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In a book I am reading, I'm following an equation that has the line:

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

but while I was working ahead I did:

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

Which is correct? Both?

Mike Flynn
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    It's best to avoid the notation $\sqrt{x}$ unless $x$ is a positive real number. – Angina Seng Mar 11 '18 at 02:43
  • It helps (in this case, at least) to peruse the Related column on the right-hand side, first. This question is essentially the same as $,i = \sqrt{-1}=\sqrt{\dfrac{1}{-1}}= \dfrac{\sqrt{1}}{\sqrt{-1}}=\dfrac{1}{i}=-i,$, which in turn is essentially the same as Simple Complex Number Problem: $1 = -1$, which is at the very top of the Related list. – dxiv Mar 11 '18 at 03:14

1 Answers1

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The second one is correct. Implicit in the assumptions in the first is using an identity like

$$\frac{1}{\sqrt x} = \sqrt{\frac 1 x}.$$

Although this is correct for $x \in \mathbb{R}^+$, it does not extend to negative or to complex numbers. There are quite a few false proofs based on the premise that $\sqrt{ab} = \sqrt a \cdot \sqrt{b}$ holds unconditionally!