Given a real number $$a$$ we define the absolute value of $$a$$ denoted $$|a|$$, as the biggest number between $$a$$ and $$-a$$: $$$|a|=max(a,-a)$$$
$$|\sqrt{2}|=max(\sqrt{2},-\sqrt{2})=\sqrt{2}$$
$$|-\sqrt{2}|=max(-\sqrt{2},-(-\sqrt{2}))=max(-\sqrt{2},\sqrt{2})=\sqrt{2}$$
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, $$$|a|= \left\{ \begin{array}{c} a, \ \ \mbox{if} \ a \geq 0 \\ -a, \ \ \mbox{if} \ a < 0 \end{array} \right.$$$
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 $$$-r < a < r.$$$