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Math Help - Imaginary numbers

  1. #1
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    Imaginary numbers

    In the following, I don't understand how the expression on the LHS equals to the one on the RHS:

    Im \left( \frac{(-e^{\pi}-1)(1-i)}{(1+i)(1-i)} \right) =\frac{e^{\pi}+1}{2}

    ("Im" means "imaginary")

    Of course if on LHS we cancel out the (1-i) term on the numerator with the one on denominator we get

    Im \left( \frac{(-e^{\pi}-1)}{(1+i)} \right)

    And if i=1, then the denominator becomes 2. But I can't work out the rest. Any help is appreciated.
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  2. #2
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    Quote Originally Posted by demode View Post
    In the following, I don't understand how the expression on the LHS equals to the one on the RHS:

    Im \left( \frac{(-e^{\pi}-1)(1-i)}{(1+i)(1-i)} \right) =\frac{e^{\pi}+1}{2}

    ("Im" means "imaginary")

    Of course if on LHS we cancel out the (1-i) term on the numerator with the one on denominator we get

    Im \left( \frac{(-e^{\pi}-1)}{(1+i)} \right)

    And if i=1, then the denominator becomes 2. But I can't work out the rest. Any help is appreciated.
    You need to convert the whole complex number into the form a + ib so that you can read off the imaginary part.

    \frac{(-e^{\pi} - 1)(1 - i)}{(1 + i)(1 - i)} = \frac{-e^{\pi} + e^{\pi}i - 1 + i}{1 + 1}

     = \frac{-e^{\pi} - 1 + (e^{\pi} + 1)i}{2}

     = \frac{-e^{\pi} - 1}{2} + \left(\frac{e^{\pi} + 1}{2}\right)i.


    So what is the imaginary part of this complex number?
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  3. #3
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    Quote Originally Posted by demode View Post
    In the following, I don't understand how the expression on the LHS equals to the one on the RHS:

    Im \left( \frac{(-e^{\pi}-1)(1-i)}{(1+i)(1-i)} \right) =\frac{e^{\pi}+1}{2}

    ("Im" means "imaginary")

    Of course if on LHS we cancel out the (1-i) term on the numerator with the one on denominator we get

    Im \left( \frac{(-e^{\pi}-1)}{(1+i)} \right)

    And if i=1, then the denominator becomes 2. But I can't work out the rest. Any help is appreciated.
    What do you mean "if i= 2". Do you solve problems with real numbers by saying "if 1= 2"? You seem to think that "i" is a variable- it is not!

    If you are going to do problems with imaginary and complex numbers, it would be a really good idea to first learn what "imaginary" and "complex numbers" are.
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