Isn't sqrt(-1) = i an axiom though, and not a theorem? In other words, if you just started with the integers and the basic rules of arithmetic, you will never run into the need for imaginary numbers. It's only after you posit their existence that a lot of cool results flow from it.
I don't really know how to interpret your sentence. Complex numbers are a field in which you do 'basic rules of arithmetic'. You can't do the 'basic rules of arithmetic' on integers.
i is defined as sqrt(-1), if that answers your question.
I should have said "Real Numbers" instead of "Integers", sorry.
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.
As the "Road to Reality" explains, if you study Quantum mechanics, complex numbers become every bit as "real" as the integers or real numbers. You just can't explain some of the quantum mechanical phenomena or concepts without using complex numbers at all.
it's as you say. it is contrived. but no more contrived than the basic integers. these are abstract systems that we have contrived in ways that are most useful to us. the natural numbers just seem more "real" because everyone in most cultures deals with "quantity" and "magnitude"
Actually, even the definition of this form requires i, because it's
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.
Am I wrong there?