Talk:Quaternion type: Difference between revisions
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:: But also, quaternions, and other Cayley–Dickson classes of numbers, are not a full generalization of simpler numbers. Complex numbers do not have some properties which real numbers have. (For example, complex numbers can not be ordered on a line.) Quaternions do not have some properties which complex numbers have (for example quaternion multiplication is not commutative). Octonions lose some properties which quaternions have (for example: octonion multiplication is not associative). So you have to decide if you are willing to deal with the problems introduced by the additional dimensions. (And even that can be risky: I have seen too many mathematical "proofs" which assume that quaternion multiplication is commutative -- which means they are about as meaningful as proofs which assume that 0 divided by 0 is unique.) --[[User:Rdm|Rdm]] 19:42, 3 August 2010 (UTC)
::: Mm. So we're talking about N and M needing to be powers of 2? (I haven't peeked at NevilleDNZ's link—no time—but that's what's coming to mind.) And that as N and M go up, the operations lose properties. Interesting and weird at the same time; it suggests to me that there's a way to linearly map a number like N or M to a property set and make prediction, but I think it ought to be getting obvious I never went very far in higher math. :-| Very, very interesting info, though. --[[User:Short Circuit|Michael Mol]] 22:58, 3 August 2010 (UTC)
:::: Actually, if we want to stick with the most obvious meanings, N=1 and M is an element of the set {0, 1, 3, 7}. That is, M must be less than 8 and must be 1 less than a positive integral power of 2. However, The Cayley-Dickson approach might let us argue about degrees of imaginaryness, and we can represent quaternions (and so on) with or without explicit use of imaginary numbers. --[[User:Rdm|Rdm]] 01:47, 4 August 2010 (UTC)
::Hi Michael, The wp article does mention [[wp:Octonian|Octonians]], and I also read [[wp:Division algebra]] enough to know that from reals to complex to quaternions to octonians; things seem to get a little less useful. The octonians seeming to have 480 ways to multiply for example. --[[User:Paddy3118|Paddy3118]] 16:28, 3 August 2010 (UTC)
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