Absolute value

In mathematics, the absolute value, or modulus (UK), of a number is that number without a negative sign. So, for example, 3 is the absolute value of both 3 and -3.

It can be defined as follows: For any real number a, the absolute value of a (denoted |a|) is equal to 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. In other words, the solution to |a| < 0 is that a is equal to the empty set, as there is no quantity which has a negative absolute value.

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.

The absolute value has the following properties:

1. |a| ≥ 0
2. |a| = 0 if and only if a = 0.
3. |ab| = |a||b|
4. |a/b| = |a| / |b| (if b ≠ 0)
5. |a+b| ≤ |a| + |b|
6. |a-b| ≥ ||a| - |b||
7. <math>\left| a \right| = \sqrt{a^2}</math>
8. |a| ≤ b if and only if -b ≤ a ≤ b

This last property is often used in solving inequalities; for example:

|x - 3| ≤ 9

-9 ≤ x-3 ≤ 9

-6 ≤ x ≤ 12

The absolute value function f(x) = |x| is continuous everywhere and differentiable everywhere except for x = 0.

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 operation is not reversible[?] because either negative or non-negative number or becomes the same non-negative number.

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.