend to shareholders. Any amount not distributed ...
has more factors of two than the divisor.


History

The ancient Greeks considered 1, the monad, to be neither fully odd nor fully even. Some of this sentiment survived into the 19th century: 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 geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
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 that is 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 between two Player (game), players. It is sometimes called international chess or Western chess to distinguish it from chess variant, related games, such as xiangqi (Chinese chess) and shogi (Japanese chess). The current ...
, where the parity of a square is indicated by its color:
bishops A bishop is an ordained clergy member who is entrusted with a position of Episcopal polity, authority and oversight in a religious institution. In Christianity, bishops are normally responsible for the governance of dioceses. The role or offic ...
are constrained to moving between squares of the same parity, whereas
knights A knight is a person granted an honorary title of knighthood by a head of state (including the Pope) or representative for service to the monarch, the Christian denomination, church or the country, especially in a military capacity. Knighthood ...
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 posed by Max Black in 1946 that asks: Suppose a standard 8×8 chessboard (or checkerboard) has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoe ...
: 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 mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring proper ...
and let ''I'' be an ideal of ''R'' whose
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 ...
is 2. Elements of the coset 0+I may be called even, while elements of the coset 1+I may be called odd. As an example, let be the localization of Z at the
prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic_properties, integers. The prime ideals for the integers are the sets that contain all the ...
(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 in the ring of integers, but the odd numbers do not—this is clear from the fact that the identity 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 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 wa ...
s are odd, with one exception: the prime number 2. All known
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...
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 list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even natural number greater than 2 is the s ...
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 carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as Computer program, pr ...
calculations have shown this conjecture to be true for integers up to at least 4 × 1018, but still no general proof has been found.


Group theory

The parity of a permutation (as defined in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
) 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 Three-dimensional space, 3-D combination puzzle originally invented in 1974 by Hungarians, Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik t ...
,
Megaminx 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 of the Rubik's Cube. History The Megaminx, or Magic Dodecahedron, ...
, 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 Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
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 mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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 Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players ...
, 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. The
parity function In Boolean algebra (logic), Boolean algebra, a parity function is a Boolean function whose value is one 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. T ...
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 In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean algebra, Boolean complement of the sequenc ...
, 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 information. The field was originally established by the works of Harry Nyquist a ...
, a
parity bit A parity bit, or check bit, is a bit added to a string of binary code. Parity bits are a simple form of Error detection and correction, error detecting code. Parity bits are generally applied to the smallest units of a communication protocol, ...
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 that contains some type of resonator (usually a tube) in which a column of air is set into vibration by the player blowing into (or over) a mouthpiece set at or near the end of the resonator. The pitch ...
s with a cylindrical bore and in effect closed at one end, such as the
clarinet The clarinet is a musical instrument in the woodwind family. The instrument has a nearly cylindrical bore (wind instruments), bore and a flared bell, and uses a Single-reed instrument, single reed to produce sound. Clarinets comprise a Family ...
at the mouthpiece, the
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
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 of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a note that is perceived as ...
. (With cylindrical pipes open at both ends, used for example in some
organ stop An organ stop is a component of a pipe organ The pipe organ is a musical instrument that produces sound by driving pressurized air (called ''wind'') through the organ pipes selected from a keyboard. Because each pipe produces a single pitc ...
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 harmonics, musical tones, or pure tones whose frequency is an integer multiple of a ''fundamental frequency''. Definite pitch, Pitched musical instruments are often based on an acoustic ...
. 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 Address (geography), postal address. The te ...
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 hi ...
, even numbers primarily indicate east–west highways while odd numbers primarily indicate north–south highways. Among airline
flight number In the aviation industry, a flight number or flight designator is a code for an airline service consisting of two-character airline designator and a 1 to 4 digit number. For example, "BA 98" is a British Airways British Airways (BA) is th ...
s, even numbers typically identify eastbound or northbound flights, and odd numbers typically identify westbound or southbound flights..


See also

*
Divisor In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
* Half-integer


References

{{reflist, 30em Elementary arithmetic Mathematical concepts