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I know that $\dfrac{1}{i}=-i$. However, $\sqrt{\dfrac{-1}{1}}=\sqrt{\dfrac{1}{-1}}$, so $i=\dfrac{1}{i}$. What am I missing here?

Thuong Vo
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    It is not true that $\sqrt{a/b}=\sqrt{a}/\sqrt{b}$ in the complex numbers. – Thomas Andrews May 31 '22 at 02:10
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    It’s also not true that $\sqrt{ab}=\sqrt{a}\sqrt b.$ For example, try $a=b=-1.$ Alternative, you can define $\sqrt z$ as a “multi-valued function.” Then $i$ and $-i$ are both values for $\sqrt{-1},$ and in general $\sqrt{a}\sqrt{b}$ is one value of $\sqrt {ab}.$ – Thomas Andrews May 31 '22 at 02:18
  • In general when dealing with square roots problems like this arrive. $\sqrt{-1}=i$ or $=-i$. – herb steinberg May 31 '22 at 02:19

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