# Additive inverse

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In mathematics, the **additive inverse** of a number a is the number that, when added to a, yields zero. This number is also known as the **opposite** (number),^{[1]} **sign change**,^{[2]} and **negation**.^{[3]} 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 a is denoted by unary minus: −*a* (see also § Relation to subtraction below).^{[4]} 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.

Similarly, the additive inverse of *a* − *b* is −(*a* − *b*) which can be simplified to *b* − *a*. The additive inverse of 2*x* − 3 is 3 − 2*x*, because 2*x* − 3 + 3 − 2*x* = 0.^{[5]}

The additive inverse is defined as its inverse element under the binary operation of addition (see also § Formal definition below), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect: −(−*x*) = *x*.

## Common examples[edit]

For a number (and more generally in any ring), the additive inverse can be calculated using multiplication by −1; that is, −*n* = −1 × *n*. Examples of rings of numbers are integers, rational numbers, real numbers, and complex numbers.

### Relation to subtraction[edit]

Additive inverse is closely related to subtraction, which can be viewed as an addition of the opposite:

*a*−*b*=*a*+ (−*b*).

Conversely, additive inverse can be thought of as subtraction from zero:

- −
*a*= 0 −*a*.

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[edit]

In addition to the identities listed above, negation has the following algebraic properties:

- −(−
*a*) =*a*, it is an Involution operation - −(
*a*+*b*) = (−*a*) + (−*b*) - −(
*a*−*b*) =*b*−*a* *a*− (−*b*) =*a*+*b*- (−
*a*) ×*b*=*a*× (−*b*) = −(*a*×*b*) - (−
*a*) × (−*b*) =*a*×*b*- notably, (−
*a*)^{2}=*a*^{2}

- notably, (−

## Formal definition[edit]

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

If + is associative, i.e., (*x* + *y*) + *z* = *x* + (*y* + *z*) for all x, y, z, then an additive inverse is unique. To see this, let x′ and x″ each be additive inverses of x; then

*x′*=*x′*+*o*=*x′*+ (*x*+*x″*) = (*x′*+*x*) +*x″*=*o*+*x″*=*x″*.

For example, since addition of real numbers is associative, each real number has a unique additive inverse.

## Other examples[edit]

All the following examples are in fact abelian groups:

- Complex numbers: −(
*a*+*bi*) = (−*a*) + (−*b*)*i*. 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 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):
- Sequences, matrices and nets are also special kinds of functions.
- In a vector space, the additive inverse −
**v**is often called the opposite vector of**v**; 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 x is also defined: it is the number a such that a + x ≡ 0 (mod n). This additive inverse always exists. For example, the inverse of 3 modulo 11 is 8 because it is the solution to 3 +*x*≡ 0 (mod 11).

## Non-examples[edit]

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[edit]

- −1
- Absolute value (related through the identity |−
*x*| = |*x*|). - Additive identity
- Inverse function
- Involution (mathematics)
- Multiplicative inverse
- Reflection symmetry
- Semigroup
- Monoid
- Group (mathematics)

## Notes and references[edit]

**^**Tussy, Alan; Gustafson, R. (2012),*Elementary Algebra*(5th ed.), Cengage Learning, p. 40, ISBN 9781133710790.**^**Brase, Corrinne Pellillo; Brase, Charles Henry (1976).*Basic Algebra for College Students*. Houghton Mifflin. p. 54. ISBN 978-0-395-20656-0....to take the additive inverse of the member, we change the sign of the number.

**^**The term "negation" bears a reference to negative numbers, which can be misleading, because the additive inverse of a negative number is positive.**^**Weisstein, Eric W. "Additive Inverse".*mathworld.wolfram.com*. Retrieved 2020-08-27.**^**"Additive Inverse".*www.learnalberta.ca*. Retrieved 2020-08-27.