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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. 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.

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||
  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.