Arlo: > ...Would it not be better to say, "are there number(data?)-structures that > provide for interesting algebras not yet considered?" >
Yes indeed. I was fumbling for a way to say that but ran out of steam! Roger Critchlow: > http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf > Now that is interesting, and nice to know this is a broader conversation than I had known. GA's .. gotta look into them and their unification of complex numbers and vectors. Roger/Carl: > Suspect you are about to pop out of algebra and end up someplace else as > interesting. As you say, I think this is the more fruitful approach. All: The Cayley Dickson generalizations discussed in wikipedia: R C H O did present an "answer" in that there are successful numeric extensions, that complex numbers "are not alone". As much as I wish computer graphics had used them for their transformations rather than 4-tuples (homogeneous coordinates) and 4-matrices, I'm not sure just how quaternions differ in theory from linear algebra, which simply started in on generalized n-tuples. In other words, simple n-tuple algebras might have put all these generalizations from R into a single framework. Why *aren't* complex numbers simply our first use of 2-tuples, unified with the rest of linear algebra. Possibly the answer is that, yes linear algebras uses n-tuples, but they focus on very different matters such as linear independence, spanning sets, projections, subspaces, null spaces and so on. Fun! So now I hope I can find some interesting problems that ONLY can be handled with some of these non linear algebraic higher number systems. Interestingly enough, I believe all of the extensions mentioned, as well as all of linear algebra, have the same cardinality .. the continuum, right? Thanks for the insights, -- Owen
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