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Thread: Complex Numbers

  1. #1
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    Complex Numbers

    Solve for Z when

    a) (Z+i)(1-i)=2+3i

    b) 1/z + 1/(2-i) = 3/(1+i)
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  2. #2
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    Part A

    Quote Originally Posted by djmccabie View Post
    Solve for Z when

    a) (Z+i)(1-i)=2+3i
    $\displaystyle
    (Z+i)=(2+3i)/(1-i)
    $
    $\displaystyle Z+i=[(2+3i)(1+i)] / [(1-i)(1+i)]$ (multiply top and bottom by 1+i)
    $\displaystyle

    Z+i=(2+2i+3i+3i^2) / (1-i^2)
    $ (difference of two squares)

    $\displaystyle
    Z+i=(-1+5i)/2
    $
    $\displaystyle
    Z=-(1/2)+(3/2)i
    $
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  3. #3
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    Part B

    Quote Originally Posted by djmccabie View Post
    Solve for Z when

    b) 1/z + 1/(2-i) = 3/(1+i)


    $\displaystyle
    (1/z)=3/(1+i)-1/(2-i)

    $
    $\displaystyle
    1/z=[3(2-i)-(1+i)]/[(1+i)(2-i)]
    $
    $\displaystyle
    1/z=(5-2i)/[(1+i)(2-i)]
    $
    $\displaystyle
    z=(1+i)(2-i)/(5-2i)
    $
    $\displaystyle
    z=(1+i)(2-i)(5+2i)/[(5-2i)(5+2i)]
    $
    $\displaystyle
    z=(-1+17i)/29
    $
    $\displaystyle
    z=-(1/29)+(17/29)i
    $
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