Am I wrong there?
i is defined as sqrt(-1), if that answers your question.
My point is that sqrt(-1) = i is an extra assumption that you don't need to make in order to derive the math you do in school. If you don't make that assumption, you can still derive a lot of math. If you do make that assumption, then you can also derive the theorems of complex analysis.
I think that's true, anyway. I could be wrong.
It just seems strange to me that someone would be bothered by the lack of negative square roots, since their existence is never derived, only assumed. But then again, people's minds work very differently, especially in Mathematics.
I was the opposite in Math class. I fought accepting "i" when they tried to teach it to us since it seemed so arbitrary and contrived to me. This was before I discovered that all Math is arbitrary and that there is no "real math", anyway.
sqrt(-1) = sqrt(e^Pi) = e^(Pi/2) = i
So the new concept was really a 2D plane for numbers, not a new definition for sqrt of negative integers.
e^(i)(pi/2) = i.
The simple assumption that sqrt(-1) = i is problematic because you can get something like
i^2 = (sqrt(-1))^2 = sqrt((-1)^2) = 1.
The wonder and surprise of complex numbers is that assuming seemingly arbitrary properties of a constant like i lead to a number of deep and beautiful results.