Absolute value - Definition and Overview

In mathematics, the absolute value (or modulus) of a number is the difference between that number and 0. Simply speaking, it is the number without a negative sign. So, for example, 3 is the absolute value of both 3 and −3.

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.

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 R¹ is equal to the absolute value: ||x||=||1||.|x|

Properties

The absolute value has the following properties:

1. |a| ≥ 0
2. |a| = 0 iff a = 0.
3. |ab| = |a||b|
4. |a/b| = |a| / |b| (if b ≠ 0)
5. |a+b| ≤ |a| + |b| (the triangle inequality)
6. |a−b| ≥ ||a| − |b||
7. <math>\left| a \right| = \sqrt{a^2}<math>
8. |a| ≤ b iff −b ≤ a ≤ b
9. |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| = √(a² + b²) = √ (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.

Absolute value and complex numbers

<math>\mbox{if }c = a + bi \mbox{ then }|c| = \sqrt{a^2 + b^2}\,\!<math> (the modulus)

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;
begin
  read (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:

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.