For a polar representation of a complex number, why the principal argument is to be considered in $(-\pi, \pi]$ or how can it arise?
I read some books where they just maintained it, but they don't give any reason. Also I read in Wikipedia page where they maintain: Some authors define the range of the principal value as being in the closed-open interval $[0, 2\pi)$, but why? Further I search in MSE and get two same problems(links are 1, 2). But i am not satisfied with such comments only. I expect for a detailed explanation. My special focus is how did it assumed. Any help is highly appreciated.
The reason that makes it feel natural for me is that the value of the argument in $(-\pi,\pi)$ corresponds to the shortest distance you have to walk, from the origin at $1$, to reach a given point on the circle, where the sign tells you when the shortest path involves walking backwards.
The fact that you choose $\pi$ for the principal argument of $-1$, instead of $-\pi$, is a matter of choice.
Note: this is assuming walking along a unit circle, so that the angle actually corresponds to the arc-length.