Parity

Parity is a property used to classify objects into two complementary groups, often called even and odd. In mathematics the most common use is the parity of integers: an integer n is even if it is divisible by 2, and odd otherwise. The parity of n is often captured by n mod 2, taking values 0 (even) or 1 (odd).

Arithmetic rules: parity is compatible with addition and multiplication mod 2. The sum of two even numbers or two odd numbers is even; the sum of an even and an odd is odd. A product is even if at least one factor is even; the product of two odd numbers is odd. These rules extend to congruences modulo 2 and underlie simple parity arguments in proofs.

Parity of a permutation: the sign of a permutation is either +1 (even) or −1 (odd). A permutation is even if it can be decomposed into an even number of transpositions and odd otherwise. This parity determines the sign of the permutation in the determinant of a matrix.

In physics and geometry, parity refers to spatial inversion: the parity transformation maps coordinates x to −x. A system is parity-symmetric if its laws are invariant under this transformation. Parity is a conserved quantum number in strong and electromagnetic interactions, but it is violated in the weak interaction, a discovery that has deep implications in particle physics.

In computer science, parity is used in error detection with parity bits. An even-parity bit ensures an even number of 1s; an odd-parity bit ensures an odd number of 1s. XOR operations calculate parity and are used in parity checks in data transmission and storage. The notion of parity also appears in various mathematical and combinatorial contexts modulo 2.