Additive inverse
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The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. The additive inverse of n is denoted −n.
For example:
- The additive inverse of 7 is −7, because 7 + (−7) = 0;
- The additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0.
Thus by the last example, −(−0.3) = 0.3.
The additive inverse of a number is its inverse element under the binary operation of addition. It can be calculated using multiplication by −1; that is, −n = −1 × n.
Types of numbers with additive inverses include:
Types of numbers without additive inverses (of the same type) include:
But note that we can construct the integers out of the natural numbers by formally including additive inverses. Thus 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.
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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.
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).
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),
Universal construction
to do: symmetrization of an abelian semigroup