Parity (mathematics)
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In [[mathematics]], the '''parity''' of an object states whether it is even or odd.
This concept begins with [[integer]]s. An '''even number''' is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an '''odd number''' is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "[[Divisibility|divisible]]".)
A formal definition of an odd number is that it is an integer of the form ''n'' = 2''k'' + 1, where ''k'' is an integer. An even number has the form ''n'' = 2''k'' where ''k'' is an [[integer]].
Examples of even numbers are −4, 8, 0, and 42. Examples of odd numbers are −3, 9, 1, and 5. A fractional number like 1/2 or 3.141 is neither even nor odd.
The [[set]] of even numbers can be written:
: {Evens} = 2'''Z''' = {..., −6, −4, −2, 0, 2, 4, 6, ...},
where '''Z''' is the set of all integers. The set of odd numbers can be shown like this:
: {Odds} = 2'''Z''' + 1 = {..., −5, −3, −1, 1, 3, 5, ...}.
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.
That is, if the last digit is 1, 3, 5, 7, or 9, then it's odd; otherwise it's even. 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 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.
== Arithmetic on even and odd numbers ==
The following laws can be verified using the properties of [[divisibility]]. 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.
=== [[Addition]] and [[subtraction]] ===
*even ± even = even;
* even ± odd = odd;
* odd ± odd = even;
Rules analogous to these for divisibility by 9 are used in the method of [[casting out nines]].
===[[Multiplication]]===
* even × even = even;
* even × odd = even;
* odd × odd = odd.
=== [[Division (mathematics)|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 isn't even ''or'' odd, since the concepts even and odd apply only to integers.
But when the quotient is an integer, it will be even [[if and only if]] the [[division (mathematics)|dividend]] has more factors of two than the divisor.
==History==
The ancient Greeks considered 1 to be neither fully odd nor fully even. Some of this sentiment survived into the 19th century: [[Friedrich Wilhelm August Froebel]]'s 1826 ''[[The Education of Man]]'' instructs the teacher to drill students with the claim that 1 is neither even nor odd, to which Froebel attaches the philosophical afterthought,
{{blockquote|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.}}
In the 18th century, some mathematicians wrote that [[infinity]] was neither even nor odd while arguing that [[Grandi's series]] {{nowrap|1=1 − 1 + 1 − 1 + · · ·}} equaled 1/2.
==Music theory==
In [[wind instrument]]s which are cylindrical and in effect closed at one end, such as the [[clarinet]] at the mouthpiece, the [[harmonic]]s produced are odd multiples of the [[fundamental frequency]]. (With cylindrical pipes open at both ends, used for example in some [[organ stop]]s such as the [[Flue pipe#Diapason|open diapason]], the harmonics are even multiples of the same frequency, but this is the same as being all multiples of double the frequency and is usually perceived as such.) See [[harmonic series (music)]].
==Higher mathematics==
The even numbers form an [[ring ideal|ideal]] in the [[ring (algebra)|ring]] of integers, but the odd numbers do not — this is clear from the fact that the [[Identity (mathematics)|identity]] element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 [[modular arithmetic|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]]s are odd, with one exception: the prime number 2. All known [[perfect number]]s are even; it is unknown whether any odd perfect numbers exist.
The squares of all even numbers are even, and the squares of all odd numbers are odd. Since an even number can be expressed as 2x, (2x)<sup>2</sup> = 4x<sup>2</sup> which is even. Since an odd number can be expressed as 2x + 1, (2x + 1)<sup>2</sup> = 4x<sup>2</sup> + 4x + 1. 4x<sup>2</sup> and 4x are even, which means that 4x<sup>2</sup> + 4x + 1 is odd (since even + odd = odd).
[[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<sup>14</sup>, but still no general [[mathematical proof|proof]] has been found.
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.
==Parity for other objects==
Parity is also used to refer to a number of other properties.
*The parity of a [[permutation]] (as defined in [[abstract algebra]]) is the parity of the number of [[Transposition (mathematics)|transposition]]s 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. See the article on [[even and odd permutations]] for an elaboration.
*The [[Even and odd functions|parity of a function]] describes how its values change when its arguments are exchanged with their negatives. An even function, such as an even power of a variable, gives the same result for positive or negative arguments. An odd function, such as an odd power of a variable, gives the negative of its result when given the negative of an 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.
==See also==
*[[Evenness of zero]]
*[[Even and odd functions]]
*[[Even and odd permutations]]
*[[Thue-Morse sequence]]
*[[Flight number]]
*[[House numbering]]
*[[United States Numbered Highways]]
[[Category:Parity|*]]
[[Category:Elementary arithmetic]]
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