Additive inverse

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Additive inverse

In mathematics, the additive inverse, or opposite, of a number

Number

A number is a mathematical object used in counting and measurement. A notational symbol which represents a number is called a Numeral system, but in common usage the word number is used for both the abstract object and the symbol, as well as for the numeral for the number....
n is the number that, when added

Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
to n, yields zero

0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
. 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.

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Encyclopedia

In mathematics, the additive inverse, or opposite, of a number

Number

Addition

0 (number)

Inverse element

In mathematics, the idea of inverse element generalises the concepts of additive inverse, in relation to addition, and Multiplicative inverse, in relation to multiplication....
under the binary operation

Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
of addition. It can be calculated using multiplication

Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
by −1; that is, −n = −1 × n.

Integer

Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
s, rational number

Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
s, real number

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s, and complex number

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
all have additive inverses, as they contain negative as well as positive numbers. Natural number

Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
s, cardinal number

Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of Set ....
s, and ordinal number

Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-order. They are usually identified with hereditarily transitive sets....
s, 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

Closure (mathematics)

In mathematics, a Set is said to be closed under some operation if the Operation on members of the set produces a member of the set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 7 are both natural numbers, but the result of 3 − 7 is not....
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

Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
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

Associativity

In mathematics, associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order that the operations are performed does not matter as long as the sequence of the operands is not changed....
((x+y)+z = x+(y+z) for all x,y,z), then an additive inverse is unique

= (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 group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
s:

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
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
valued functions (not necessarily linear),

sequence
Sequence

In mathematics, a sequence is an ordered list of objects . Like a Set , it contains Element , and the number of terms is called the length of the sequence....
s, matrices
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
and nets
Net (mathematics)

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....
are also special kinds of functions.
In a vector space additive inversion corresponds to scalar multiplication
Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . Note that scalar multiplication is different from scalar product which is an inner product between two vectors....
by −1. For Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
, it is inversion in the origin
Inversion in a point

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 modular arithmetic
Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus....
, 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).

Additive inverse

General definition

Other examples

See also