Negation (algebra)

Negation is the mathematical operation that reverses the sign of a number. Thus the negation of a positive number is negative, and the negation of a negative number is positive. The negation of zero is zero. The negation of a number a, usually denoted −a, is often also called the opposite of a.

More generally, negation may refer to any operation that replaces an object with its additive inverse. For example, the negation of a vector is another vector with the same magnitude and the exact opposite direction.

Relation to other operations

Negation is closely related to subtraction, which can be viewed as a combination of addition and negation:

a − b  =  a + (−b).

Similarly, negation can be thought of as subtraction from zero:

−a  =  0 − a.

Negation can also be thought of as multiplication by negative one:

(−1) × a  =  −a.

Double negation

The negation of a negative number is positive. For example,

−(−3) = +3.

In general, double negation has no net effect on a number. That is,

−(−a) = a

for any real number a.

This sometimes leads to somewhat confusing notation. For example, the absolute value of a real number a is usually defined by the formula:

$|a| = \begin{cases} a, & \text{if } a \ge 0 \\ -a, & \text{if } a < 0. \end{cases}$

When a is negative, the absolute value of a is defined to be −a, which is always positive, being the negation of a negative number.

Additive inverse

The negation of a real number is an additive inverse for that number. That is,

a + (−a) = 0

for any real number a. (Note that 0 is the additive identity.)

In abstract algebra, certain binary operations in algebraic structures are commonly written as addition. In this case, the identity element with respect to the binary operation is usually thought of as 0, and the inverse of any element is thought of as its negation.

Other properties

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

−(a + b) = (−a) + (−b)

a − (−b) = a + b

(−a) × b = a × (−b) = −(a × b)

(−a) × (−b) = a × b

Generalizations

Negation may also be defined for complex numbers by the formula

−(a + bi)  =  (−a) + (−b)i

On the complex plane, this operation rotates a complex number 180 degrees around the origin.

Similarly, the negation of a Euclidean vector can be obtained by rotating the vector 180 degrees. Thus the negation of a vector has the same magnitude as the original, but the exact opposite direction. (Vectors in exactly opposite directions are sometimes referred to as antiparallel.) In terms of vector components

−(x, y, z)  =  (−x, −y, −z)

or more generally

−(x₁, ..., x_n)  =  (−x₁, ..., −x_n)

Negation of vectors has the same effect as scalar multiplication by −1.

In abstract algebra, negation may refer to any operation that takes the additive inverse of an element of an abelian group (or any invertible element of an additive magma).

See also

• Negative and non-negative numbers
• Sign (mathematics)