Finding Argument of Complex Number Given Equations

[NEILS BOHR]

Homework Statement

let z , w be complex nos. such that z + i ( conjugate of w ) = 0 and zw = pi . Then find arg z..

Homework Equations

The Attempt at a Solution

well i m unable to understand wat is meant by zw=pi...

 

[eumyang]

You sure you wrote the problem correctly? I found one very similar to this by searching via Google. The problem goes like this:
"Let z, w be complex numbers such that

$$\bar{z} + i\bar{w} = 0$$

and

arg(zw) = π.

Find arg(z)."

 

[NEILS BOHR]

hmmm
well in the quesn it is given like this only...

but yeah without arg it doesn't make much of a sense...

 

[Char. Limit]

It is possible for z*w = pi, but it's more likely that arg(z*w) = pi.

 

[NEILS BOHR]

can u pleasez elaborate a little??

 

[HallsofIvy]

If $z= r_ze^{i\theta_z}$ and $w= r_we^{i\theta_w}$ then [itex]arg(zw)= \theta_z+ \theta_w= \pi[itex] so $\theta_z$ and $\theta_w$ are supplementary angles.

Saying that $z+ i\overline{w}= 0$ means that $z= -i\overline{w}$ and so $arg(z)= arg(i\overline{w})- \pi$.

Now, taking the conjugate of a complex number multiplies its argument by -1 and multiplying by i adds $\pi/2$ to the argument. That is, if w has argument $\theta$, then $i\overline{w}$ has argument $\pi/2- \theta$. From the previous paragraph, $arg(z)= arg(i\overline{w})+ \pi= \pi2- \theta+ \pi= -(\theta+ \pi/2)$. More than that, I don't believe you can say.