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Encyclopedia > Additive inverse

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. A number is an abstract entity that represents a count or measurement. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... 0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. ...


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. In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, multiplication is an elementary arithmetic operation. ...


Types of numbers with additive inverses include:

Types of numbers without additive inverses (of the same type) include: The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

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. In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...


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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...


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: In mathematics, an abelian group is a commutative group, i. ...

  • 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),

In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*. In other words, the vector from X to P is the same as the vector from...

See also


  Results from FactBites:
 
math lessons - Additive inverse (409 words)
The additive inverse of n is denoted −n.
The additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0.
The additive inverse of a number is its inverse element under the binary operation of addition.
PlanetMath: minus one times an element is the additive inverse in a ring (146 words)
PlanetMath: minus one times an element is the additive inverse in a ring
"minus one times an element is the additive inverse in a ring" is owned by alozano.
This is version 6 of minus one times an element is the additive inverse in a ring, born on 2004-03-09, modified 2005-11-24.
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