![Parity of 5 and 6 Cuisenaire rods](https://upload.wikimedia.org/wikipedia/commons/e/e1/Parity_of_5_and_6_Cuisenaire_rods.png)
In
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 consume, alter, share, r ...
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 languag ...
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
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
In mathematics, zero is an even number. In other words, its parity—the quality of an integer 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 re ...
is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the
decimal numeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.
The same sequence of symbo ...
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 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
where ''k'' is an integer; an odd number is an integer of the form
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:
and an odd number is not:
The
sets of even and odd numbers can be defined as following:
The set of ''even'' numbers is a normal subgroup of
and create the factor group
. Parity can then be defined as a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from
to
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, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
, 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 of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of 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 b ...
the
dividend
A dividend is a distribution of 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 is taken to be re-i ...
has more
factors of two than the divisor.
History
The ancient Greeks considered 1, the
monad
Monad may refer to:
Philosophy
* Monad (philosophy), a term meaning "unit"
**Monism, the concept of "one essence" in the metaphysical and theological theory
** Monad (Gnosticism), the most primal aspect of God in Gnosticism
* ''Great Monad'', a ...
, to be neither fully odd nor fully even. Some of this sentiment survived into the 19th century:
Friedrich Wilhelm August Fröbel Friedrich may refer to:
Names
* Friedrich (surname), people with the surname ''Friedrich''
* Friedrich (given name), people with the given name ''Friedrich''
Other
* Friedrich (board game), a board game about Frederick the Great and the Seven Year ...
'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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 ''D
n''
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 for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...
, where the parity of a square is indicated by its color:
bishops are constrained to moving between squares of the same parity, whereas
knights alternate parity between moves. This form of parity was famously used to solve the
mutilated chessboard problem: 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 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 considere ...
of ''R'' whose
index is 2. Elements of the
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) ...
may be called even, while elements of the coset
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 (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 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, but the odd numbers do not—this is clear from the fact that 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), ...
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 ways ...
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.
...
s are even; it is unknown whether any odd perfect numbers exist.
Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern
computer calculations have shown this conjecture to be true for integers up to at least 4 × 10
18, but still no general
proof has been found.
Group theory
![Rubiks revenge solved](https://upload.wikimedia.org/wikipedia/commons/5/52/Rubiks_revenge_solved.jpg)
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,
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 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, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
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 player ...
, an ''evil number'' is a number that has an even number of 1's in its
binary representation
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 ...
, 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 game in combinatorial game theory, invented by Henry Dudeney in 1908. Given a row of imagined bowling pins, players take turns to knock out either one pin, or two adjacent pins, until all the pins are gone. Using the ...
. The
parity function In 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 XOR function.
The parity function is notable for its r ...
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 complement of the sequence obtained thus ...
, 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, a
parity bit 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 pitc ...
s with a cylindrical bore and in effect closed at one end, such as the
clarinet at the mouthpiece, the
harmonics 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 waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...
. (With cylindrical pipes open at both ends, used for example in some
organ stops 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).
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 num ...
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 ...
, 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 service from Toronto-Pearson to ...
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 ...
*
Half-integer
References
{{reflist, 30em
Elementary arithmetic
Mathematical concepts