Revision as of 19:42, 3 August 2010 (view source) Rdm (talk \| contribs) (quaternions are not so general) ← Older edit		Revision as of 19:43, 3 August 2010 (view source) Rdm (talk \| contribs) mNo edit summary Newer edit →
Line 5: :: In short: no. :: First, only certain values of N and M "work". Complex numbers would be N=2 orand M=0 OR N=1 and M=1. Quaternions would be N=4 OR N=2 and M=2 OR N=1 and M=3 (depending on exactly what you meant by "imaginary dimensions"). But you can not do anything non-trivially useful with N=3 and M=0 (nor N=1 and M=2). To my knowledge, only 1, 2, 4 and 8 dimensions work here with this kind of arithmetic. :: But also, quaternions (and other such classes of numbers) are not a full generalization of complex 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)

Talk:Quaternion type: Difference between revisions

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Revision as of 19:43, 3 August 2010