How to simplify $\frac x{|x|}

Question

Is there a simplification for this relation?

$$ \frac{x}{\left| x\right| } $$

where $x=a+i b$, $a$ and $b$ are reals.

Answer 1

By polar form $x=|x|e^{i\theta}$ we obtain

$$\frac{x}{\left| x\right| }=e^{i\theta}$$

with $\theta = \operatorname {atan2} \left(b, a \right)$ (see atan2).

Answer 2

Assuming $x \neq 0$, you can reduce it to:-

$e^{i\theta}$, where

$\theta = \tan^{-1}(\frac b a)$ if $a > 0$

$\theta = \tan^{-1}(\frac b a) + \pi$ if $a<0$

$\theta = \frac \pi 2$ if $a=0$ and $b>0$

$\theta = -\frac \pi 2$ if $a=0$ and $b<0$

Answer 3

any complex number say '$z$' can be written in the form of : $z = a + ib$

so we define a function $f(z) = \frac{z}{|z|} \,\, (z \not= 0)\,\,$ then we can show that : $$f(a+ib) = \frac{a}{\sqrt{a^2 +b^2}} + i\frac{b}{\sqrt{a^2 +b^2}} = cos(\theta) + sin(\theta) = \cos(\tan^{-1}(\frac{b}{a})) + i\sin(tan^{-1}(\frac{b}{a})) $$