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Determining Variables Involving Complex Numbers

  1. Sep 18, 2011 #1
    1. The problem statement, all variables and given/known data

    Let a, b in R, not both zero. Find c, d in R such that (a+bi)^-1 = c+di

    2. Relevant equations

    i^2=-1

    R is the set of all real numbers

    3. The attempt at a solution
    I have a feeling I'm approaching this problem incorrectly but:

    1 = (a + bi)(c + di)
    =ac + adi + cbi + bdi^2 but i^2=-1
    so 1 = ac - bd + (ad + bc)i^2

    This is as far as I've attempted becuase I realised that my solution really isn't going anywhere. Maybe someone could just give me a hint to start me off on the right track.
     
  2. jcsd
  3. Sep 18, 2011 #2

    LCKurtz

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    You are on the right track, but that last equation should have i instead of i2. It should read:

    1 + 0i = (ac-bd)+(bc+ad)i

    Set the real and imaginary parts equal to each other, giving two equations in the two unknowns c and d.
     
  4. Sep 18, 2011 #3
    Oops, I had i written down but I just misstyped it as i^2 here.

    So I do:

    1-ac+bd=(bc+ad)i + 0i

    but I dont see how that gives me equations to solve for c and d.
     
  5. Sep 18, 2011 #4

    LCKurtz

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    What I meant is to set the real parts equal to each other and ditto the imaginary parts.
     
  6. Sep 18, 2011 #5
    so from:

    1+0i=ac-bd+(ad+bc)i

    Am I allowed to say that:

    ac-bd=1 and (ad+bc)i =0i ???

    If so, is this because the imaginary numbers of the equation cannot affect the real numbers and vice versa?
     
  7. Sep 18, 2011 #6

    LCKurtz

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    So ad + bc = 0, you don't need the i in that equation. But yes, two equations in the unknowns c and d.
     
  8. Sep 18, 2011 #7
    Ok, thank you very much
     
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