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Find the solution of the following inequality. $$\arg{z}\leq\frac{\pi}{4}$$

My first answer: Any number $z=x+iy$ in $y\leq x$, $y\geq 0$ and $x\not=0$ satisfies $\arg{z}\leq\frac{\pi}{4}$.

My additional answer:

I am confused in determining whether or not we are allowed to subtract an integer multiple of $2\pi$ from an angle when comparing two angles. If it is allowed then for example, $z=i$ can also be the solution because $\frac{\pi}{2} - 2\pi =-\frac{3\pi}{2}<\frac{\pi}{4}$. Thus the solution is all complex number excluding $z=0$.

Question

What is the correct solution of $\arg{z}\leq\frac{\pi}{4}$?

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The "correct" answer to this question depends on the definition of $\arg$. If the convention is that it is single valued with an answer in $[0,2\pi)$ then your first answer is correct. If it's multivalued as you describe it, then the second answer is correct.

It's likely that your instructor (if this is for a course) wants the first answer.

As an instructor, I'd be delighted to get a paper that gave both answers along with a discussion of the ambiguity.

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    $\begingroup$ Another convention is $(-\pi,\pi]$. $\endgroup$
    – GEdgar
    Feb 27 at 15:39

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