I am just learning more about complex numbers and a question popped up I can't figure out on my own, so I've posted it here. I already know $i^2=-1$ and $i=\sqrt{-1}$ (isn't it even true that $\pm i=\sqrt{-1}$?)
I know $\sqrt{a} \sqrt{b} = \sqrt{ab}$ is only defined or valid if at least one of the two is a positive number. One of the both being negative should work as well, see
(1) $\sqrt{3} \sqrt{2} = \sqrt{6}$
(2) $\sqrt{3} \sqrt{-2} = \sqrt{-6} = \sqrt{(-1)6}=\pm\sqrt{6}i$
Now the tricky part:
What about
(3) $\sqrt{-3i} = \sqrt{(-1)3i}$
(4) $\sqrt{(-1)3i}=\sqrt{-1}\sqrt{3i}=i\sqrt{3i}$
The third (3) is true (obviously), but Wolfram Alpha says the fourth (4) is not true anymore. Can anyone tell me why? Assuming $a=-1$ (negative) and $b=3i$ (positive) the formula above should be working, or am I wrong?
Best regards!
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$\begingroup$First, how would you define positive and negative for complex numbers? For example, is $-1+2i$ positive of negative?
Second, when dealing with complex numbers, we often define the square root to be multivalued. $$\sqrt{-3i}=\pm \sqrt{3} i \sqrt{i}$$ is true.
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