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Concept
Additive inverse
Summary
In mathematics, the additive inverse of a number a (sometimes called the opposite of a) is the number that, when added to a, 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 a is denoted by unary minus: −a (see also below). 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.
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: −(−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:
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 "−".
In addition to the identities listed above, negation has the following algebraic properties:
−(−a) = a, it is an Involution operation
−(a + b) = (−a) + (−b)
−(a − b) = b − a
a − (−b) = a + b
(−a) × b = a × (−b) = −(a × b)
(−a) × (−b) = a × b
notably, (−a)2 = a2
The notation + is usually reserved for commutative binary operations (operations where x + y = y + x for all x, y).
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