Inverse (mathematics)

In many contexts in mathematics the term inverse indicates the opposite of something. This word and its derivatives are used widely in mathematics, as illustrated below.

• Inverse element of an element x with respect to a binary operation * with identity element e is an element y such that x * y = y * x = e. In particular,
  • the additive inverse of x is –x;
  • the multiplicative inverse or reciprocal of x is x^–1.
  • inverse function — inverse element with respect to function composition: a function that "reverses" the action of a given function: f^–1(f(x)) = x.
• Inversion in a point — a geometric transform.
• Circle inversion — another particular geometric transformation of a plane that maps the outside of a circle to the inside and vice-versa.
• Inverse limit — a notion in abstract algebra.
• Inverse (logic) — ~p → ~q is the inverse of p → q.
• Inverse matrix — inverse element with respect to matrix multiplication.
• Pseudoinverse, a generalization of the inverse matrix.
• Inverse proportion, also inversely proportional — a relationship between two variables x and y characterized by the equation y = k / x.
• Inverse problem — the task of identifying model parameters from observed data; see for example
  • inverse scattering problem
  • inverse kinematics
  • inverse dynamics.
• Inverse perspective — the further the objects, the larger they are drawn.
• Inversive ring geometry — classical projective geometry extended by ring theory
• Inverse semigroup
• Inverse of an element in a semigroup
• Inverse-square law — the magnitude of a force is proportional to the inverse square of the distance.
• Inverse transform sampling — generate some random numbers according to a given probability distribution.
• Inverse chain rule method — related to integration and differentiation.
• Inversion of elements, a pair of adjacent out-of-order elements of a permutation (viewed as a list).
• Inverse relation
• Inversion transformation, an extension of Poincaré transformation.

See also Inversion.