# Parity (mathematics)

For other uses, see Parity (disambiguation). Parity (mathematics)_sentence_0

"Odd number" redirects here. Parity (mathematics)_sentence_1

For the 1962 Argentine film, see Odd Number (film). Parity (mathematics)_sentence_2

In mathematics, parity is the property of an integer of whether it is even or odd. Parity (mathematics)_sentence_3

An integer's parity is even if it is divisible by two with no remainders left and its parity is odd if its remainder is 1. Parity (mathematics)_sentence_4

For example, -4, 0, 82, and 178 are even because there is no remainder when dividing it by 2. Parity (mathematics)_sentence_5

By contrast, -3, 5, 7, 21 are odd numbers as they leave a remainder of 1 when divided by 2. Parity (mathematics)_sentence_6

Even and odd numbers have opposite parities, e.g. 22 (even number) and 13 (odd number) have opposite parities. Parity (mathematics)_sentence_7

In particular, zero's parity is even. Parity (mathematics)_sentence_8

A formal definition of an even number is that it is an integer of the form n = 2k, where k is an integer; it can then be shown that an odd number is an integer of the form n = 2k + 1 (or alternately, 2k - 1). Parity (mathematics)_sentence_9

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. Parity (mathematics)_sentence_10

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. Parity (mathematics)_sentence_11

The sets of even and odd numbers can be defined as following: Parity (mathematics)_sentence_12

A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. Parity (mathematics)_sentence_13

That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even. Parity (mathematics)_sentence_14

The same idea will work using any even base. Parity (mathematics)_sentence_15

In particular, a number expressed in the binary numeral system is odd if its last digit is 1; it is even if its last digit is 0. Parity (mathematics)_sentence_16

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. Parity (mathematics)_sentence_17

## Arithmetic on even and odd numbers Parity (mathematics)_section_0

The following laws can be verified using the properties of divisibility. Parity (mathematics)_sentence_18

They are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. Parity (mathematics)_sentence_19

As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, and multiplication is distributive over addition. Parity (mathematics)_sentence_20

However, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic. Parity (mathematics)_sentence_21

### Addition and subtraction Parity (mathematics)_section_1

Parity (mathematics)_unordered_list_0

• even ± even = even;Parity (mathematics)_item_0_0
• even ± odd = odd;Parity (mathematics)_item_0_1
• odd ± odd = even;Parity (mathematics)_item_0_2

### Multiplication Parity (mathematics)_section_2

Parity (mathematics)_unordered_list_1

• even × even = even;Parity (mathematics)_item_1_3
• even × odd = even;Parity (mathematics)_item_1_4
• odd × odd = odd;Parity (mathematics)_item_1_5

The structure ({even, odd}, +, ×) is in fact a field with just two elements. Parity (mathematics)_sentence_22

### Division Parity (mathematics)_section_3

The division of two whole numbers does not necessarily result in a whole number. Parity (mathematics)_sentence_23

For example, 1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even and odd apply only to integers. Parity (mathematics)_sentence_24

But when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor. Parity (mathematics)_sentence_25

## History Parity (mathematics)_section_4

The ancient Greeks considered 1, the monad, to be neither fully odd nor fully even. Parity (mathematics)_sentence_26

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, Parity (mathematics)_sentence_27

## Higher mathematics Parity (mathematics)_section_5

### Higher dimensions and more general classes of numbers Parity (mathematics)_section_6

Integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. Parity (mathematics)_sentence_28

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. Parity (mathematics)_sentence_29

This feature manifests itself in chess, where the parity of a square is indicated by its color: bishops are constrained to squares of the same parity; knights alternate parity between moves. Parity (mathematics)_sentence_30

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. Parity (mathematics)_sentence_31

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. Parity (mathematics)_sentence_32

### Number theory Parity (mathematics)_section_7

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. Parity (mathematics)_sentence_33

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. Parity (mathematics)_sentence_34

All prime numbers are odd, with one exception: the prime number 2. Parity (mathematics)_sentence_35

All known perfect numbers are even; it is unknown whether any odd perfect numbers exist. Parity (mathematics)_sentence_36

Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. Parity (mathematics)_sentence_37

Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 10, but still no general proof has been found. Parity (mathematics)_sentence_38

### Group theory Parity (mathematics)_section_8

The parity of a permutation (as defined in abstract algebra) is the parity of the number of transpositions into which the permutation can be decomposed. Parity (mathematics)_sentence_39

For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). Parity (mathematics)_sentence_40

It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Parity (mathematics)_sentence_41

Hence the above is a suitable definition. Parity (mathematics)_sentence_42

In Rubik's Cube, 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. Parity (mathematics)_sentence_43

The Feit–Thompson theorem states that a finite group is always solvable if its order is an odd number. Parity (mathematics)_sentence_44

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. Parity (mathematics)_sentence_45

### Analysis Parity (mathematics)_section_9

The parity of a function describes how its values change when its arguments are exchanged with their negations. Parity (mathematics)_sentence_46

An even function, such as an even power of a variable, gives the same result for any argument as for its negation. Parity (mathematics)_sentence_47

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. Parity (mathematics)_sentence_48

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. Parity (mathematics)_sentence_49

The Taylor series 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. Parity (mathematics)_sentence_50

### Combinatorial game theory Parity (mathematics)_section_10

In combinatorial game theory, 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. Parity (mathematics)_sentence_51

The parity function 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. Parity (mathematics)_sentence_52

The Thue–Morse sequence, 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. Parity (mathematics)_sentence_53

In information theory, a parity bit appended to a binary number provides the simplest form of error detecting code. Parity (mathematics)_sentence_54

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. Parity (mathematics)_sentence_55

In this way, all single-bit transmission errors may be reliably detected. Parity (mathematics)_sentence_56

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. Parity (mathematics)_sentence_57

In wind instruments 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. Parity (mathematics)_sentence_58

(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.) Parity (mathematics)_sentence_59

See harmonic series (music). Parity (mathematics)_sentence_60

In some countries, house numberings 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. Parity (mathematics)_sentence_61

Similarly, among United States numbered highways, even numbers primarily indicate east-west highways while odd numbers primarily indicate north-south highways. Parity (mathematics)_sentence_62

Among airline flight numbers, even numbers typically identify eastbound or northbound flights, and odd numbers typically identify westbound or southbound flights. Parity (mathematics)_sentence_63