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