# Thread: Complex numbers

1. Hi!
I was sitting rehearsing calculations with complex numbers when I suddenly encountered a problem. I was supposed to calculate the argument for the complex number .
So this is what I did:
Let's say that the argument is v. Then   So then I reckon that the argument is plus n*(pi), since But it says in the key that the right answer is + n*2(pi).
What am I missing here?  2.

3. Originally Posted by thyristor
Hi!
I was sitting rehearsing calculations with complex numbers when I suddenly encountered a problem. I was supposed to calculate the argument for the complex number .
So this is what I did:
Let's say that the argument is v. Then   So then I reckon that the argument is plus n*(pi), since But it says in the key that the right answer is + n*2(pi).
What am I missing here?
I have not tried to work through you trigonometric identies, but here is what is going on from a geometric perspective.

First I assume that you really mean .

So now you are adding two complex numbers,namely and . Both of those are complex numbers of modulus 1. The first has argument 0 and the second argument . Now adding complex numbers is just addition of vectors in 2 real dimensions, so what you get for the sum is determined by the parallelogram law. In particular, since each term has the same modulus (length) the vector sum bisects the angle between the two. So that angle is which of course is the argument and which is only determined modulo . So your book is right.  4. Thanks a lot! Now I understand what went wrong. Indeed But this is equal to , which means that we "lose" information of where are complex number is. From your geometrical explanation, I know realize that it has to be .  Bookmarks
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