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Math Help - Complex Conjugation Confusion

  1. #1
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    Question Complex Conjugation Confusion

    Hi

    I seem to have contradicting rules on how to treat complex conjugates

    where we have

    a = x + iy
    a*= x - iy

    but in an example showing how you can find whether a complex expression is real or imaginery we have

    b = i(cd* - c*d)
    b* = -i(c*d - cd*)

    so to compare to my previous expression

    if we say x = 0 and y = (cd* - c*d)

    we have the first example saying

    a = x + iy
    a*= x - iy

    and the second saying

    b = x + iy
    b*= x - iy*

    is it simply that the * is dropped when a variable is known to be real ?

    I am new to this subject and any insights would be most welcome !
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  2. #2
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    Staten Island, NY
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    I'm not quite sure what you're asking, but I'll try to answer you anyway.

    If a is a complex number, then a can be written as a=x+iy where x and y are real numbers. The conjugate of a is a*=x-iy. A complex number a is real if and only if a=a* (or equivalently, y=0).


    If you want to make sense of what b is in terms of real numbers, then you need to let c=x+iy and d=z+iw for real numbers x,y,z,w and then simplify those expressions.
    Saying y=(cd*-c*d) doesn't make sense until you are sure that cd*-c*d is real (which I believe it is, but it must be checked first).

    I'm not sure if I understood your question but I hope some of this helps.
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  3. #3
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    Hi, thanks for your awnser

    Yes sorry I seem to have left out some details! I will try and phrase it again, thanks for bearing with me!

    I have 2 examples of conjugates taken of 2 different complex numbers and they seem to be create different definitions.

    Where a and b are complex numbers I have the definitions of
    a = x + iy
    a*= x - iy
    and
    b = i(cd* - c*d)
    b* = -i(c*d - cd*)

    If I substitue (cd* - c*d) for y and 0 for x in the second equation then the conjugate of b looks like

    b*= x - iy*
    not
    b*= x - iy

    which is defined by the first equation where

    a*= x - iy

    I am asking why the y is turned into the conjugate of y on the second example and not on the first where i is simply reversed and y is left alone.

    Thanks again
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  4. #4
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    y is a real number. So y*=y
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  5. #5
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    Of course ...... sorry I was being stupid !

    thanks I could have gone round in circles for days over that
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