# Additive inverse

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**, and **negation**.[2] For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself.

The additive inverse of a is denoted by unary minus: −a (see the discussion below). 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 .

The additive inverse is defined as its inverse element under the binary operation of addition (see the discussion 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

For a number and, 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

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 "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:

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

- notably, (−

## Formal definition

The notation **+** is usually 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 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 (( *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

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

Natural numbers, cardinal numbers, and ordinal numbers, 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* under taking additive inverses.

## See also

- Multiplicative inverse
- Additive identity
- Absolute value (related through the identity | −
*x*| = |*x*| ). - Involution (mathematics)
- Reflection symmetry

## Footnotes

- Tussy, Alan; Gustafson, R. (2012),
*Elementary Algebra*(5th ed.), Cengage Learning, p. 40, ISBN 9781133710790. - The term "negation" bears a reference to negative numbers, which can be misleading, because the additive inverse of a negative number is positive.

## References

- Margherita Barile. "Additive Inverse".
*MathWorld*.