Determine what is the angle of an complex number without imaginary value

Question

Okay, so this is bugging me now.

I know this:

while tg angle = Y / X

for y > 0, angle is an element of < 0, PI >

for y < 0, angle is an element of < PI, 2PI >

for y = 0, angle is 0 or PI or 2PI.

Okay, but how to determine what out of those 3 values angle actually is ?

So far this is what I know:

If X is negative, angle should be PI, and if X is positive, angle should be either 0 or 2PI... And thats bugging me now, how do I know if its 2PI or 0 ?

EDIT:

I am asking this because if I assume that angle is 0, while calculating root of complex number I would use:

r * cis ((0 + 2*k*PI) / n)

and if I assume that angle is 2PI, it would be

r * cis ((2PI + 2*k*PI) / n)

What should I assume my angle is when y = 0 and x > 0 ?:/

Answer 1

If you replace k in the second expression by k-1, you will get the first expression for r*cis.

Answer 2

The choice of which among the numbers $\phi + 2k\pi$ should be "the" angle is a matter of convention only. Whichever value of $k$ you choose you will always get the same complex number from $r\text{ cis} (\phi $).

For complex numbers off the positive real axis the convention is (as you know) to use an angle strictly between $0$ and $2\pi$. For numbers on the axis the usual convention is that the angle is $0$.

To summarize: it's just a convention and you can stop letting it bug you.

Answer 3

So you are asking if the solution to $z^4=1$ should be $z=1$ from $k=0$ or $z=i$ from $k=2.$ But there are also $z=-1$ and $z=-i$ from $k=4$ and $k=6.$ Actually, $z=-i$ results from $(6+8j)\pi.$ One might consider $i$ the "nicest" solution. In general, the solution coming from $2\pi$ is sometimes informally referred to as the principal primitive root.

If you wanted to look at complex numbers with $x\gt 0$ it would be most convenient to use angles between $-\frac{\pi}2$ and $\frac{\pi}2$ as , for example, with $\arctan.$

Sometimes it is convenient to make a certain choice but the others are equally valid.

By the way, at $(0,0)$ the angle is totally undetermined, but not undefined. Any choice at all is valid, but it doesn't really matter.