Editing Additive inverse (section)

== Formal definition ==
Given an algebraic structure defined under addition <math>(S, +)</math> with an additive identity <math>e \in S</math>, an element <math>x \in S</math> has an additive inverse <math>y</math> if and only if <math>y \in S</math>, <math>x + y = e</math>, and <math>y + x = e</math>.<ref name=":1" />

Addition is typically only used to refer to a [[Commutative property|commutative]] operation, but it is not necessarily [[Associative property|associative]]. When it is associative, so <math>(a + b) + c = a + (b + c)</math>, the left and right inverses, if they exist, will agree, and the additive inverse will be unique. In non-associative cases, the left and right inverses may disagree, and in these cases, the inverse is not considered to exist.

The definition requires [[Closure (mathematics)|closure]], that the additive element <math>y</math> be found in <math>S</math>. However, despite being able to add the natural numbers together, the set of natural numbers does not include the additive inverse values. This is because the additive inverse of a natural number (e.g., <math>-3</math> for <math>3</math>) is not a natural number; it is an [[integer]]. Therefore, the natural numbers in set <math>S</math> do have additive inverses and their associated inverses are [[negative number]]s.