# Thread: Argument of a complex number

1. ## Argument of a complex number

Just a quick couple of questions, when working out the argument of a complex number, does the result always have to be between $\displaystyle \pi$ and $\displaystyle -\pi$?

Also, if the number was $\displaystyle 1 - i$ for example, is the answer, $\displaystyle \tan^{-1}\frac{-1}{1}$ or $\displaystyle \tan^{-1}\frac{1}{1}$?

Thanks for your help, just forgot a couple of things from earlier in the year

Craig

2. ## Re :

Originally Posted by craig
Just a quick couple of questions, when working out the argument of a complex number, does the result always have to be between $\displaystyle \pi$ and $\displaystyle -\pi$?

Also, if the number was $\displaystyle 1 - i$ for example, is the answer, $\displaystyle \tan^{-1}\frac{-1}{1}$ or $\displaystyle \tan^{-1}\frac{1}{1}$?

Thanks for your help, just forgot a couple of things from earlier in the year

Craig

From what i learnt, the angle $\displaystyle \theta$ has to be expressed in radians with domain $\displaystyle -\pi<\theta\leq\pi$ .

The argument for $\displaystyle z=1-i$ is $\displaystyle tan^{-1}\frac{-1}{1}$

remember , for the argument for a complex number z=x+yi , its argument will be $\displaystyle tan^-1\frac{y}{x}$

remember , for the argument for a complex number z=x+yi , its argument will be $\displaystyle \color{red} tan^-1\frac{y}{x}$
That is true only if $\displaystyle x>0$.
Just a quick couple of questions, when working out the argument of a complex number, does the result always have to be between $\displaystyle \pi$ and $\displaystyle -\pi$?
How you define the principle argument is a matter of convention. eg. $\displaystyle 0 \leq \theta < 2 \pi$ is equally valid.