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Math Help - proof of -1 = +1

  1. #16
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    Quote Originally Posted by Jhevon View Post
    yes, proof by counter-example is obvious here. i was hoping for something else. because this seems to imply a larger problem, namely, the distribution of powers over a product. if it doesn't work for the 1/2 power, that would mean it probably won't work for all or some other powers, right? a proof i would like to see is when this move is illegal, or is it always illegal? is (ab)^3 = a^3b^3 false for a,b \in \mathbb{C}? etc
    The equation (ab)^n = a^nb^n is true for a,b\in \mathbb{C}, ~ n\in \mathbb{Z}.
    Furthermore, (a^b)^n = a^{bn}.
    However, (a^b)^c = a^{bc} --> fails!
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  2. #17
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    there is an error because
    example x^2=(-)*(-)=+
    or=(+)*(+)=+
    but you can say that redical -1 * -1 =1
    then -1 is undifined.
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  3. #18
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by lebanon View Post
    there is an error because
    example x^2=(-)*(-)=+
    or=(+)*(+)=+
    but you can say that redical -1 * -1 =1
    then -1 is undifined.
    ??

    see post #16, it tells you the rule you are using does not work in general for complex numbers.
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  4. #19
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    Let f:\mathbb{C}\to \mathbb{C} be defined as f(x)=x^\frac{1}{2}. Then f is not a well defined map, because there exists x\in\mathbb{C} such that f(x) maps to two different elements in \mathbb{C}

    This doesn't really apply to your proof, but in general be careful when you use the square root with complex numbers, as you'll often get multiple answers. In fact, when you use the square root with real numbers, there are multiple possible answers, but it's convention to ignore the negative parts. I could do the same proof without complex numbers.

    We know that (-6)^2=36 and the square root and square operations are inverses so they cancel and \sqrt{36}=\sqrt{(-6)^2}=-6 .

    And we know that \sqrt{a}\sqrt{b} = \sqrt{ab} .
    example:  \sqrt{4}\sqrt{9} = \sqrt{4(9)} = \sqrt{36}

    So 6 = 2(3) = \sqrt{4}\sqrt{9} = \sqrt{(4)(9)} = \sqrt{36} .

    So 6=-6. gg.

    Where is the problem with this?

    Quote Originally Posted by mnova View Post
    [I hope I did the LaTex correctly.]

    We know that \textit{i}^2 = \sqrt{-1}^2 and the square root and square operations are inverses so they cancel and \textit{i}^2 = -1 .

    And we know that \sqrt{a}\sqrt{b} = \sqrt{ab} .
    example: 2\textit{i} = \sqrt{4}\sqrt{-1} = \sqrt{4(-1)} = \sqrt{-4}

    So \textit{i}^2 = \textit{ii} = \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{+1} = +1 .

    So -1 = +1
    QED

    Where's the problem with this?
    Last edited by Yendor; February 2nd 2009 at 10:24 PM.
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  5. #20
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    Quote Originally Posted by Yendor View Post
    Let f:\mathbb{C}\to \mathbb{C} be defined as f(x)=x^\frac{1}{2}. Then f is not a well defined map, because there exists x\in\mathbb{C} such that f(x) maps to two different elements in \mathbb{C}
    Yes it is well-defined. There are two ways to define the square root, but the one used in post #1 is the principal square root: \sqrt{z} = \sqrt{|z|}e^{i\arg(z)},z\not = 0. Once you settle with this definition the square root function is well-defined.
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  6. #21
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    another proof

    another one...find out whats wrong...





    perhaps its easier to prove -1=1 than to prove 1=1. lol.
    Last edited by adhyeta; May 4th 2009 at 07:50 AM. Reason: the last line
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  7. #22
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    \infty - \infty is an indeterminate expression.
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  8. #23
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    Quote Originally Posted by Chop Suey View Post
    The property \sqrt{ab} = \sqrt{a} \sqrt{b} is defined for a, b \geq 0. And hence, as TPH mentioned, your step here:
    <br />
\textit{i}^2 = \textit{ii} = \sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{+1} = +1<br />

    is erroneous.
    Yes, that's right and the actual solution is:


    {i}^2 = \textit{ii} = \sqrt{-1}\sqrt{-1} = \textit{(-1)}^ {0.5+0.5}= -1
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