It looks like there are different intervals in which the argument of a complex number can be.
- 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.