Additive Inverse - Other Examples

Other Examples

All the following examples are in fact abelian groups:

• addition of real valued functions: here, the additive inverse of a function f is the function −f defined by (−f )(x) = − f (x) , for all x, such that f + (−f ) = o , the zero function ( o(x) = 0 for all x ).
• more generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the identity element of this group):
  • complex valued functions,
  • vector space valued functions (not necessarily linear),
• sequences, matrices and nets are also special kinds of functions.
• In a vector space the additive inverse −v is often called the opposite vector of v. Additive inversion corresponds to scalar multiplication by −1. For Euclidean space, it is inversion in the origin.
• In modular arithmetic, the modular additive inverse of x is also defined: it is the number a such that a + x ≡ 0 (mod n). This additive inverse does always exist. For example, the inverse of 3 modulo 11 is 8 because it is the solution to 3 + x ≡ 0 (mod 11).