"Odd number" redirects here. The term may also refer toOdd Number (film).
Property of being an even or odd number
Cuisenaire rods: 5 (yellow)cannot be evenly divided in 2 (red) by any 2 rods of the same color/length, while 6 (dark green)can be evenly divided in 2 by 3 (lime green).
Inmathematics,parity is theproperty of aninteger of whether it iseven orodd. An integer is even if it isdivisible by 2, and odd if it is not.[1] For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 61 are odd numbers.
The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. 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, theparity of zero is even.[2] Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in thedecimalnumeral system 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 thebinary 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.[3]
An even number is an integer of the formwherek is an integer;[4] an odd number is an integer of the form
An equivalent definition is that an even number isdivisible by 2:and an odd number is not:
Thesets of even and odd numbers can be defined as following:[5]
The set ofeven numbers is aprime ideal of and thequotient ring is thefield with two elements. Parity can then be defined as the uniquering homomorphism from to where odd numbers are 1 and even numbers are 0. The consequences of this homomorphism are covered below.
The following laws can be verified using the properties ofdivisibility. They are a special case of rules inmodular arithmetic, 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.
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 evennor odd, since the concepts of even and odd apply only to integers. But when thequotient is an integer, it will be evenif and only if thedividend has morefactors of two than the divisor.[6]
The ancient Greeks considered 1, themonad, to be neither fully odd nor fully even.[7] Some of this sentiment survived into the 19th century:Friedrich Wilhelm August Fröbel's 1826The 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,
It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two relatively different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and even numbers one number (one) which is neither of the two. Similarly, in form, the right angle stands between the acute and obtuse angles; and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws.[8]
Each of the whitebishops is confined to squares of the same parity; the blackknight can only jump to squares of alternating parity.
Integer coordinates of points inEuclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, theface-centered cubic lattice and its higher-dimensional generalizations (theDnlattices) consist of all of the integer points whose coordinates have an even sum.[9] This feature also manifests itself inchess, where the parity of a square is indicated by its color:bishops are constrained to moving between squares of the same parity, whereasknights alternate parity between moves.[10] This form of parity was famously used to solve themutilated 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.[11]
LetR be acommutative ring and letI be anideal ofR whoseindex is 2. Elements of thecoset may be calledeven, while elements of the coset may be calledodd.As an example, letR =Z(2) be thelocalization ofZ at theprime ideal (2). Then an element ofR is even or odd if and only if its numerator is so inZ.
The even numbers form anideal in thering of integers,[13] but the odd numbers do not—this is clear from the fact that theidentity element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2.
Allprime numbers are odd, with one exception: the prime number 2.[14] All knownperfect numbers are even; it is unknown whether any odd perfect numbers exist.[15]
Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. Moderncomputer calculations have shown this conjecture to be true for integers up to at least 4 × 1018, but still no generalproof has been found.[16]
Theparity of a permutation (as defined inabstract algebra) is the parity of the number oftranspositions into which the permutation can be decomposed.[17] 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. InRubik'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 theconfiguration space of these puzzles.[18]
TheFeit–Thompson theorem states that afinite 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.[19]
Theparity 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 casef(x) = 0, to be both odd and even.[20] TheTaylor 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.[21]
Incombinatorial game theory, anevil number is a number that has an even number of 1's in itsbinary representation, and anodious 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 gameKayles.[22] Theparity 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. TheThue–Morse sequence, an infinite sequence of 0's and 1's, has a 0 in positioni wheni is evil, and a 1 in that position wheni is odious.[23]
Ininformation theory, aparity bit appended to a binary number provides the simplest form oferror 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.[24] 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.[25]
Inwind instruments with a cylindrical bore and in effect closed at one end, such as theclarinet at the mouthpiece, theharmonics produced are odd multiples of thefundamental frequency. (With cylindrical pipes open at both ends, used for example in someorgan stops such as theopen 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.) Seeharmonic series (music).[26]
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.[27] Similarly, amongUnited States numbered highways, even numbers primarily indicate east–west highways while odd numbers primarily indicate north–south highways.[28] Among airlineflight numbers, even numbers typically identify eastbound or northbound flights, and odd numbers typically identify westbound or southbound flights.[29]
^Owen, Ruth L. (1992),"Divisibility in bases"(PDF),The Pentagon: A Mathematics Magazine for Students,51 (2):17–20, archived fromthe original(PDF) on 2015-03-17.
^Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio (2013), "Empirical verification of the even Goldbach conjecture, and computation of prime gaps, up to 4·1018",Mathematics of Computation,83 (288):2033–2060,doi:10.1090/s0025-5718-2013-02787-1. In press.
^Bender, Helmut; Glauberman, George (1994),Local analysis for the odd order theorem, London Mathematical Society Lecture Note Series, vol. 188, Cambridge: Cambridge University Press,ISBN978-0-521-45716-3,MR1311244;Peterfalvi, Thomas (2000),Character theory for the odd order theorem, London Mathematical Society Lecture Note Series, vol. 272, Cambridge: Cambridge University Press,ISBN978-0-521-64660-4,MR1747393.
^Guy, Richard K. (1996), "Impartial games",Games of no chance (Berkeley, CA, 1994), Math. Sci. Res. Inst. Publ., vol. 29, Cambridge: Cambridge Univ. Press, pp. 61–78,MR1427957. See in particularp. 68.
