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Math Help - Complex numbers

  1. #1
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    Complex numbers

    Prove the following properties of any complex number z:

    Re[z]=(z+z(with line on top))/2





    Im[z]=(z+z(with line on top))/2i
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  2. #2
    Senior Member MaxJasper's Avatar
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    Re: Complex numbers

    z = x+i y
    z(with line on top) = x-i y
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  3. #3
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    Re: Complex numbers

    let z = a+bi

    then z* (i am writing z* for z-conjugate, rather than \overline{z}) to avoid extra typing. so i'm lazy. sue me.) is a-bi.

    now z+z* = (a+bi) + (a-bi) = (a+a) + (b-b)i = 2a + 0i = 2a, so (z+z*)/2 = a = Re(z).

    and: z-z* = (a+bi) - (a-bi) = (a-a) + (b-(-b))i = 0 + 2bi = 2bi, so (z-z*)/(2i) = 2bi/(2i) = b = Im(z).
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  4. #4
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    Re: Complex numbers

    Quote Originally Posted by MaxJasper View Post
    z = x+i y
    z(with line on top) = x-i y
    In LaTeX code [TEX] \overline{a+bi}[/TEX] gives  \overline{a+bi}
    Last edited by Plato; September 13th 2012 at 08:46 AM.
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  5. #5
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    Re: Complex numbers

    if you look closely, i used that code at least once in my post :P
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