Talk:Quaternion type: Difference between revisions

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:: First, only certain values of N and M "work". Complex numbers would be N=2 and 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.
:: First, only certain values of N and M "work". Complex numbers would be N=2 and 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)
:: 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)


::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)
::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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