Supposedly, $\frac1i=-i$. However, my simplification reaches another result: $$\frac1i=\frac{\sqrt1}{\sqrt{-1}}=\sqrt{\frac1{-1}}=\sqrt{-1}=i\text.$$ What's wrong with my process?
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Well, I'd say it makes about as much sense as $1=\sqrt{1}=-1$. – Nov 04 '20 at 20:23
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This has been asked and answered here many times (with slight variations); here is the standard one: Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$? – xxxxxxxxx Nov 04 '20 at 20:31
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The formula $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ is only guaranteed to be valid when $a,b$ are positive real numbers.
For complex numbers, all bets are off.
Lee Mosher
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