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For $z\in\mathbb{C}$, $arg(z)\in[0;2\pi)$ OR $arg(z)\in(-\pi;\pi]$. Which one do I use in problem solving?

Consider $A(a),~B(b)$. I noticed that $\angle AOB=arg(b)-arg(a)$ for $A-O-B$ clockwise and $\angle AOB=arg(a)-arg(b)$ for $A-O-B$ anti-clockwise. I also noticed people usually use the second version without specifying the context.

If I want to prove, for example, $AB\perp CD\Leftrightarrow Re\frac{a-b}{c-d}=0$, do I need to draw a figure and based on that consider the clockwise / anti-clockwise cases?

Also, does the $[0;2\pi)$ / $(-\pi;\pi]$ have any influence on this?

Thankss.

Neox
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    $\arg$ is a multi-valued function, and the principal value $\text{Arg}$ is the restriction to a $2 \pi$-long interval, usually $(-\pi,\pi],$. The choice does not matter when taking the difference of two $\arg$ angles, or for expressions that involve only trig functions since they all have $2\pi$ as a period. Geometrically, $\arg$ is an oriented angle (see for example 1) and $\angle AOB = \arg a - \arg b$ is always true when considering $\angle AOB$ as an oriented angle. – dxiv Jul 08 '21 at 16:41
  • @dxiv So "oriented" angles can have negative values? That means, if $\angle AOB$ is non-trigonometrical (clockwise) it has negative value, and trigonometrical (anti-clockwise) angles are positive? That would explain everything, and if it's not like that, then my theory collapses and I'm left in a dense fog :c – Neox Jul 08 '21 at 17:19
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    Yes, oriented angles can be negative (or in the $,[\pi,2\pi),$ range, depending on convention). Main point is that an equality like $,\angle AOB + \angle BOC = \angle AOC \pmod{2\pi},$ holds always true when using oriented angles, but not plain unsigned angles. Not directly related, but there are other quantities in geometry that are used in dual signed/unsigned mode, for example signed segments (in the Menelaus' theorem), or oriented areas (in the shoelace formula). – dxiv Jul 08 '21 at 17:36
  • @dxiv When I learned Menelaus' theorem I noticed that my teacher sometimes used to put arrows above the segments and when I asked her what is the difference between arrow and non-arrow mode she said that it's trivial. That's why I don't like school and I want to self study. Also just today I've learned the shoelace formula and when I tried to prove it by induction (beginning with the area of a triangle) I was confused by the abs value, I didn't know how to "open" it but then I read that abs value can be ignored in the case of a trigonometrical arranged set of points. Thank you a lot! – Neox Jul 08 '21 at 17:56

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