- Additive inverse
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.
In other words, the additive inverse of a number is the number's negative. For example, the additive inverse of 8 is −8, the additive inverse of 10002 is −10002 and the additive inverse of x² is −(x²).
Integers, rational numbers, real numbers, and complex number all have additive inverses, as they contain negative as well as positive numbers. Natural numbers, cardinal numbers, and ordinal numbers, on the other hand, do not have additive inverses within their respective sets. Thus, for example, we can say 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.
The notation '+' is reserved for commutative binary operations, i.e. such that 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). If then, for a given x, there exists x' such that x + x' (= x' + x) = o, then x' is called an additive inverse of x.
If '+' is associative ((x+y)+z = x+(y+z) for all x,y,z), then an additive inverse is unique
- ( x" = x" + o = x" + (x + x') = (x" + x) + x' = o + x' = x' )
– y instead of x + (–y).
For example, since addition of real numbers is associative, each real number has a unique additive inverse.
All the following examples are in fact abelian groups:
- addition of real 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):
- complex valued functions,
- vector space valued functions (not necessarily linear),
- sequences, matrices and nets are also special kinds of functions.
- In a vector space additive inversion corresponds to scalar multiplication by −1. For Euclidean space, it is inversion in the origin.
- 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 does always exist. For example, the inverse of 3 modulo 11 is 8 because it is the solution to 3+x ≡ 0 (mod 11).
Wikimedia Foundation. 2010.
Look at other dictionaries:
additive inverse — the number that must be added to a number to equal zero [−5 is the additive inverse of 5] … English World dictionary
additive inverse — noun (mathematics) one of a pair of numbers whose sum is zero; the additive inverse of 5 is +5 • Topics: ↑mathematics, ↑math, ↑maths • Hypernyms: ↑inverse, ↑opposite * * * noun … Useful english dictionary
additive inverse — noun The inverse with respect to addition; the opposite. The additive inverse of 12 is −12. Syn: negative, opposite … Wiktionary
additive inverse — noun Date: 1953 a number that when added to a given number gives zero < the additive inverse of 4 is 4 > compare opposite 3 … New Collegiate Dictionary
additive inverse — Math. the number in the set of real numbers that when added to a given number will yield zero: The additive inverse of 2 is 2. [1955 60] * * * … Universalium
Additive — may refer to:* Additive function, a function which preserves addition * Additive inverse, an arithmetic concept * Additive category, a preadditive category with finite biproducts * Additive rhythm, a larger period of time constructed from smaller … Wikipedia
Inverse (mathematics) — Inverse is the opposite of something. This word and its derivatives are used greatly in mathematics, as illustrated below. * Inverse element of an element x with respect to a binary operation * with identity element e is an element y such that x… … Wikipedia
Additive category — In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A 1,..., A n of C have a biproduct A 1 ⊕ ⋯ ⊕ A n in C. (Recall that a category C is preadditive if all its… … Wikipedia
Additive identity — In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x . One of the most familiar additive identities is the number 0 from elementary… … Wikipedia
Inverse element — In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can undo the effect of combination with… … Wikipedia