# How should I find the argument when a equals zero with complex numbers?

I know how to plot an Argand diagram and I think I know how to find the argument usually, but I've done some looking around to be certain of my answers and found what seems to be conflicting information regarding the argument of purely imaginary numbers (for example, $-2i$).

My textbook gives the example of $\arg(-5j) = -\pi/2$, which makes sense to me as I was told to take the angle as a negative, measured clockwise and in radians. On the diagram, a pure negative imaginary number would be $-90$ degrees ($-\pi/2$ radians) from the real numbers line.

However, I've done some looking online and found two sources which insist it's $3*\pi/2$, and I'm not sure why. Links provided: http://www.convertalot.com/complex_arithmetical_calculator.html http://www.mathamazement.com/Lessons/Pre-Calculus/06_Additional-Topics-in-Trigonometry/complex-numbers-in-polar-form.html (Example 1 d).

I'm guessing my textbook is correct as it seems to make the most sense and would be the more trustworthy source, but I have a feeling I'm missing something important considering two different sources came to the same conclusion.

• $\frac{3\pi}{2} = \frac{-\pi}{2}$ subtract $2 \pi$ from $\frac{3\pi}{2}$ – bigfocalchord Mar 28 '16 at 9:48
• Would there be a preferred answer? Wouldn't the different value of 3pi/2 and -pi/2 cause problems? – Dalekcaan1963 Mar 28 '16 at 9:55
• There really isn't that big of a problem. As long as we know we are working in modulo $2\pi$ per se, we know exactly which angle the argument corresponds to. It merely is a matter of convenience. The intervals $[0,2\pi)$ and $(-\pi,\pi]$ are much easier to work with than say, $[\pi/4,\pi/4+2\pi).$ – quasicoherent_drunk Mar 28 '16 at 10:17

We had a little chat about this in our class as well, because the $arg(z)$ function was defined as $\theta$ for which $\dfrac{z}{|z|}=e^{i\theta}$, but then we know that $e^{i\theta}$ is a periodic function, because it can be written as $isin(\theta)+cos(\theta)$, which are $2\pi$-periodic functions. So there is still a huge argument about this, and some textbooks like to work with $[0,2\pi)$ while others like to work with $[-\pi,\pi)$, because regardless of which interval you take, the value of $\theta$ is uniquely determined if the length of the interval is exactly $2\pi$. I,like you, normally work with the latter , but there are some others like your sources who work with the former. You think that $90^o$ counter clockwise is like rotating backward, hence you use the minus sign to denote that. On the other hand, the people who have designed the links you have visited must have thought that it's the same as going $270^o$ degrees forward, hence they gave it $3\pi/2$ argument. It's all a matter of taste, there is no problem with either approach otherwise. However I have noticed that the former of the two approaches is more popular.