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Thread: Complex numbers 3

  1. #1 Complex numbers 3 
    Forum Masters Degree thyristor's Avatar
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    Sorry to bother you again guys, but I'm stuck on another problem and I would be grateful if somebody could help me.
    The roots of the equation (where z is a complex number) lie on the vertices of an equilateral triangle of sidelength 3. Determine what the complex numbers a and b are (not necessarily a certain number, but it could also be some sort of circle in the complex plane or another geometric object)
    Any suggestions?


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  3. #2 Re: Complex numbers 3 
    . DrRocket's Avatar
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    Quote Originally Posted by thyristor
    Sorry to bother you again guys, but I'm stuck on another problem and I would be grateful if somebody could help me.
    The roots of the equation (where z is a complex number) lie on the vertices of an equilateral triangle of sidelength 3. Determine what the complex numbers a and b are (not necessarily a certain number, but it could also be some sort of circle in the complex plane or another geometric object)
    Any suggestions?



    So, take the following complex numbers



    Let
    and

    Then and are solutions of where there are no constraints on c.

    All other solutions are obtained by translating by a fixed complex number or multiplying them by a fixed complex number of modulus 1, or a combination of both and then using the above formulas to calculate . and .

    To see this note that are points on an equilateral triangle centered at the origin with side of length 3, and that this applies also if translations and rotations are applied to the set.

    Now we need to add the additional constraint that .

    For the complex numbers that we have selected, and any rotations of them To obtain simply add to each number in any set of solutions yielding points on a circle in the complex plain centered at (1/3,0).


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  4. #3  
    Forum Masters Degree thyristor's Avatar
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    Okay, thanks for yout help, but in the key it says that a=1 and your solution gives, unless I'm mistaken, that a is not 1.
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