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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.

SAHEB PAL
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1 Answers1

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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.

Arnaud Mortier
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