In mathematics, the additive inverse of a number is the number that, when added to, yields zero. This number is also known as the opposite, sign change, and negation. For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself. The additive inverse of is denoted by unary minus: −. For example, the additive inverse of 7 is −7, because 7 + = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0 . The additive inverse is defined as its inverse element under the binary operation of addition, which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect:. , are mutually opposite
Additive inverse is closely related to subtraction, which can be viewed as an addition of the opposite: Conversely, additive inverse can be thought of as subtraction from zero: Hence, unary minus sign notation can be seen as a shorthand for subtraction with "0" symbol omitted, although in a correct typography there should be no space after unary "−".
Other properties
In addition to the identities listed [|above], negation has the following algebraic properties:
The notation + is usually reserved for commutativebinary operations; i.e., such that, for all, . If such an operation admits an identity element , then this element is unique. For a given , if there exists such that , then is called an additive inverse of. If + is associative, then an additive inverse is unique. To see this, let and each be additive inverses of ; then For example, since addition of real numbers is associative, each real number has a unique additive inverse.
Other examples
All the following examples are in fact abelian groups:
addition of real- and complex-valued functions: here, the additive inverse of a function is the function − defined by , for all, such that , the zero function.
more generally, what precedes applies to all functions with values in an abelian group :
In a vector space the additive inverse is often called the opposite vector of ; it has the same magnitude as the original and opposite direction. Additive inversion corresponds to scalar multiplication by −1. For Euclidean space, it is point reflection in the origin. Vectors in exactly opposite directions are sometimes referred to as antiparallel.
* vector space-valued functions,
In modular arithmetic, the modular additive inverse of is also defined: it is the number such that. This additive inverse always exists. For example, the inverse of 3 modulo 11 is 8 because it is the solution to.
Non-examples
s, cardinal numbers, and ordinal numbers, do not have additive inverses within their respective sets. Thus, for example, we can say that natural numbersdo 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.
