# -i = i [Disproven]

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Hey all,

Where am I making a mistake?

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

Cheers,

Gabe

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I think your mistake is that:

$(-1)(\sqrt{-1})\neq\sqrt{(-1)(-1)(-1)}$

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You are attempting to apply the relation $a^ma^n=a^{m+n}$ to a situation where this relation does not apply: complex numbers. You are also abusing the square root symbol, which strictly applies only to non-negative reals.

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Hey all,

Where am I making a mistake?

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

Cheers,

Gabe

$(-1)(\sqrt{-1})\neq\sqrt{(-1)(-1)(-1)}$ because $\sqrt{a^2}=\left|a \right|$

aspon myslim...

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If $\sqrt{-1}$ is illegal, how can $i$ be defined?

aspon myslim...

I told you, no Czech; only English

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The error in the original post ultimately lies with the implicit use of $\sqrt{(-1)^2}=-1$. There is no need for imaginary numbers here. For example, using $(-1)^4 = 1 = ((-1)^2)^2$,

\aligned 1 &= \sqrt{1} \\ &= \sqrt{(-1)^4} \\ &= \sqrt{(-1)^2}\sqrt{(-1)^2} \\ &= \sqrt{(-1)^2}\sqrt{1} \\ &= (-1)\times 1 \\ &= -1 \endaligned

The square root function takes a branch cut at x=0. Proofs that -x=x involve an abuse of this branch cut. Don't do that!

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Ah, I see now. My bad. Thanks for explaining guys ;-)

Cheers,

Gabe

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• 2 weeks later...

You can proove a negative very easily, by saying,

$x=zy^2$

$2x=zy^2+x$

Divide by 2 on both sides

$x= \frac{1}{2} (zy^2+x)$

Manipulating through algebra gives:

$\frac{1}{2} x-x= \frac{1}{2} (zy^2)$

Solve the equation;

$\frac{1}{2} x= -\frac{1}{2} zy^2$

Replace $\frac{1}{2}$ x with $\frac{1}{2}$ zy^2 then

$\frac{1}{2} x= -\frac{1}{2} x$

Which is my own derivation.

Edited by MolecularEnergy
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$\frac{1}{2} x-x= \frac{1}{2} (zy^2)$

It should be:

$x= \frac{1}{2} (zy^2) + \frac{1}{2} x$

$x - \frac{1}{2} x= \frac{1}{2} (zy^2)$

Which works out fine.

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Why... i thought the algebra i used was fine?

Obviously not though, if you are adament it is wrong.

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But the algebra wasn't fine. You made a mistake. Check it again. Klaynos showed you the step you made the mistake on.

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I know, i see it now. I am sorry.

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