Additive inverse

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number),[1] sign change,[2] and negation.[3] 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 a is denoted by unary minus: a (see also § Relation to subtraction below).[4] For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0.

Similarly, the additive inverse of ab is −(ab) which can be simplified to ba. The additive inverse of 2x − 3 is 3 − 2x, because 2x − 3 + 3 − 2x = 0.[5]

The additive inverse is defined as its inverse element under the binary operation of addition (see also § Formal definition below), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect: −(−x) = x.

For a number (and more generally in any ring), the additive inverse can be calculated using multiplication by −1; that is, n = −1 × n. Examples of rings of numbers are integers, rational numbers, real numbers, and complex numbers.

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 "−".

In addition to the identities listed above, negation has the following algebraic properties:

The notation + is usually reserved for commutative binary operations (operations where 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, i.e., (x + y) + z = x + (y + z) for all x, y, z, then an additive inverse is unique. To see this, let x′ and x″ each be additive inverses of x; then

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

Natural numbers, cardinal numbers and ordinal numbers do not have additive inverses within their respective sets. Thus one can say, for example, 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.