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**Mod note**: Fixed all of the radicals. The expressions inside the radical need to be surrounded with braces -- { }

(This question is probably asked a lot but I could not find it so I'll just ask it myself.)

Does the square root of negative numbers exist in the complex field? In other words is ##\sqrt{(-a)}=\sqrt

{(a)}*i## or is it undefined? Recently in a forum I saw a guy claiming that the square root of -10 is undefined and his proof was something like this:

If ##\sqrt{-100}=10i## then ##\sqrt{-100}*\sqrt{-100}=10i*10i=-100##, but

##\sqrt{-100}*\sqrt{-100}=\sqrt{-100*-100}=\sqrt{10,000}=\sqrt{100*100}=\sqrt{100}*\sqrt{100}=100##

Which implies ##100=-100##, a paradox. And so ##\sqrt{-100}## is not ##10i## but is instead undefined.

I feel like he is making some sort of assumption here that I am not seeing because the square root of negatives is undefined in the complex field that kind of defeats the point of having the complex field in the first place, so is he right?

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