Number that, when added to the original number, yields the additive identity
In mathematics, the additive inverse of an element x, denoted -x, is the element that when added to x, yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero element.
In elementary mathematics, the additive inverse is often referred to as the opposite number, or its negative. The unary operation of arithmetic negation is closely related to "subtraction" and is important in solving algebraic equations. Not all sets where addition is defined have an additive inverse, such as the natural numbers.
Common examples.
When working with integers, rational numbers, real numbers, and complex numbers, the additive inverse of any number can be found by multiplying it by −1.
The concept can also be extended to algebraic expressions, which is often used when balancing equations.
Relation to subtraction.
The additive inverse is closely related to subtraction, which can be viewed as an addition using the inverse:
"a" − "b"  =  "a" + (−"b").
Conversely, the additive inverse can be thought of as subtraction from zero:
−"a" = 0 − "a".
This connection lead to the minus sign being used for both opposite magnitudes and subtraction as far back as the 17th century. While this notation is standard today, it was met with opposition at the time, as some mathematicians felt it could be unclear and lead to errors.
Formal definition.
Given an algebraic structure defined under addition formula_1 with an additive identity formula_2, an element formula_3 has an additive inverse formula_4 if and only if formula_5, formula_6, and formula_7.
Addition is typically only used to refer to a commutative operation, but it is not necessarily associative. When it is associative, so formula_8, the left and right inverses, if they exist, will agree, and the additive inverse will be unique. In non-associative cases, the left and right inverses may disagree, and in these cases, the inverse is not considered to exist.
The definition requires closure, that the additive element formula_4 be found in formula_10. This is why despite addition being defined over the natural numbers, it does not an additive inverse for its members. The associated inverses would be negative numbers, which is why the integers do have an additive inverse.
Notes and references.
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