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Properties of Complex Numbers (phasor notation)

  1. Jul 12, 2009 #1
    1. Statement:
    The Real Part of a "Complex Number is expressed as the following:
    [tex]Real(A) = \frac{1}{2}(A + A*) = \frac{1}{2}(|A|e^{j\alpha} + |A|e^{-j\alpha}) = \frac{1}{2}|A|(2cos(\alpha)) = |A|cos(\alpha)[/tex]. (#1)

    The Imaginary Part of a "Complex Number" is expressed as the following:
    [tex]Imag(A) = \frac{1}{2}(A - A*) = \frac{1}{2}(|A|e^{j\alpha} - |A|e^{-j\alpha}) = \frac{1}{2}|A|(2jsin(\alpha)) = j|A|sin(\alpha)[/tex]. (#2)


    2. Questions:
    I was just curious how [tex]\frac{1}{2}|A|(2cos(\alpha))[/tex] was derived in equation (#1), and how [tex]\frac{1}{2}|A|(2jsin(\alpha))[/tex] was derived in equation (#2)?

    thanks,


    Jeff
     
  2. jcsd
  3. Jul 13, 2009 #2

    tiny-tim

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    e = cosα + jsinα
    e-jα = cosα - jsinα

    so e + e-jα = 2cosα
    e - e-jα = 2jsinα
     
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