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1 = -1 where is the error

  1. Nov 18, 2005 #1
    I've tried and tried and I cannot find the error in the reasoning below. It's probably something simple and I'll feel like an idiot when someone explains it.
    i = i
    sqrt(-1) = sqrt(-1)
    sqrt(1/-1) = sqrt (-1/1)
    sqrt(1)/sqrt(-1) = sqrt(-1)/sqrt(1)
    sqrt(1) * sqrt(1) = sqrt(-1) * sqrt(-1)
    [sqrt(1)]^2 = [sqrt(-1)]^2
    1 = -1

    Does it have something to do with sqrt(-1/1) = sqrt(-1)/sqrt(1)? Does complex numbers not obey this property?
     
  2. jcsd
  3. Nov 18, 2005 #2
    doesnt [sqrt(a)]^2 = |a| ? maybe thats just in the reals
     
    Last edited: Nov 18, 2005
  4. Nov 18, 2005 #3
    [tex]\frac{\sqrt{1}}{\sqrt{-1}}[/tex] pretty much sums up what's wrong with the train of thought. This is a good reason why when you divide a real number by an imaginary number that you must first multiply by the conjugate on the numerator and denominator. I can't nail down a good reason other than that.

    I should note that this inconsitency does not exist if you do the division with sqrt(1) and sqrt(-1) in polar form.
     
  5. Nov 19, 2005 #4

    matt grime

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    square rooting is 1 to 2, so you need to pick a choice of square root. You've not done so consitently.
     
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