Thread: Find solutions of z bar = z^2

1. Find solutions of z bar = z^2

Im trying find all solutions for: $\bar{z}=z^2$

Should I convert z into $x +iy$ or $r(cos(\theta) +isin(\theta))$ if I want to interpret my result geometrically?

What I have done so far is:

$x-iy = (x+iy)^2$

$x-iy = x^2+i2xy-y^2$

$x - iy = x^2 - y^2 + i2xy$

Therefore $x = x^2 -y^2$ and $y = 2xy$

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 $6 + i\sqrt3$ etc. Not sure which direction to head to solve for this equation so I can interpret geometrically.

Thanks.

2. Re: Find solutions of z bar = z^2

Originally Posted by terrorsquid
Im trying find all solutions for: $\bar{z}=z^2$

Should I convert z into $x +iy$ or $r(cos(\theta) +isin(\theta))$ if I want to interpret my result geometrically?

What I have done so far is:

$x-iy = (x+iy)^2$

$x-iy = x^2+i2xy-y^2$

$x - iy = x^2 - y^2 + i2xy$

Therefore $x = x^2 -y^2$ and $\color{red}y = 2xy$
You have a mistake. It should be:
$-y=2xy$

3. Re: Find solutions of z bar = z^2

So I get $x = -\frac{1}{2}$ and $x = x^2 -y^2$

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

4. Re: Find solutions of z bar = z^2

Originally Posted by terrorsquid
So I get $x = -\frac{1}{2}$ and $x = x^2 -y^2$

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.

5. Re: Find solutions of z bar = z^2

Originally Posted by alexmahone
Can you find y from the second equation?
Also, don't forget that y = 0 is also a solution of -y = 2xy.
There are only two solutions to this question:
$z=0\text{ or }z=1$.

Note that $z=\bar{z}$ if and only if $\text{Im}(z)=0$ or $z\in\mathbb{R}$.

What real numbers have the property that $x=x^2~?$

6. Re: Find solutions of z bar = z^2

Originally Posted by Plato
There are only two solutions to this question:
$z=0\text{ or }z=1$.
How about $z=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$?

7. Re: Find solutions of z bar = z^2

Originally Posted by alexmahone
Can you find y from the second equation?

Also, don't forget that y = 0 is also a solution of -y = 2xy.
$y = \sqrt{x^2-x}$

$\therefore$ y = 0 when x = 0

and

$y = \sqrt{\frac{3}{4}} = \frac{\sqrt3}{2}$ when $x = -\frac{1}{2}$

8. Re: Find solutions of z bar = z^2

Originally Posted by alexmahone
How about $z=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$?
You are right. Good catch.

9. Re: Find solutions of z bar = z^2

Originally Posted by Plato
You are right. Good catch.
I threw you off with my initial mistake, obviously

10. Re: Find solutions of z bar = z^2

Originally Posted by terrorsquid
...so I can interpret geometrically.
Since you mentioned this, here's another approach:

Take absolute values of both sides of $\bar{z}=z^2$, getting $|\bar{z}|=|z|=|z|^2$, 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 $z\bar{z}=z^3$. Do you know an identity relating $z\bar{z}$ 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).

11. Re: Find solutions of z bar = z^2

Originally Posted by LoblawsLawBlog
Since you mentioned this, here's another approach:

Take absolute values of both sides of $\bar{z}=z^2$, getting $|\bar{z}|=|z|=|z|^2$, 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 $z\bar{z}=z^3$. Do you know an identity relating $z\bar{z}$ 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: $z=-\frac{1}{2}-i\frac{\sqrt{3}}{2}$

12. Re: Find solutions of z bar = z^2

I know that $\bar{z}z$ gives you $|z|^2$ right? the positive real part.

13. Re: Find solutions of z bar = z^2

Originally Posted by terrorsquid
I know that $\bar{z}z$ gives you $|z|^2$ right? the positive real part.
Yes. And $|z|=1$, so $z^3=1$.

14. Re: Find solutions of z bar = z^2

So applying this geometrically I would have a solution set of $cos(0+\frac{2\pi k}{3})$ where $k = 0, 1, 2$

Which is the same as the $z = 1$, $z=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$ and $z=-\frac{1}{2}-i\frac{\sqrt{3}}{2}$

But where does the z = 0 fit in?

15. 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.

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