Talk:Bicomplex number/temp

message: Notice that the product jj should be equal to -1 and not 1 as it is shown on page.  The current expression is a contradiction if we want to hold associativity for the product of numbers.
*No, there is no error in this regard and associativity is not broken. Eric119 02:03, 23 Mar 2005 (UTC)
replies: text: *True - if I understand you correctly, these would be the complexes, which are indeed a solution; though a degenarate one, since effectively only two dimensional. The (4-dim.) bicomplexes can be understood as consisting of two copies of the complexes, \mathbb{C} \oplus \mathbb{C} which are joined through a plane of zero-divisors. In the notation here, this plane would be spanned by \{ 1+j , 1-j \}. Thanks, Jens Koeplinger (talk) 15:08, 23 November 2008 (UTC)
thread_title: Bicomplex number

message: Hello - as far as I can see, several isomorphisms exist to what is here called "bicomplex number": Coquaternions / split-quaternion / hyperbolic quaternion from Musean hypernumbers. How many references to "bicomplex numbers" as described in this article do we have in published works? If we only have isolated incidents, I would recommend to disambiguate and redirect e.g. to coquaternions since they appear to be the oldest; there a reference to the use of the term "bicomplex number" could be provided (just like it is already existing in the coquaternion article for the newer term "split-quaternion"). If I'm not mistaken, in the late 18s the term "bicomplex number" was also used synonymously with quaternions (built on three roots of -1). Any comment is welcome. Thanks, Jens Koeplinger 14:22, 4 August 2006 (UTC)
replies: text: Oops, I'm wrong, sorry. These bicomplex numbers contain two roots of -1 and one root of +1 (j). And multiplication here is defined both commutative and associative. I should read closer, sorry about that, not sure what prompted me to overlook all that. These numbers are then distinct. Do we have more references? Thanks, Jens Koeplinger 20:26, 4 August 2006 (UTC)
replies: text:  Ok, just wrote down the multiplication table, and they are isomorphic to tessarines (w. complex number coefficients) and conic quaternions. Sorry for the mixup. That's the correct isomorphism which I wanted to write about at the beginning, but I mixed it all up. In the tessarine article, the ~j there can be mapped to the ~-j here. In the conic quaternion article, the ~\varepsilon there maps to ~-j here, and the ~i_0 there maps to ~i_2 here. The algebraic properties match again (commutativity, associativity, etc). Thanks, Jens Koeplinger 20:39, 4 August 2006 (UTC)
thread_title:  Several isomorphisms exist 

message: No comments so far; I'll double-check and make sure the isomorphism with tessarines holds, and then propose to the math community to have the term "bicomplex number" redirect to "tessarine", and then list isomorphisms there (carry-forward the reference from this article, and detail how it maps. Along these lines. We would still need more references to the term "bicomplex number" in this use, thought, because in the 19th century some people (e.g. Maxwell when inventing his formulation of electromagnetism) used the term "bicomplex number" for Hamilton's quaternions. Thanks, Jens Koeplinger 12:13, 8 August 2006 (UTC)
replies: text: PS: Another use of the term "bicomplex", in a different program (see e.g. http://arxiv.org/abs/math.DG/0512383 references [1, 7-10]). Thanks, Jens Koeplinger 15:40, 8 August 2006 (UTC)
replies: text: PS: More "bicomplex", in another different program: http://arxiv.org/abs/hep-th/0105015 references listed under [7]. I'm getting the impression that maybe the article "bicomplex number" should become a disambiguation article, which refers to all these different programs? Jens Koeplinger 15:59, 8 August 2006 (UTC)

text: I've added a multicomplex number stub to Wikipedia, which appears to be the parent number program for the current "bicomplex number" article and terminology. I think now that we really cannot redirect anymore to tessarines, but have to provide isomorphism links between all these numbers. Oh well: tessarines are isomorphic to bicomplex numbers (from multicomplex numbers program), and conic quaternions (from the hypernumbers program). I guess that's the best we can do. At least we're providing a backbone for future first-use references, etc. Thanks, Jens Koeplinger 01:05, 9 August 2006 (UTC)
replies: text: if they are really all the same, then the articles should be merged. Theorigin of the different terminology can be explained in a history section, but we won't have to duplicate the math. --MarSch 16:55, 12 December 2006 (UTC)
thread_title:  Propose to merge content into [[tessarine]]s article