# Absolute value

In mathematics, the

**absolute value**(or

**modulus**) of a number is that number without a negative sign. So, for example, 3 is the absolute value of both 3 and -3.

Table of contents |

2 Properties 3 Algorithm |

## Definition

It can be defined as follows: For any real number*a*, the

**absolute value**of

*a*(denoted

**|**) is equal to

*a*|*a*itself 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*.

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. It is thus a concept useful to scientists, for whom it serves as a measure of the *magnitude* of any quantity, whether scalar or vector.

## Properties

The absolute value has the following properties:- |
*a*| ≥ 0 - |
*a*| = 0 if and only if*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*if and only if -*b*≤*a*≤*b*

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

*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.

## Algorithm

If the absolute value would not be a standard function **Abs** in Pascal it could be easily computed using the following code:

program absolute_value; var n: integer; beginread (n); if n < 0 then n := -n; writeln (n)end.

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:

#includeThe floating-point versions are trickier, as they have to contend with special codes for infinity and not-a-numbers.int abs(int i) { if (i < 0) return -i; else return i; }