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"
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.