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Math Help - complexe

  1. #1
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    complexe

    Hello Guys Hope you are Fine


    How to determine the Numbers complexes Z such as:
    Z, Z (2), 1+Z has the same Module?

    Thanks in Advance Guys
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  2. #2
    MHF Contributor red_dog's Avatar
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    What is the second number?
    Is Z^2, \ 2Z, \ Z-2, \ Z+2 or what? I don't understand.
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  3. #3
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    Quote Originally Posted by red_dog View Post
    What is the second number?
    Is Z^2 or what? I don't understand.
    it's Z^2
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  4. #4
    MHF Contributor red_dog's Avatar
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    So, we have to find Z such as |Z|=|Z^2|=|Z+1|.
    The first equality can be written as |Z|=|Z|^2\Rightarrow |Z|=0 or |Z|=1.
    If |Z|=0\Rightarrow Z=0\Rightarrow |Z+1|=1, but |Z+1| must be 0.

    If |Z|=1=|Z+1| and Z=x+yi, we have to solve the system:
    \displaystyle\left\{\begin{array}{ll}\sqrt{x^2+y^2  }=1\\\sqrt{(x+1)^2+y^2}=1\end{array}\right. or

    \displaystyle\left\{\begin{array}{ll}x^2+y^2=1\\x^  2+y^2+2x=0\end{array}\right.
    Substracting the first equation from the second we get x=-\frac{1}{2}.
    Now, plug x in the first equation and solve for y.
    We get y=-\frac{\sqrt{3}}{2} or y=\frac{\sqrt{3}}{2}.

    So, there are two complex numbers satisfying the problem:
    Z=-\frac{1}{2}-\frac{\sqrt{3}}{2}i and Z=-\frac{1}{2}+\frac{\sqrt{3}}{2}i
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  5. #5
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    ok Many Thanks i was the only one who made it
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