Can different arguments be correct for the same complex number?

[Maxo]

Homework Statement

Calculate argument of complex number
$$-1-\sqrt{3}i$$

Homework Equations

The Attempt at a Solution

The argument of this is -120 degrees but why couldn't we as well say it's 240 degrees? Since going 240 degrees will go to the same point as -120 degrees. Why is this false?

 

[HallsofIvy]

Why do you say it is false? Yes, -120 degrees is the same as 360- 120= 240. If your textbook gives -120 as the argument that does not necessarily mean 240 is not. But perhaps your text is using a particular convention here: in order to avoid ambiguity some texts require that angles be given between -180 and 180 degrees, others between 0 and 360 degrees.

(Personally, I would have said that the argument was $4\pi/3$. I am a little surprised you are using degrees rather than radians.)

 

[Mentallic]

The answer can be any multiple of $2\pi$ radians, but it's just been chosen so that the principal argument (smallest angle) is in the range $(-\pi,\pi]$. It's just a custom really.

 

[Maxo]

Well for example here: http://www.wolframalpha.com/input/?i=arg(−1−sqrt(3)i)=4pi/3 it says false.

 

[Mentallic]

Well for example here: http://www.wolframalpha.com/input/?i=arg(−1−sqrt(3)i)=4pi/3 it says false.

Wolfram Alpha is abiding by the principal argument custom. When you use calculators, you sometimes need to have an understanding of what their unexpected results could mean.

Like this one:

http://www.wolframalpha.com/input/?i=%28-1%29%5E%281%2F3%29

Notice it says "Assuming the principal root". There are 3 roots, and the one with the smallest argument which is was returned to the user, while many people would expect the answer to be -1.