Additive Inverse - General Definition

General Definition

The notation + is usually reserved for commutative binary operations, i.e. such that x + y = y + x, for all x, y . If such an operation admits an identity element o (such that x + o ( = o + x ) = x for all x), then this element is unique ( o′ = o′ + o = o ). For a given x , if there exists x′ such that x + x′ ( = x′ + x ) = o , then x′ is called an additive inverse of x.

If + is associative (( x + y ) + z = x + ( y + z ) for all x, y, z), then an additive inverse is unique

x″ = x″ + o = x″ + (x + x′) = (x″ + x) + x′ = o + x′ = x′

We often write x − y as x + (−y).

For example, since addition of real numbers is associative, each real number has a unique additive inverse.