Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

So it's obvious geometrically that the argument of $z=i$ is $\pi/2$.

However the method of getting the argument is $\arctan(y/x)$. And when in the case of $z=i$, $y/x = 1/0$ which is undefined...

So when you want to find the argument of a complex number is this the correct process -

  1. $\operatorname{Argument}(z) = \arctan(y/x)$ if $x\neq0$.
  2. If $x = 0$, then $\operatorname{Argument}(z) = \pi/2\text{ or }-\pi/2$

Is that the way I should be approaching it?

share|improve this question
5  
You will find this wiki article very relevant. –  Sasha Jan 24 '12 at 1:12
2  
Are you sure the arctangent rule is the definition of "argument" in the presentation you're following? That would be unusual; commonly one defines the argument in some different way, and then the arctangent rule is just a computational technique that works if (and only if) the real part is positive. –  Henning Makholm Jan 24 '12 at 1:38
add comment

1 Answer

Although 1/0 is undefined, $\lim_{x \to 0^+}\operatorname{tan}(y/x)=\pi/2$ for any y. This corresponds with the geometric justification. One considers the limit as x goes to 0 from above because this is what happens when x rotates counterclockwise from 1 to i. To get the argument of -i, one still considers the counterclockwise rotation, which leads to the limit from below, which gives instead $-\pi/2$. The arguments of multiples of i and -i work out the same way.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.