Absolute value

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:

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. |a| = √ (a²)
8. |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

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

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.

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