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Math Help - Find the error!

  1. #1
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    Find the error!

    x \in \mathbb{R}, e^{x} = e^{x \frac{2 \pi i}{2 \pi i}} = e^{2 \pi i \frac{x}{2 \pi i}} = (e^{2 \pi i})^{\frac{x}{2 \pi i}} = (cos(2 \pi) + i \text{ } sin(2 \pi))^{\frac{x}{2 \pi i}} = 1
    Last edited by Lord Voldemort; September 7th 2013 at 10:07 PM.
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  2. #2
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    Re: Find the error!

    I think the error is in the step e^{x\frac{2i\pi}{2i\pi}}=e^{2i\pi\frac{x}{2i\pi} For similar reasons to why this is incorrect:
    e^{i\pi}=-1
    e^{2i\pi}=1
    ln(e^{2i\pi})=ln(1)
    2i\pi=0
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  3. #3
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    Re: Find the error!

    Quote Originally Posted by Lord Voldemort View Post
    x \in \mathbb{R}, e^{x} = e^{x \frac{2 \pi i}{2 \pi i}} = e^{2 \pi i \frac{x}{2 \pi i}} = (e^{2 \pi i})^{\frac{x}{2 \pi i}} = (cos(2 \pi) + i \text{ } sin(2 \pi))^{\frac{x}{2 \pi i}} = 1
    When x is real valued we cannot say that 1^x = 1 for all x. For example, what happens when x = 1/2?

    -Dan
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  4. #4
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    Re: Find the error!

    Quote Originally Posted by Shakarri View Post
    I think the error is in the step e^{x\frac{2i\pi}{2i\pi}}=e^{2i\pi\frac{x}{2i\pi} For similar reasons to why this is incorrect:
    e^{i\pi}=-1
    e^{2i\pi}=1
    ln(e^{2i\pi})=ln(1)
    2i\pi=0
    I don't understand though. Isn't that step just the associative property for multiplication?

    Quote Originally Posted by topsquark View Post
    When x is real valued we cannot say that 1^x = 1 for all x. For example, what happens when x = 1/2?

    -Dan
    I'm confused: 1^(1/2) = sqrt(1) = 1, right?
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  5. #5
    Forum Admin topsquark's Avatar
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    Re: Find the error!

    Quote Originally Posted by Lord Voldemort View Post
    I'm confused: 1^(1/2) = sqrt(1) = 1, right?
    What about 1^(1/2) = -1? Mind you the exponent in your problem is going to be imaginary so this isn't a direct example...but my point is that 1^x is, in general, multi-valued for x \in \mathbb{C}.

    -Dan
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    Re: Find the error!

    Greetings mathematicians,

    On a much lighter note -
    The error is here:
    Find the error!-brain.jpg

    XD
    fu456789
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