Additive inverse

In mathematics, the additive inverse of a number $a$ (sometimes called the opposite of $a$ )^[1] is the number that, when added to $a$ , yields zero. The operation taking a number to its additive inverse is known as sign change^[2] or 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.

"Opposite number" redirects here. For other uses, see analog and counterpart.

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 2x − 3 is 3 − 2x, because 2x − 3 + 3 − 2x = 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: $-(- x) = x$ .

$-(- a) = a$ , it is an

Involution operation

$-(a + b) = (- a) + (- b)$

$-(a - b) = b - a$

$a - (- b) = a + b$

$(- a) \times b = a \times (- b) = -(a \times b)$

(−a) × (−b) = a × b

²

: $-(a + bi) = (- a) + (- b) i$ . On the complex plane, this operation rotates a complex number 180 degrees around the origin (see the image above).

Complex numbers

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):

matrices and nets are also special kinds of functions.

Sequences

opposite vector

In , the modular additive inverse of $x$ is also defined: it is the number $a$ such that $a + x \equiv 0 (mod n)$ . This additive inverse always exists. For example, the inverse of 3 modulo 11 is 8 because it is the solution to $3 + x \equiv 0 (mod 11)$ .

modular arithmetic

All the following examples are in fact abelian groups:

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.

−1

(related through the identity $|- x | = | x |$ ).

Additive inverse

Involution operation

²

Complex numbers

Sequences

opposite vector

modular arithmetic

Non-examples[edit]

−1

Absolute value

Additive identity

Group (mathematics)

Monoid

Inverse function

Involution (mathematics)

Multiplicative inverse

Reflection (mathematics)

Reflection symmetry

Semigroup