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I understand why $1/i$ is supposed to equal $-i$:

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

and that $\frac{1}{i}=i$ is definitely wrong, since that would imply $i^2=1$, which is obviously false, however, I don't know what is wrong with the argument:

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

user819128
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    How do you justify that $$\frac{1}{\sqrt{-1}}=\sqrt{\frac{1}{-1}}$$ – Dietrich Burde Aug 25 '20 at 14:15
  • $$\sqrt{\frac{a}b}=\frac{\sqrt{a}}{\sqrt{b}}$$is only certain for $a,b\gt0$. – Peter Foreman Aug 25 '20 at 14:15
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    In my opinion, writing $i=\sqrt{-1}$ is a really bad idea, because there are two complex numbers $z$ such that $z^2=-1$. By doing that, you make a confusion between these two numbers. – TheSilverDoe Aug 25 '20 at 14:20
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    Please just avoid $\sqrt{\phantom{-1}}$ on negative and non-real numbers. $i=\sqrt{-1}$ is entirely the wrong entry point into complex numbers, but unfortunately the most common one. $i^2=-1$ is miles better. – Arthur Aug 25 '20 at 14:20

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