What is the interval for possible values of the argument of a complex number? It looks like there are different intervals in which the argument of a complex number can be.


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*Some say it goes from $-\pi$ to $+\pi$

*others say it goes from $0$ to $2\pi$.


For the most part, both ways look compatible to each other.
However, if one states that $arg z \lt \pi$, the result appears to be different depending on what the interval of possible values looks like.For the first interval, that means all possible values, but for the second one only half of them.
The above statement is just a statement (my tautology club member number is my tautology club member number). It will hold true for some $z$ but not for others. It will also hold true for the same $z$ given the first definition, but won't hold true when using the other definition, even if it's the same number.
The numbers for which the statement holds true are different depending on what definition is used. I think this is a problem because it should be clear what numbers it's true for.
Edit: I'm not sure if I can deal with the deamons that I summoned. I added the complex-analysis tag, as this is apparently what I'm doing here. It was pointed out to me that this aesthetically pleasing for math majors.
I thought I asked about something as simple as an angle. 
I'm not a math major. I couldn't even order a pizza in analysis. Please keep it simple.
 A: Welcome to complex analysis! I hope you stay around. It's quite a nice subject, and I think it's the most aesthetically pleasing undergraduate course offered to math majors. 
You're not limited to those values. I can say the argument of $-i$ is $-\pi/2$, $3\pi/2$, $7\pi/2$, etc. So what's the answer? Well, I just choose whichever one I want and say, pick the other arguments that make my $Arg$ function continuous. But even then there's still another problem. In your example of choose $\[0,2\pi\)$, our argument function won't be continuous for positive real numbers, if we choose $\(-\pi,\pi\]$ then argument function won't be continuous at any negative real number. 
Each argument function you create by specifying a domain and range is a "branch" of the argument function. This choice occurs because argument is a "multi-valued" function (there are many reasonable answers as to what the argument of a complex number is). 
There is a way around this problem for a lot of cases. Instead of thinking of the argument function as living on the complex-plane, we can create a complex manifold (a space with a weird shape the looks smooth) and define an argument function on that shape that is continuous everywhere and suddenly there's no more problem. In some sense this weird shape is where the argument function "naturally" lives.Here's the actual shape . 
It's kind of like a spiral staircase, and whether the argument of $i$ is $3\pi/2$ or $-\pi/2$ is determined by what "floor" or "level" you're looking at. 
This area isn't my strong suit and I haven't studied it in awhile. So I might be conflating terminology, and I'm sure there's someone who can explain this better.  
A: I don't see how you arrive at that last statement.  Of course, taking $\theta$ between $0$ and $\pi$ would certainly give only "half" the results of $0$ to $2\pi$.  But the values for $-\pi$ to $0$ give the other half.
