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.

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.

These complex numbers, two of eight values of 8√1, are mutually opposite

Common examples

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

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

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 = a2

Formal definition

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

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, but not necessarily the same magnitude, are sometimes referred to as antiparallel vectors.
  • 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 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

Notes and references

  1. ↑ Tussy, Alan; Gustafson, R. (2012), Elementary Algebra (5th ed.), Cengage Learning, p. 40, ISBN 9781133710790.
  2. ↑ 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.
  3. ↑ The term "negation" bears a reference to negative numbers, which can be misleading, because the additive inverse of a negative number is positive.
  4. ↑ Weisstein, Eric W. "Additive Inverse". mathworld.wolfram.com. Retrieved 2020-08-27.
  5. ↑ "Additive Inverse". www.learnalberta.ca. Retrieved 2020-08-27.
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