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 elementInverse elementIn 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 inverseAdditive inverseIn 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 inverseMultiplicative inverseIn 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 functionInverse functionIn 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.
- the additive inverse
- 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 limitInverse limitIn 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 p → q. - Inverse matrix — inverse element with respect to matrix multiplication.
- PseudoinversePseudoinverseIn 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 problemInverse problemAn 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 problemInverse scattering problemIn 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 kinematicsInverse kinematicsInverse 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 dynamicsInverse dynamicsInverse 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 scattering problem
- Inverse perspectivePerspective (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 geometryInversive ring geometryIn 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 semigroupInverse semigroupIn 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 semigroupSemigroupIn 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 lawInverse-square lawIn 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 relationInverse relationIn 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'...