inverse

Dictionary: in·verse (ĭn-vûrs', ĭn'vûrs') pronunciation

adj.

Reversed in order, nature, or effect.
Mathematics. Of or relating to an inverse or an inverse function.
Archaic. Turned upside down; inverted.

n. (ĭn'vûrs', ĭn-vûrs')

Something that is opposite, as in sequence or character; the reverse.
Mathematics. One of a pair of elements in a set whose result under the operation of the set is the identity element, especially:
1. The reciprocal of a designated quantity. Also called multiplicative inverse.
2. The negative of a designated quantity. Also called additive inverse.

[Middle English, from Latin inversus, past participle of invertere, to invert. See invert.]

inversely in·verse'ly adv.

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IN BRIEF: Exactly opposite, reversed.

The compulsion to take ourselves seriously is in inverse proportion to our creative capacity. When the creative flow dries up, all we have left is our importance. — Eric Hoffer (1902-1983), American writer & philosopher.

Wikipedia: Additive inverse

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Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2007)

In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, 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.

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).

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.

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 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' )

and denoted by (– x), and one can write x – y instead of x + (– y).

For example, since addition of real numbers is associative, each real number has a unique additive inverse.

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 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).

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inverse

General definition

Other examples

See also