Real modulus/Introduction/Section

Definition

For a real number ${\displaystyle {}x\in \mathbb {R} }$, the modulus is defined in the following way.

${\displaystyle {}\vert {x}\vert ={\begin{cases}x\,,{\text{ if }}x\geq 0\,,\\-x,\,{\text{ if }}x<0\,.\end{cases}}\,}$

So the modulus (also called the absolute value) is never negative and has only at ${\displaystyle {}x=0}$ the value ${\displaystyle {}0}$, elsewhere it is always positive. The mapping

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \vert {x}\vert ,}$

is called the modulus function. Its graph consists of two half lines; such a function is called piecewisely linear.

Lemma

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \vert {x}\vert ,}$
fulfills the following properties (${\displaystyle {}x,y}$ are arbitrary real numbers).
1. ${\displaystyle {}\vert {x}\vert \geq 0}$.
2. ${\displaystyle {}\vert {x}\vert =0}$ if and only if ${\displaystyle {}x=0}$.
3. ${\displaystyle {}\vert {x}\vert =\vert {y}\vert }$ if and only if ${\displaystyle {}x=y}$ or ${\displaystyle {}x=-y}$ holds.
4. ${\displaystyle {}\vert {y-x}\vert =\vert {x-y}\vert }$.
5. ${\displaystyle {}\vert {xy}\vert =\vert {x}\vert \vert {y}\vert }$.
6. For ${\displaystyle {}x\neq 0}$ we have ${\displaystyle {}\vert {x^{-1}}\vert =\vert {x}\vert ^{-1}}$.
7. We have ${\displaystyle {}\vert {x+y}\vert \leq \vert {x}\vert +\vert {y}\vert }$ (triangle inequality for the modulus).
8. ${\displaystyle {}\vert {x+y}\vert \geq \vert {x}\vert -\vert {y}\vert }$.

Proof

${\displaystyle \Box }$