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For z = x+iy find the relationship between x and y

  1. Feb 2, 2016 #1
    1. The problem statement, all variables and given/known data
    For z = x+iy find the relationship between x and y so that (Imz2) / z2 = -i

    2. The attempt at a solution
    I attempted this in a few different ways (i.e. looking at the exponential and trig forms of complex numbers)... I settled on simple FOIL which gave me the following:

    (x+iy)^2 = x^2 + i2xy - y^2

    The imaginary part is 2xy; so:

    2xy / (x^2 + i2xy - y^2) = -i <-- from original problem

    from here, multiplying -i by the denominator gives:

    2xy = -ix^2 + 2xy + iy^2


    Cancel out 2xy to get zero on the left side, and factor out i, leaving:

    x^2 = y^2

    However.... this does not seem to produce any solutions resulting in -i.

    For example, if x=2 and y=2, z^2 = 8i ... for x=-2 and y=2, z^2 = -8i ...

    The problem is, there never seems to be a sign change for Im(z^2) over z^2.

    Is this problem flawed, or am I missing something obvious...?
     
  2. jcsd
  3. Feb 2, 2016 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    What is wrong with those examples?

    Im(8i)=8. What is 8/(8i)?
    Im(-8i)=-8. What is -8/(-8i)?
     
  4. Feb 2, 2016 #3
    Ah, I didn't realize that 1/i equals -i.

    Thanks for the quick response!
     
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