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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 ...

number
is the number that, when
added
added
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 ...

zero
. 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 ...

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 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 ...

generalization
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
no net effect
no net effect
: .


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 ...

multiplication
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 ...

complex number
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 ...

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 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 ...

typography
, 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 ...

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 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 ...

Complex number
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 ...

Sequence
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 ...

scalar multiplication
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 ...

Absolute value
(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 ...

Multiplicative inverse
*
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 ...

Reflection symmetry


Notes and references

{{reflist Abstract algebra Arithmetic Elementary algebra