4 ms·
Sure we can locally visualize a conformal mapping. But in the historical development of complex numbers, or pedagogically in a college algebra class… this would
by dls2016 5y ago
Sure we can locally visualize a conformal mapping. But in the historical development of complex numbers, or pedagogically in a college algebra class… this would be putting the cart before the horse.
Again why would anyone posit that a pair of numbers suddenly appears when trying to solve x^2+k=0 as k goes from negative to positive?
- lordnacho 5y agoI guess you mean a pair of pairs, as we expected to have two solutions that were normally real numbers. Yeah it's one of those times when the veil lifts and you find out you'd not seen the imaginary part until now. I guess enough has been said about completeness by other commentators, but there's no obvious answer as to why adding just one number solves your problem. Why won't it simply create new problems once you use those new complex numbers as coefficients in an expression? Surprisingly it's all we need.
- dls2016 5y agoI hate to “pull rank” but I have a PhD in mathematics so I’m personally well aware of all the nuances. Again my point is historical/pedagogical. Why should “pair numbers” or “double numbers” be the answer as the OP suggested? It’s not straightforward without getting into conformal mappings. And why is two dimensions enough for third and higher degree equations? Were there a simpler geometric connection, you’d probably have a nicer proof of the Jordan curve theorem… but you don’t. Not sure what you mean by pair of pairs… the OP said that calling imaginary numbers “double numbers” would clear up a lot of issues but that is not clear at all to me.
- lordnacho 5y agoI'm don't doubt that you know more than me about this, just coming at it from my very basic knowledge. By pair of pairs I just mean that you're looking for two solutions, and they turn out to be computed numbers, both of them.