The number of complex numbers that satisfy the equation

[Chuckster]

Homework Statement

So, i have this equation, and it is asked of me to find the number of complex numbers that satisfy the equation. (z=x+iy)

Homework Equations

$$z-\overline{z}+|z-i|=4-2i$$

The Attempt at a Solution

I tried replacing the numbers and i got something like this
$$x+iy-x+iy+\sqrt{x^{2}+(y-1)^{2}}=4-2i}$$

After that, the calculus gets a little complicated, so I'm wondering if I'm going in the right direction?

 

[spamiam]

Your approach is fine, but you made several silly errors. If $z = x+iy$, then $\overline{z} = x - iy$. Also z - i = x + iy - i = x + i(y-1).

Anyway, there's actually an easy way to see the answer to this problem. Recall that the real part of z can be calculated by

$$\text{Re}(z) = \frac{z + \overline{z}}{2}$$

So what kind of a number is $z + \overline{z}$? And |z - i|? (Don't just say complex.)

 

[Chuckster]

Your approach is fine, but you made several silly errors. If $z = x+iy$, then $\overline{z} = x - iy$. Also z - i = x + iy - i = x + i(y-1).

Anyway, there's actually an easy way to see the answer to this problem. Recall that the real part of z can be calculated by

$$\text{Re}(z) = \frac{z + \overline{z}}{2}$$

So what kind of a number is $z + \overline{z}$? And |z - i|? (Don't just say complex.)

I made an error while copying the original equation, and partly copying my idea, fixed it in the original post now.
I think my first step is okay now, having in mind changes i made?

It's important that this is the right way. I'll just finish it, i guess i made a mistake in the calculus somewhere along the way.

 

[spamiam]

Ah okay, that changes things! For your original equation, the left-hand side was all real numbers, so the equation had no solutions.

Yes, your revised attempt looks good. Now you just have to relate the real and imaginary parts on the right- and left-hand sides.