Additive inverse

In mathematics, the additive inverse of a

number

is the number that, when

added

to , yields

zero

. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive.

Zero

is the additive inverse of itself.

The additive inverse of is denoted by unary minus: (see also below). For example, the additive inverse of 7 is −7, because , and the additive inverse of −0.3 is 0.3, because .

Similarly, the additive inverse of is which can be simplified to . The additive inverse of is , because .

The additive inverse is defined as its inverse element under the binary operation of addition (see also below), which allows a broad

generalization

to mathematical objects other than numbers. As for any inverse operation, double additive inverse has

no net effect

: .

Common examples

For a number (and more generally in any ring), the additive inverse can be calculated using

multiplication

by −1; that is, . Examples of rings of numbers are integers, rational numbers, real numbers, and

complex number

s.

Relation to subtraction

Additive inverse is closely related to

subtraction

, which can be viewed as an addition of the opposite:
:.
Conversely, additive inverse can be thought of as subtraction from zero:
:.
Hence, unary minus sign notation can be seen as a shorthand for subtraction (with the "0" symbol omitted), although in a correct

typography

, there should be no

space

after unary "−".

Other properties

In addition to the identities listed above, negation has the following algebraic properties:
* , it is an Involution operation
*
*
*
*
*
*: notably,

Formal definition

The notation + is usually reserved for commutative binary operations (operations where for all , ). If such an operation admits an identity element (such that for all ), then this element is unique (). For a given , if there exists such that , then is called an additive inverse of .

If + is associative, i.e., for all , , , then an additive inverse is unique. To see this, let and each be additive inverses of ; then

:.

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:

*

Complex number

s: . On the complex plane, this operation rotates a complex number 180 degrees around the origin (see the image above).
* Addition of real- and complex-valued functions: here, the additive inverse of a function is the function defined by , for all , such that , the zero function ( for all ).
* More generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the identity element of this group):
*

Sequence

s, matrices and nets are also special kinds of functions.
* In a vector space, the additive inverse is often called the opposite vector of ; it has the same magnitude as the original and opposite direction. Additive inversion corresponds to

scalar multiplication

by −1. For Euclidean space, it is point reflection in the origin. Vectors in exactly opposite directions (multiplied to negative numbers) are sometimes referred to as antiparallel.
**vector space-valued functions (not necessarily linear),
* In modular arithmetic, the modular additive inverse of is also defined: it is the number such that . This additive inverse always exists. For example, the inverse of 3 modulo 11 is 8 because it is the solution to .

Non-examples

Natural numbers, cardinal numbers and ordinal numbers do not have additive inverses within their respective sets. Thus one can say, for example, 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.

See also

* −1
*

Absolute value

(related through the identity ).
* Additive identity
* Inverse function
* Involution (mathematics)
*

Multiplicative inverse

*

Reflection symmetry

Notes and references

{{reflist

Abstract algebra
Arithmetic
Elementary algebra