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 $-a$: $$|a|=max(a,-a)$$

$$|\sqrt{2}|=max(\sqrt{2},-\sqrt{2})=\sqrt{2}$$

$$|-\sqrt{2}|=max(-\sqrt{2},-(-\sqrt{2}))=max(-\sqrt{2},\sqrt{2})=\sqrt{2}$$

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 $$-r < a < r.$$