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Complex numbers finding a and b

  1. Apr 15, 2014 #1
    Hello,

    1. The problem statement, all variables and given/known data

    The complex numbers [itex]z_{1} = \frac{a}{1 + i}[/itex] and [itex]z_{2} = \frac{b}{1+2i}[/itex] where a and b are real, are such that [itex]z_{1} + z_{2} = 1. Find a and b.

    2. Relevant equations



    3. The attempt at a solution

    This looked like a time for partial fractions to me, so I went down that road;

    [itex] \frac{a}{1 + i} + \frac{b}{1+2i} = 1[/itex]

    [itex]\frac{a(1+2i)}{(1 + i)(1+2i)} + \frac{b(1+i)}{(1+2i)(1+i)} = 1[/itex]

    [itex]a(1+2i) + b(1+i) = (1+i)(1+2i)[/itex]

    Expanding the brackets gives me;

    [itex]a(1+2i) + b(1+i) = (1+i)(1+2i)[/itex]

    [itex]a(1+2i) + b(1+i) = -1 + 3i[/itex]

    ∴ [itex]a + b = -1[/itex]

    And now i'm stuck...

    Is this the right approach? And, how do I move forward?

    Thanks!
     
  2. jcsd
  3. Apr 15, 2014 #2

    vela

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    What about the imaginary part? What equation does that give you?
     
  4. Apr 15, 2014 #3
    Aha, woops.

    2a + b = 3

    Which allows me to work out via simultaneous equations that a = 4 and b = -5.

    Thanks.
     
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