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Math Help - -1 = 1 Proof ! What is the mistake done to prove it ?

  1. #1
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    -1 = 1 Proof ! What is the mistake done to prove it ?

    Untitled Page check out this picture !

    i want to know what was the wrong step that guides us to -1=1 , thnx
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    Re: -1 = 1 Proof ! What is the mistake done to prove it ?

    The square root has more than one solution.
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    Re: -1 = 1 Proof ! What is the mistake done to prove it ?

    On the third step you can't just split (-1)^(2/2) into ((-1)^2)^(1/2), because that's only done for positive numbers.
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    Re: -1 = 1 Proof ! What is the mistake done to prove it ?

    the problem, as other have pointed out, is this step:

    \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)}

    -1 actually has TWO square roots, i and -i. unfortunately, there is no way to tell "which is the positive one".

    when you write out what this actually means you get:

    (\pm i)(\pm i) = \pm 1 which is far less controversial.

    if one writes out these complex numbers in polar form:

    (\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right))^2 = \cos(\pi) + i\sin(\pi) = \cos\left(\frac{2\pi}{2}\right) + i\sin\left(\frac{2\pi}{2}\right)

    that is, the "proper" square root of 1 to take on the RHS is the negative one, which leads to -1 = -1, which is hardly surprising.

    put another way, "angles" aren't uniquely defined by a number, we have to restrict the range of these numbers. this can lead to confusing results, when talking about "half-angles"

    (half of 360 degrees is 180 degrees, but half of 0 is 0, and these two angles (0 degrees and 360 degrees) represent "the same direction").
    Thanks from topsquark
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