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Old 01-28-2008   #9 (permalink)
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sanctus
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Re: Arithmatic expression vs. set description connotation of the square root

First of all it took me 30min and not 3min to figure it out.
Second I agree with you, the way I saw it is:
1)
\sqrt{-x}\cdot\sqrt{-y}=\sqrt{(-1)\cdot x \cdot(-1)\cdot y}= \sqrt{1}\cdot\sqrt{xy}=\pm \sqrt{xy} since \sqrt{1}=\pm 1

2)
\sqrt{-x}\cdot\sqrt{-y}= \pm i\cdot\sqrt{x}\cdot \pm i\cdot\sqrt{y}=\mp i^2\sqrt{xy}=\pm\sqrt{xy}
Since \sqrt{-1}=\pm i, but this is strange as it implies that the definition of the imaginary unit is not unique.


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