Even and odd numbers

In mathematics, the parity of an object refers to whether it is even or odd. In particular, any integer is either even or odd according to the following rules: If it is a multiple of two, it is an even number; otherwise, it is an odd number. Examples of even numbers are −4, 8, 0, and 70. Examples of odd numbers are −5, 1, and 71. The number zero is even, because it is equal to two multiplied by zero.

The set of even numbers can be written:

Evens = 2Z = {..., −6, −4, −2, 0, 2, 4, 6, ...}.

The set of odd numbers can be shown like this:

Odds = 2Z + 1 = {..., −5, −3, −1, 1, 3, 5, ...}.

Results using parity

A number 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 or odd according to the sum of its digits - it is even if and only if the sum of its digits is even.

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. 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.

All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even; it is unknown whether any odd perfect numbers exist.

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 × 10¹⁴, but still no general 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.

In wind instruments which are cylindrical and in effect closed at one end, such as the clarinet at the mouthpiece, the harmonics produced are odd multiples of the fundamental frequency. (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, but this is the same as being all multiples of double the frequency and is usually perceived as such.) See harmonic series (music).

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

The rules analogous to these for divisibility by 9 are used in the method of casting out nines.

even ± even = even;
even ± odd = odd;
odd ± odd = even.

Multiplication

These rules only hold because 2 is a prime number; the analogous rules for divisibility by a composite number would be more complex.

even × even = even;
even × odd = even;
odd × odd = odd.

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 iff the numerator has more factors of two than the denominator.

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 (even or odd) of the number of transpositions 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 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.