Inverse (mathematics)
Encyclopedia


In many contexts in mathematics the term inverse indicates the opposite of something. This word and its derivatives are used widely in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, as illustrated below.
  • Inverse element
    Inverse element
    In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given 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
      Additive inverse
      In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....

       of x is –x;
    • the multiplicative inverse
      Multiplicative inverse
      In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

       or reciprocal of x is x–1.
    • inverse function
      Inverse function
      In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

       — 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
    Inverse limit
    In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...

     — a notion in abstract algebra.
  • Inverse (logic)
    Inverse (logic)
    In traditional logic, an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence. Any conditional sentence has an inverse: the contrapositive of the converse. The inverse of P \rightarrow Q is thus \neg P \rightarrow \neg Q...

     — ~p → ~q is the inverse of pq.
  • Inverse matrix — inverse element with respect to matrix multiplication.
  • Pseudoinverse
    Pseudoinverse
    In mathematics, and in particular linear algebra, a pseudoinverse of a matrix is a generalization of the inverse matrix. The most widely known type of matrix pseudoinverse is the Moore–Penrose pseudoinverse, which was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951 and...

    , a generalization of the inverse matrix.
  • Inverse proportion, also inversely proportional — a relationship between two variables x and y characterized by the equation
  • Inverse problem
    Inverse problem
    An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in...

     — the task of identifying model parameters from observed data; see for example
    • inverse scattering problem
      Inverse scattering problem
      In the physics field of of scattering theory, the inverse scattering problem is that of determining characteristics of an object based on data of how it scatters incoming radiation or particles....

    • inverse kinematics
      Inverse kinematics
      Inverse kinematics is a subdomain of kinematics, which is of particular interest in robotics and computer animation. In contrast to forward kinematics, which calculates the position of a body after a series of motions, inverse kinematics calculates the motions necessary to achieve a desired...

    • inverse dynamics
      Inverse dynamics
      Inverse dynamics is an inverse problem. It commonly refers to either inverse rigid body dynamics or inverse structural dynamics. Inverse rigid-body dynamics is a method for computing forces and/or moments of force based on the kinematics of a body and the body's inertial properties...

      .
  • Inverse perspective
    Perspective (graphical)
    Perspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface , of an image as it is seen by the eye...

     — the further the objects, the larger they are drawn.
  • Inversive ring geometry
    Inversive ring geometry
    In mathematics, inversive ring geometry is the extension of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio to the context of associative rings, concepts usually built upon rings that happen to be fields....

     — classical projective geometry extended by ring theory
  • Inverse semigroup
    Inverse semigroup
    In mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy...

  • Inverse of an element in a semigroup
    Semigroup
    In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...

  • Inverse-square law
    Inverse-square law
    In physics, an inverse-square law is any physical law stating that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity....

     — 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
    Inverse relation
    In mathematics, the inverse relation of a binary relation is the relation that occurs when you switch the order of the elements in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'...


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