Parity (mathematics)
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.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.By construction in the previous section, the structure ({even, odd}, +, ×) is in fact the field with two elements.[7] 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, 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.A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws.[10] 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.[15] 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 × 1018, but still no general proof has been found.It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions.[18] The Feit–Thompson theorem states that a finite group is always solvable if its order is an odd number.[19] 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.[20] 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.[23] In information theory, a parity bit appended to a binary number provides the simplest form of error detecting code.[25] 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 (disambiguation)Odd Number (film)Cuisenaire rodsmathematicspropertyintegerdivisibleparity of zerodecimalnumeral systembinary numeral systemprime idealquotient ringfield with two elementsring homomorphismdivisibilitymodular arithmeticquotientif and only ifdividendfactors of twoFriedrich Wilhelm August FröbelbishopsknightEuclidean spacesface-centered cubic latticelatticesknightsmutilated chessboard problemparity of an ordinal numbercommutative ringlocalizationidentitymoduloprime numbersperfect numbersGoldbach's conjecturecomputerRubik's Revengeparity of a permutationabstract algebratranspositionsRubik's CubeMegaminxconfiguration spaceFeit–Thompson theoremfinite groupparity of a functionTaylor seriescombinatorial game theorybinary representationKaylesparity functionmodulo 2Thue–Morse sequenceinformation theoryparity biterror detecting codewind instrumentsclarinetharmonicsfundamental frequencyorgan stopsharmonic series (music)house numberingsUnited States numbered highwaysflight numbersDivisorHalf-integerPólya, GeorgeTarjan, Robert E.Pandolfini, BruceStillwell, JohnDudley, UnderwoodMathematical CranksCameron, Peter J.Guy, Richard K.