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I was looking on Howard Eves's book "An Introduction to the History of Mathematics" and I stumbled upon a demonstration on how $-1 = 1$. The demonstration follows:

$$ \sqrt{-1} = \sqrt{-1} $$

$$ \sqrt{-1\over1} = \sqrt{1\over-1}$$

$$ {\sqrt{-1}\over\sqrt{1}} = {\sqrt{1}\over\sqrt{-1}}$$

$$ \sqrt{-1}\cdot\sqrt{-1} = \sqrt{1}\cdot\sqrt{1} $$

Thus:

$$ -1 = 1 $$

My question is simple: how can it be? Where is the error? Is that a paradox?

Thanks in Advance!

Git Gud
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Elisabeth Bracher
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    http://math.stackexchange.com/questions/438/why-sqrt-1-times-1-neq-sqrt-12 already addresses this question. – JB King Jun 15 '15 at 16:14
  • roots don't obey quite the laws you think they do (ie for real roots) when they have complex numbers under them... – danimal Jun 15 '15 at 16:24

2 Answers2

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The "rule" you use from line 2 to 3 doesn't hold for negative numbers. I.e. $$ i=\sqrt{-1}=\sqrt{\frac1{-1}}\neq\frac1{\sqrt{-1}}=-i $$

luckyrumo
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a / i != a * i

in the step 3 to 4.

user3105485
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