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Math Help - Find solutions of z bar = z^2

  1. #1
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    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.
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by terrorsquid View Post
    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
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    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.
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by terrorsquid View Post
    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.
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by alexmahone View Post
    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~?
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by Plato View Post
    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}?
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by alexmahone View Post
    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}
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by alexmahone View Post
    How about z=-\frac{1}{2}+i\frac{\sqrt{3}}{2}?
    You are right. Good catch.
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  9. #9
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by Plato View Post
    You are right. Good catch.
    I threw you off with my initial mistake, obviously
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  10. #10
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by terrorsquid View Post
    ...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).
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  11. #11
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by LoblawsLawBlog View Post
    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}
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    Re: Find solutions of z bar = z^2

    I know that \bar{z}z gives you |z|^2 right? the positive real part.
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    Re: Find solutions of z bar = z^2

    Quote Originally Posted by terrorsquid View Post
    I know that \bar{z}z gives you |z|^2 right? the positive real part.
    Yes. And |z|=1, so z^3=1.
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    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?
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  15. #15
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    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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