# What is the principal argument of $-5-5i$?

When i calculated it by $\tan^{-1}\dfrac{y}{x}$, I got $\dfrac{\pi}{4}$, then i added $\pi$ to make it in the right quadrant, so my final answer is $\dfrac{5\pi}{4}$

However, the correct answer is $-\dfrac{3\pi}{4}$...why is that?

• The "principal argument" is often taken to be normalized to lie in $(-\pi,\pi]$. Then instead of adding $\pi$, you should have subtracted $\pi$. Jan 12 '14 at 15:32
• Now thats a good answer...thanks! :) Jan 12 '14 at 15:37

## 1 Answer

Using the definition of atan2, $$\arg(-5-5i)=\arctan\left(\frac{-5}{-5}\right)-\pi=\arctan1-\pi=\frac\pi4-\pi$$

• Wish I know the mistake here that caused the down-vote. The link contains all the information why the subtraction is done and also the range, so is not included in the answer Jan 12 '14 at 15:35
• because saying that the principal argument must be between -pi and pi is a much much better and simpler answer, thanks anyway. Jan 12 '14 at 15:39
• @MohammedAlKhatib, please find "This produces results in the range" in the link Jan 12 '14 at 15:40
• Your answer is absolutely correct except the notation that you need to use $\mathrm{Arg}$. Jan 12 '14 at 15:44
• @NasuSama, thanks for your feedback. But, its really unfortunate that OP has concern with the answer Jan 12 '14 at 15:47