Definition
In mathematics, the additive inverse of a number a is the unique element ‑a such that the sum of the two elements equals the additive identity (zero):
$$ a + (-a) = 0. $$
The concept applies within any algebraic structure that possesses an addition operation and a distinguished additive identity, such as groups, rings, fields, and vector spaces.
Overview
The additive inverse provides a means of “undoing” addition. In the set of real numbers ℝ, the additive inverse of a positive number is its negative counterpart, and vice‑versa; the additive inverse of zero is zero itself. In modular arithmetic, the additive inverse of an element a modulo n is the element b satisfying a + b ≡ 0 (mod n), commonly computed as b = n – a when a ≠ 0. In abstract algebra, the existence of additive inverses is a defining property of an abelian group under addition.
Etymology/Origin
The term combines “additive,” relating to the operation of addition, with “inverse,” derived from Latin inversus meaning “turned opposite.” The notion of an additive inverse appears in early works on algebraic structures, notably in the development of group theory in the 19th century by mathematicians such as Évariste Galois and Arthur Cayley.
Characteristics
- Uniqueness: In any structure where addition is defined, each element has exactly one additive inverse.
- Involution: The additive inverse of an additive inverse returns the original element: $-(-a) = a$.
- Compatibility with scalar multiplication: In vector spaces, for any scalar λ and vector v, $-(\lambda v) = \lambda (-v) = (-\lambda) v$.
- Distribution over addition: The additive inverse of a sum equals the sum of the additive inverses: $-(a + b) = (-a) + (-b)$.
- Zero element: The additive identity (zero) is its own additive inverse.
Related Topics
- Additive identity (zero)
- Group theory, particularly abelian groups
- Rings and fields
- Modular arithmetic
- Subtraction as addition of the additive inverse
- Multiplicative inverse (reciprocal)
- Vector spaces and linear algebra