05 - Modulus

Modulus

Let $a\in \mathbb{R}$ then

$$|a|=\{\begin{array}{ll}a& \text{ if }a>0\\ 0& \text{ if }a=0\\ -a& \text{ if }a<0\end{array}$$

Also called the absolute value of $a$
Well-defined due to the trichotomy property

Properties of Modulus

Take $a,b,c\in \mathbb{R}$ then

1. $|-a|=|a|$
2. $|a|\ge 0$
3. $|a{|}^2=a^2$
4. $|ab|=|a||b|$
5. $-|a|\le a\le |a|$
6. If $c\ge 0$ then $|a|\le c⟺-c\le a\le c$
   Works similarly if we use $(<,>)$ instead of $(\le .\ge )$

Proof (not detailed)

1. By definition of modulus (do cases)
2. By definition of modulus (do cases)
3. Use definition and trichotomy (cases) and use $(-a)\cdot (-a)=a^2$
4. Use definition of modulus, trichotomy and use cases
5. If $a\ge 0$ then $-|a|\le 0\le a=|a|$
   If $a<0$ then $-|a|=a<0\le |a|$
   Hence$-|a|\le a\le |a|$
6. Proof $⟹$
   Suppose that $-c\le a\le c$
   Then by $(5)$, $-a\le c$ and $-c\le -|a|\le a\le |a|\le c$ and so done by transitivity ()

Proof $⟸$
Suppose that $-c\le a\le c$ then $-a\le c$ and $a\le c$
But $|a|=a$ or $|a|=-a$ so

$$|a|\le c$$