In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not. For example, −4, 0, and 82 are even numbers, while −7, 5, and 23 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with fractions or decimals such as 1⁄2 or 4.6978. See § Higher mathematics 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, for example 22 (an even number) and 13 (an odd number). In particular, the parity of zero is even. Any two consecutive integers have opposite parity. An 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 is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8. This concept applies to any even base system: any integer 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 if and only if the sum of its digits is even; for example, in base seven, 11 (equivalent to 8 in base ten) is even (because Σ = 2 {\displaystyle \Sigma =2} ) and 124 (equivalent to 67 in base ten) is odd (because Σ = 7 {\displaystyle \Sigma =7} ).
Definition An even number is an integer of the form
x = 2 k {\displaystyle x=2k}
where k is an integer; an odd number is an integer of the form
x = 2 k + 1. {\displaystyle x=2k+1.}
An equivalent definition is that an even number is divisible by 2:
2 | x {\displaystyle 2\ |\ x}
and an odd number is not:
2 ⧸ | x {\displaystyle 2\not |\ x}
The sets of even and odd numbers can be defined as following:
{ 2 k : k ∈ Z } {\displaystyle \{2k:k\in \mathbb {Z} \}}
{ 2 k + 1 : k ∈ Z } {\displaystyle \{2k+1:k\in \mathbb {Z} \}}
The set of even numbers is a prime ideal of Z {\displaystyle \mathbb {Z} } and the quotient ring Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } is the field with two elements. Parity can then be defined as the unique ring homomorphism from Z {\displaystyle \mathbb {Z} } to Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } 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. 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. 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. By construction in the previous section, the structure ({even, odd}, +, ×) is in fact the 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 is an integer, it will be even if and only if the dividend has more factors of two than the divisor.
History The ancient Greeks considered 1, the monad, to be neither fully odd nor fully even. 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. 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.
Higher mathematics
Higher dimensions and more general classes of numbers
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