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Absolute value

The absolute value of a number is always a non-negative number, that is, greater than or equal to zero. If we calculate the absolute value of a positive number, it will always be the same number. However, if we want to find the absolute value of a negative number, it will be the positive counterpart of the number (i.e., from x, where x < 0, the absolute value is −x). The absolute value is denoted by two vertical lines: |x|.

Basic properties

Although calculating with absolute values might seem like a piece of cake, the opposite is usually true; they can make an otherwise simple function quite unpleasant. Consider, for example, linear equations with absolute value. Let's review a few more examples:

$$\begin{eqnarray} |5| &=& 5\\ |0| &=& 0\\ |-12| &=& 12\\ |3{,}14| &=& 3{,}14\\ |-2{,}71| &=& 2{,}71 \end{eqnarray}$$

Absolute value has the following properties for the values a, b, and c 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 is often computed for complex numbers.

Graphs

Graphs of absolute value functions are characterized by producing curves that lead to a "peak". At this point, the function is not differentiable.

Graph of the function f(x)=|x| Graph the function f(x)=|x^2-2|

Calculator

If you need to calculate the absolute value, you can use a local calculator.