Inverse

Inverse or invert may refer to:

Science, engineering and mathematics

See also

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

In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) 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. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group.

The word 'inverse' is derived from Latin: inversus that means 'turned upside down', 'overturned'.

Formal definitions

In a unital magma

Let be a set with a binary operation (i.e., a magma note that a magma also has closure under the binary operation). If is an identity element of (i.e., S is a unital magma) and , then is called a left inverse of and is called a right inverse of . If an element is both a left inverse and a right inverse of , then is called a two-sided inverse, or simply an inverse, of . An element with a two-sided inverse in is called invertible in . An element with an inverse element only on one side is left invertible, resp. right invertible. A unital magma in which all elements are invertible is called a loop. A loop whose binary operation satisfies the associative law is a group.