Additive inverse
In mathematics, the additive inverse of a number (sometimes called the opposite of ) is the number that, when added to , yields zero. The operation taking a number to its additive inverse is known as sign change or negation. For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself. The additive inverse of is denoted by unary minus: (see also below). For example, the additive inverse of 7 is −7, because , and the additive inverse of −0.3 is 0.3, because . Similarly, the additive inverse of is which can be simplified to . The additive inverse of is , because . The additive inverse is defined as its inverse element under the binary operation of addition (see also below), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect: . Common examples For a number (and more generally in any ring), the additive inverse can be calculated using multiplication by −1; that is, . Examples of rings of numbers are integers, rational numbers, real numbers, and complex numbers. Relation to subtraction 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 the "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: , it is an Involution operation notably, Formal definition The notation + is usually reserved for commutative binary operations (operations where for all , ). If such an operation admits an identity element (such that for all ), then this element is unique (). For a given , if there exists such that , then is called an additive inverse of . If + is associative, i.e., for all , , , 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: Complex numbers: . On the complex plane, this operation rotates a complex number 180 degrees around the origin (see the image above). 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 ( for all ). More generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the identity element of this group): Sequences, matrices and nets are also special kinds of functions. In a vector space, the additive