In mathematics, the additive inverse, or opposite, of a number n is the number that, when added to n, yields zero. The additive inverse of n is denoted −n.
The additive inverse of a number is defined as its inverse element under the binary operation of addition. It can be calculated using multiplication by −1; that is, −n = −1 × n.
Integers, rational numbers, real numbers, and complex number all have additive inverses, as they contain negative as well as positive numbers. Natural numbers, cardinal numbers, and ordinal numbers, on the other hand, do not have additive inverses within their respective sets. Thus, for example, we can say that natural numbers do have additive inverses, but because these additive inverses are not themselves natural numbers, the set of natural numbers is not closed under taking additive inverses.
The notation '+' is reserved for commutative binary operations, i.e. such that x + y = y + x, for all x,y. If such an operation admits a neutral element o (such that x + o (= o + x) = x for all x), then this element is unique (o' = o' + o = o). If then, for a given x, there exists x' such that x + x' (= x' + x) = o, then x' is called an additive inverse of x. This additive inverse is unique for every real number.