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Imaginary number -i raised to negative power

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data

    I came across this expression in homework and for the life of me I can't figure out how this evaluates to 0: 1 - ( -i )^-4 = 1 - 1 = 0

    I know that i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. I'm just not sure how to treat the negative on the i. Do I just treat i as if it were just a regular number, i.e. (-1)^4 = 1?
    Or can I just say ( -i )*( -i )*( -i )*( -i ) = 1? Can anyone shed some light on this. I know it has to be painfully simple but for some reason I just can't see it.

    Thanks
     
  2. jcsd
  3. Feb 23, 2010 #2
    Distribute the power:
    [tex](-i)^4 = (-1)^4i^4[/tex]
     
  4. Feb 24, 2010 #3

    HallsofIvy

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    [itex]i(-i)= -i^2= -(-1)= 1[/itex] so i and -i are reciprocals. In particular, [itex](-i)^{-1}= i[/itex] and so [itex](-i)^{-4}= i^4= 1[/itex].
     
  5. Feb 24, 2010 #4
    it is very easy problem. See,

    [tex]

    1 - i^{-4}[/tex]

    = [tex]1 - \frac {1}{i^4}[/tex]

    = [tex]1 - \frac {1}{1}[/tex]

    = 1 - 1

    = 0
     
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