Saturday, 15 February 2014

Why doesn't multiplying square roots of imaginary numbers follow $sqrt{a} times sqrt{b} = sqrt{ab}$?

I've been following a series on understanding about how imaginary numbers came to be, and in the series, it mentions that imaginary numbers mostly follows the algebra rules for real numbers, such as adding or multiplying by real numbers. However, it specifically mentions this "inconsistency(?)" about multiplying square roots of imaginary numbers do not follow the rule for multiplying square roots of real numbers, namely $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$. For example, why can't I do this: $\sqrt{-2} \times \sqrt{-3} = \sqrt{(-2)(-3)} = \sqrt{6}$, which I know is the wrong answer. In the past, I've just memorized to factor out the imaginary parts first, and that $i^2 = -1.$ However, can anyone show me an explanation for this, or tell me where I could learn more about it?

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