invers

Invers is a term used across languages and disciplines to denote reversal or reciprocity, though in English the preferred form is inverse or inversion. As a standalone English word, invers is uncommon, and the term more often appears as a root within other words or as part of proper names. In discussions of mathematics, science, and computing, the concept of the inverse is central, referring to an operation, element, or transform that undoes or reverses another.

In mathematics, an inverse is something that undoes another operation or maps results back to the original input. An inverse function f^{-1} satisfies f^{-1}(f(x)) = x and f(f^{-1}(y)) = y. An additive inverse of a is -a, since a + (-a) = 0. A multiplicative inverse of a nonzero number a is 1/a, since a · (1/a) = 1. In algebraic structures with a notion of multiplication and identity, the inverse of an element a is another element b such that a·b equals the identity. In linear algebra, a square matrix A has an inverse A^{-1} when det(A) ≠ 0, with AA^{-1} = A^{-1}A = I. The concept extends to inverse problems, where one infers causes from observed effects.

In computing and digital systems, bitwise inversion is an operation that flips each bit, often denoted by the NOT operator. In signal processing and physics, inverse transforms, such as the inverse Fourier transform, reconstruct original signals from their frequency components. The word invers is also used in some languages and contexts as a cognate of inverse or inversion, and may appear in proper names.

Etymologically, invers derives from Latin inversus, meaning turned inside out. In English, inverse and inversion are the standard terms, while invers appears mainly as a linguistic form or in names. See also inverse, inversion, reciprocal.