Absolute value

The absolute value of a number is always non-negative, that is, greater than or equal to zero. If we want to find the absolute value of a positive number, it is always the same number. But if we want to find the absolute value of a negative number, the result is the opposite number (that is, for x where x < 0, the absolute value is −x). The absolute value is written with two vertical bars: |x|.

Basic properties

Although it seems that working with absolute values will be a piece of cake, the opposite is usually true, and they can make an otherwise easy function quite unpleasant. See, for example, linear equations with absolute value. Let’s look at a few more examples:

$$\begin{array}{rcl}|5|& =& 5\\ |0|& =& 0\\ |-12|& =& 12\\ |3.14|& =& 3.14\\ |-2.71|& =& 2.71\end{array}$$

The absolute value has the following properties, for values x, a, b from the set of real numbers:

1. |x| ≥ 0
2. |a · b| = |a| · |b|
3. |a + b| ≤ |a| + |b|
4. |a| ≤ b ⇔ −b ≤ a ≤ b

The absolute value of complex numbers is calculated separately.

Graphs

Graphs of functions with an absolute value are characterised by curves with a sharp point. At that point the function is not differentiable.