Absolute value of a real number

Given a real number $a$ we define the absolute value of $a$ denoted $| a |$ , as the biggest number between $a$ and $- a$ : $| a | = m a x (a, - a)$

Example

$| \sqrt{2} | = m a x (\sqrt{2}, - \sqrt{2}) = \sqrt{2}$

$| - \sqrt{2} | = m a x (- \sqrt{2}, - (- \sqrt{2})) = m a x (- \sqrt{2}, \sqrt{2}) = \sqrt{2}$

As we can see in the example, the absolute value of a positive number is the same number, while the absolute value of a negative number is its opposite. That is, $| a | = {\begin{matrix} a, if a \geq 0 \\ - a, if a < 0 \end{matrix}$

Properties of the absolute value

For any pair of real numbers $a$ and $b$ , it is satisfied that:

$| a | > 0$ if $a \neq 0$ , and $| 0 | = 0$ .
$| a | = | - a | .$
Triangle inequality: $| a + b | \leq | a | + | b | .$
$| a \cdot b | = | a | \cdot | b | .$

And if $a$ is a real number and $r$ is a positive real number, the inequality $| a | < r$ is equivalent to the chain of inequalities $- r < a < r .$