Finding a Complex Number Given Arg and Modulus

[squenshl]

Homework Statement

If $\text{arg}(w)=\frac{\pi}{4}$ and $|w\cdot \bar{w}|=20$, then what is $w$ of the form $a+bi$.

Homework Equations

The Attempt at a Solution

The only way for the argument of $w$ to be $\frac{\pi}{4}$ is when $a+bi$ where $a=b \in \mathbb{Z}$ right?

 

[andrewkirk]

That is correct.

 

[squenshl]

That is correct.

Great thanks.
I get $a=b= \pm\sqrt{10}$ so $w$ follows.

 

[Ray Vickson]

Great thanks.
I get $a=b= \pm\sqrt{10}$ so $w$ follows.

No, $a = b = -\sqrt{10}$ would not be correct; only the $+\sqrt{10}$ answer applies.

 

[RPinPA]

No, $a = b = -\sqrt{10}$ would not be correct; only the $+\sqrt{10}$ answer applies.

Do you understand Ray's point? What is $\arg(w)$ if $a = b$ and both are negative?

 

[squenshl]

Do you understand Ray's point? What is $\arg(w)$ if $a = b$ and both are negative?

I do.
$w=-\sqrt{10}-\sqrt{10}i$ is in the wrong quadrant.
$\text{arg}(w)=-\pi+\frac{\pi}{4}$.