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-1 = 1

  1. Nov 1, 2004 #1


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    heres a little problem that at a first glance is real:

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

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

    by splitting it the square root into two parts...

    [tex] \frac{i}{1} = \frac{1}{i} [/tex]
    and [tex] i^2 = 1 [/tex]

    -1 = 1

    wonder if there are any more similar "proofs"?
  2. jcsd
  3. Nov 1, 2004 #2


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    You can't split the square root into two parts. There are plenty of similar "proofs". You can search the web for them, and there are a number of them on this site alone.
  4. Nov 1, 2004 #3
    You can not take the square root of a negative number.
  5. Nov 1, 2004 #4
    [tex] \sqrt{-1}=i [/tex]

    imaginay numbers allow for negitive sqroots

    he just violated a law in the way he split up his negitive signs.
  6. Nov 1, 2004 #5
  7. Nov 1, 2004 #6
  8. Nov 2, 2004 #7


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    A thinly veiled version of the same, though the fallacy is perhaps more transparent:

    Euler's formula tells us:


    So we see that:


    taking roots gives:


    Using Euler's formula again and we get:


    or -1=1
  9. Nov 2, 2004 #8


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    Here's another (though this one cheats in a different way) :

    [tex]1 = \sqrt{1} = \sqrt{(-1)(-1)} = \sqrt{-1} \sqrt{-1} = i^2 = -1[/tex]
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