Find solutions of z bar = z^2

Im trying find all solutions for:

Should I convert z into or if I want to interpret my result geometrically?

What I have done so far is:

Therefore and

Not sure what to do with my results. I'm a little lost with the general values for z; I am fine with the definite values such as etc. Not sure which direction to head to solve for this equation so I can interpret geometrically.

Thanks.

Re: Find solutions of z bar = z^2

Quote:

Originally Posted by

**terrorsquid** Im trying find all solutions for:

Should I convert z into

or

if I want to interpret my result geometrically?

What I have done so far is:

Therefore

and

You have a mistake. It should be:

Re: Find solutions of z bar = z^2

So I get and

I'm not sure what to do with these results.

Re: Find solutions of z bar = z^2

Quote:

Originally Posted by

**terrorsquid** So I get

and

I'm not sure what to do with these results.

Can you find y from the second equation?

Also, don't forget that y = 0 is also a solution of -y = 2xy.

Re: Find solutions of z bar = z^2

Re: Find solutions of z bar = z^2

Quote:

Originally Posted by

**Plato** There are only two solutions to this question:

.

How about ?

Re: Find solutions of z bar = z^2

Re: Find solutions of z bar = z^2

Quote:

Originally Posted by

**alexmahone** How about

?

You are right. Good catch.

Re: Find solutions of z bar = z^2

Quote:

Originally Posted by

**Plato** You are right. Good catch.

I threw you off with my initial mistake, obviously :D

Re: Find solutions of z bar = z^2

Re: Find solutions of z bar = z^2

Quote:

Originally Posted by

**LoblawsLawBlog** Since you mentioned this, here's another approach:

Take absolute values of both sides of

, getting

, so |z|=0 or |z|=1. Since 0 is easily seen to be a solution, we're focused on possible solutions on the unit circle.

Now multiply your original equation by z, getting

. Do you know an identity relating

and |z|? This should lead you to consider cube roots of unity, which have a nice geometric interpretation. You might get some extraneous solutions here though (possibly, too tired to trust myself now lol).

Thanks to you, I just realized that I had missed a solution:

Re: Find solutions of z bar = z^2

I know that gives you right? the positive real part.

Re: Find solutions of z bar = z^2

Quote:

Originally Posted by

**terrorsquid** I know that

gives you

right? the positive real part.

Yes. And , so .

Re: Find solutions of z bar = z^2

So applying this geometrically I would have a solution set of where

Which is the same as the , and

But where does the z = 0 fit in?

Re: Find solutions of z bar = z^2

|z|=0 (so z=0) was one of the solutions to |z|=|z|^2. Once you verify that 0 is a solution, then you can focus on the solutions where |z|=1, which you just listed, so we have all 4 solutions.