reciprocal

Reciprocal, in mathematics, refers to the multiplicative inverse of a number: the quantity that, when multiplied by the original, yields 1. For any nonzero x, its reciprocal is 1/x. This is also called the inverse with respect to multiplication.

Examples include: the reciprocal of 2 is 1/2; the reciprocal of -3 is -1/3; the reciprocal of 0.25 is 4. For a fraction a/b with a ≠ 0, the reciprocal is b/a. The reciprocal of a product follows the rule (xy)^{-1} = x^{-1} y^{-1} for commuting quantities; with matrices or other noncommuting objects, (AB)^{-1} = B^{-1} A^{-1}. The reciprocal of a reciprocal is the original number: (1/x)^{-1} = x. The reciprocal of zero is undefined.

The reciprocal function f(x) = 1/x is a standard example in algebra and analysis. Its domain is all real numbers except 0, and its range is all real numbers except 0; it is not defined at x = 0. In linear algebra, the multiplicative inverse of a nonzero square matrix A is denoted A^{-1}, satisfying AA^{-1} = A^{-1}A = I; such an inverse exists only when det(A) ≠ 0.

Reciprocal concepts appear across mathematics and science, aiding in solving equations, simplifying fractions, and analyzing rates and units (for example, per-unit quantities). The idea of a reciprocal emphasizes that a quantity and its reciprocal cancel to give a neutral product of 1.