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.
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|
- |a-b| ≥ ||a| - |b||
- |a| = √ (a2)
- |a| ≤ b if and only if -b ≤ a ≤ b
This last property is used often in solving inequalities; for example:
|x - 3| ≤ 9
-9 ≤ x-3 ≤ 9
-6 ≤ x ≤ 12
For a complex number z = a + ib, one defines the absolute value or modulus to be |z| = √ (a2 + b2) = √ (z z*). 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.