In mathematics, the additive inverse of a
number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

is the number that, when
to , yields
zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

. This number is also known as the opposite (number), sign change, and negation. For a
real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, it reverses its
sign
A sign is an object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Entity, something that is tangible and within the grasp of the senses
** Object (abstract), an object which does not exist at ...
: the additive inverse (opposite number) of a
positive number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is negative, and the additive inverse of a
negative number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is positive.
Zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

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
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
under the
binary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of addition (see also below), which allows a broad
generalization
A generalization is a form of abstraction
Abstraction in its main sense is a conceptual process where general rules and concept
Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They ...

to mathematical objects other than numbers. As for any inverse operation,
double
Double may refer to:
* Look-alike, a person who closely resembles another person
* Double (filmmaking), someone who substitutes for the credited actor of a character
* Doppelgänger, ghostly double of a living person
* Polish Enigma doubles, repl ...
additive inverse has
: .
Common examples
For a number (and more generally in any
ring), the additive inverse can be calculated using
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

by
−1
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
; that is, . Examples of rings of numbers are
integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...
s,
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s,
real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, and
complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s.
Relation to subtraction
Additive inverse is closely related to
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

, 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
Typography is the art and technique of arranging type to make written language
A written language is the representation of a spoken or gestural language
A language is a structured system of communication used by humans, including ...

, there should be no
space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Gre ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
binary operations (operations where for all , ). If such an operation admits an
identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
(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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, 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 group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s:
*
Complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s: . On the
complex plane
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, this operation
rotates a complex number 180
degrees
Degree may refer to:
As a unit of measurement
* Degree symbol (°), a notation used in science, engineering, and mathematics
* Degree (angle), a unit of angle measurement
* Degree (temperature), any of various units of temperature measurement ...
around the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
(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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s,
matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
and
nets are also special kinds of functions.
* In a
vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, the additive inverse is often called the opposite vector of ; it has the same
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
as the original and opposite direction. Additive inversion corresponds to
scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module (mathematics), module in abstract algebra). In common geometrical contexts, scalar multiplication of a re ...

by −1. For
Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, it is
point reflection
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
in the origin. Vectors in exactly opposite directions (multiplied to negative numbers) are sometimes referred to as antiparallel.
**
vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
-valued functions (not necessarily linear),
* In
modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...
, 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 number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s,
cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
s and
ordinal number
In set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...
s 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
*
Absolute value
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

(related through the identity ).
*
Additive identity In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
*
Inverse function
In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
*
Involution (mathematics)
Image:Involution.svg, An involution is a function f:X\to X that, when applied twice, brings one back to the starting point.
In mathematics, an involution, or an involutory function, is a function (mathematics), function that is its own inverse fun ...
*
Multiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

*
Reflection symmetry
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

Notes and references
{{reflist
Abstract algebra
Arithmetic
Elementary algebra