Additive inverse
From Wikipedia, the free encyclopedia
 |
This article does not cite any references or sources. (December 2007) Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed. |
In mathematics, the additive inverse, or opposite, of a number n is the number that, when added to n, yields zero. The additive inverse of F is denoted −F.
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.
The additive inverse of a number is defined as its inverse element under the binary operation of addition. It can be calculated using multiplication by −1; that is, −n = −1 × n.
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.
[edit] General definition
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 a neutral 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. This additive inverse is unique for every real number.
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' )
and denoted by (– x), and one can write x – y instead of x + (– y).
[edit] Other examples
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 null function (constantly equal to zero, for all arguments).
- more generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the neutral 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).
