In mathematics, the **absolute value** (or **modulus**) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3, and 0 is the only absolute value of 0.

## Definition

It can be defined as follows: For any real number *a*, the **absolute value** of *a*, denoted |*a*|, is equal to *a* if *a* ≥ 0, and to −*a*, if *a* < 0 (see also: inequality). |a| is never negative, as absolute values are always either positive or zero. Put another way, |*a*| < 0 has no solution for *a*. Also, it is not necessary that |-*a*| = *a* since *a* can be negative.

The absolute value can be regarded as the *distance* of a number from zero; indeed the notion of distance in mathematics is a generalisation of the properties of the absolute value. When real numbers are considered as one-dimensional vectors, the absolute value is the magnitude, and the *p*-norm for any *p*. Up to a constant factor, every norm in **R**^{1} is equal to the absolute value: ||x||=||1||.|x|

## Properties

The absolute value has the following properties:

- |
*a*| ≥ 0
- |
*a*| = 0 iff *a* = 0.
- |
*ab*| = |*a*||*b*|
- |
*a/b*| = |*a*| / |*b*| (if *b* ≠ 0)
- |
*a*+*b*| ≤ |*a*| + |*b*| (the triangle inequality)
- |
*a*−*b*| ≥ ||*a*| − |*b*||
- |
*a*| ≤ *b* iff −*b* ≤ *a* ≤ *b*
- |
*a*| ≥ *b* iff *a* ≤ −*b* **or** *b* ≤ *a*

The last two properties are often used in solving inequalities; for example:

- |
*x* − 3| ≤ 9
- −9 ≤
*x*−3 ≤ 9
- −6 ≤
*x* ≤ 12

For real arguments, the absolute value function *f*(*x*) = |*x*| is continuous everywhere and differentiable everywhere except for *x* = 0. For complex arguments, the function is continuous everywhere but differentiable *nowhere* (One way to see this is to show that it does not obey the Cauchy-Riemann equations).

For a complex number *z* = *a* + *ib*, one defines the absolute value or *modulus* to be |*z*| = √(*a*^{2} + *b*^{2}) = √ (*z* *z*^{*}) (see square root and complex conjugate). This notion of absolute value shares the properties 1-6 from above. If one interprets *z* as a point in the plane, then |*z*| is the distance of *z* to the origin.

It is useful to think of the expression |*x* − *y*| as the *distance* between the two numbers *x* and *y* (on the real number line if *x* and *y* are real, and in the complex plane if *x* and *y* are complex). By using this notion of distance, both the set of real numbers and the set of complex numbers become metric spaces.

The function is not invertible, because a negative and a positive number have the same absolute value.

(the modulus)

## Algorithm

In the C programming language, the `abs()`

, `labs()`

, `llabs()`

(in C99), `fabs()`

, `fabsf()`

, and `fabsl()`

functions compute the absolute value of an operand. Coding the integer version of the function is trivial, ignoring the boundary case where the largest negative integer is input:

int abs(int i)
{
if (i < 0)
return -i;
else
return i;
}

The floating-point versions are trickier, as they have to contend with special codes for infinity and not-a-numbers.

Last updated: 09-12-2005 02:39:13