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Math Help - Complex Numbers

  1. #1
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    Complex Numbers

    I need to find the rectangular form of i ^ (-3) + i ^ (-2)
    i = imaginary number
    The answer in my book says -1 + i but i do not know how they got it
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  2. #2
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    Quote Originally Posted by OzzMan View Post
    I need to find the rectangular form of i ^ (-3) + i ^ (-2)
    i = imaginary number
    The answer in my book says -1 + i but i do not know how they got it
    i^{-3} = \frac{1}{i^3} = \frac{1}{i \times i \times i} = \frac{1}{-i}

    Multiply top and bottom by i:

    = \frac{i}{-i \times i} = \frac{i}{1} = i.

    i^{-2} = \frac{1}{i^2} = \frac{1}{i \times i} = \frac{1}{-1} = -1.

    Therefore ......
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  3. #3
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    Quote Originally Posted by OzzMan View Post
    I need to find the rectangular form of i ^ (-3) + i ^ (-2)
    i = imaginary number
    The answer in my book says -1 + i but i do not know how they got it
    Hello,

    i^{-3}+i^{-2}= \frac{1}{i^3} + \frac{1}{i^2}= \frac{1}{(-1) \cdot i} + \frac{1}{-1}= \frac{1}{(-1) \cdot i} \cdot \frac{i}{i} - 1  = \frac{i}{(-1) \cdot (-1)} - 1 = i-1
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  4. #4
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    why do you multiply top and bottom by i ?
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  5. #5
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    oh is it because the conjugate of -i is i?
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  6. #6
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    Quote Originally Posted by OzzMan View Post
    I need to find the rectangular form of i ^ (-3) + i ^ (-2)
    [snip]
    Quote Originally Posted by mr fantastic
    [snip]
    Multiply top and bottom by i:

    = \frac{i}{-i \times i} = \frac{i}{1} = i.
    [snip]
    Quote Originally Posted by OzzMan View Post
    why do you multiply top and bottom by i ?
    Quote Originally Posted by OzzMan View Post
    oh is it because the conjugate of -i is i?
    Well, if you have to give an answer in rectangular form, then \frac{1}{-i} is not really in the form you want, is it? So what would you do .....?

    Returning to

    Quote Originally Posted by OzzMan View Post
    why do you multiply top and bottom by i ?
    Quote Originally Posted by OzzMan View Post
    oh is it because the conjugate of -i is i?
    Well, yes it is. But it also happens to be the easiest way to express \frac{1}{-i} in rectangular form .....
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