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I was messing around in Geometry class today and found a very odd 'proof'.

It relies on only two facts, $1^2=1$ and $i=\sqrt{-1}$

From here I did this:

$$i = \sqrt{-1}$$ $$i^2 = -1$$ $$-i^2 = 1$$

therefore, since $-i^2 = 1$ and $1^2 = 1$:

$$-i^2=1^2$$ $$-i=1$$ $$i=-1$$

but this is obviously inconsistent, since $-1^2 \neq i^2$

what in the world (of math) did I do wrong, since this result is clearly impossible??

Also, any tags I should add to this question?

AlphaModder
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    An equally interesting fallacy: $-1=(-1)^{2/2}=[(-1)^2]^{1/2}=\sqrt{1}=1$. As others have mentions in their answers the mistake here is at the step: $\sqrt{1}=1$ actually $\sqrt{1}$ is defined as the positive root of the equation $x^2=1$, but there is also a negative root here (because $(-1)^2=1$). – Pantelis Sopasakis Dec 15 '14 at 23:59

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Not quite; at the line where you have $-i^2=1^2$ and you take a square root, you should have $$\sqrt{-i^2} = \sqrt{-1}\sqrt{i^2} = i \cdot i = -1 \neq 1$$

graydad
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    And even this can lead you astray: $(-1)^2 = 1^2$, but "taking the square root of both sides" doesn't prove $-1 = 1$. – BaronVT Dec 15 '14 at 23:49
  • Ah, I see. I was under the impression that since $i^2 = i * i$ then $-1(i^2)$ could be distributed to $-i * -i$ which the square root of would be $-i$ making it $-i$. Whoops. – AlphaModder Dec 15 '14 at 23:58
  • I should point out that technically $\sqrt{i^2} = \pm i $. There are two solutions to $\sqrt{i^2}$, and one of them would make your equality true, namely using $\sqrt{i^2}=-i$ But the other solution (which I used in my answer) will cause your equality to fail, so you have to conclude overall that there is no equality. As for distributing the negative sign, is it true that $-4 = -2 \cdot -2$ simply because $2^2=4$? – graydad Dec 16 '14 at 00:08
  • @AlphaMCubed Try plugging in other numbers there - there's nothing 'special' about $i$ in that regard. $-1(4^2)\neq (-4)\cdot(-4)$! – Steven Stadnicki Dec 16 '14 at 00:30
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The flaw is that you can't conclude $-i = 1$ from $-i^2 = 1^2$.

user2566092
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