In mathematics, the additive inverse of a numbera (sometimes called the opposite of a)[1] is the number that, when added to a, yields zero. The operation taking a number to its additive inverse is known as sign change[2] or 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 unaryminus: −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 a − b is −(a − b) which can be simplified to b − a. The additive inverse of 2x − 3 is 3 − 2x, because 2x − 3 + 3 − 2x = 0.[5]
Additive inverse is closely related to subtraction, which can be viewed as an addition of the opposite:
a − b = a + (−b).
Conversely, additive inverse can be thought of as subtraction from zero:
−a = 0 − a.
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:
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 elemento (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
Addition of real- and complex-valued functions: here, the additive inverse of a function f is the function −f defined by (−f )(x) = − f (x), for all x, such that f + (−f ) = o, the zero function (o(x) = 0 for all x).
More generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the identity element of this group):
In modular arithmetic, the modular additive inverse of x is also defined: it is the number a such that a + x ≡ 0 (mod n). This additive inverse always exists. For example, the inverse of 3 modulo 11 is 8 because it is the solution to 3 + x ≡ 0 (mod 11).
Non-examples
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.
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