Talk:Bicomplex number/temp

Bicomplex number[edit]

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)
- If that was no error, then there are distinct complex roots to the simultaneous equations in the article.

i₁ = i
i₂ = -i
and j = 1 solves all of them at once. Mthw2vc (talk) 14:48, 23 November 2008 (UTC)[reply]

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)[reply]

Several isomorphisms exist[edit]

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)[reply]

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)[reply]

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)[reply]

Propose to merge content into tessarines article[edit]

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)[reply]

Which usage, if either, is now more common? We're not paleontologists, stuck with apatosaur because it happens to have been published first.
- I can only answer for the term "bicomplex number": I don't know. The URL link provided on this article appears not sufficient for a historic reference, or use reference. To the least, the term "bicomplex number" is historically overloaded with two different algebras. They are one type of algebra contained in Kantor's and Solodovnikov's definition of "hypercomplex number" (I.L. Kantor, A.S. Solodovnikov, "Hypercomplex numbers: an elementary introduction to algebras"; translated by A. Shenitzer (original in Russian). New York: Springer-Verlag, c1989). They are also isomorphic to conic quaternions from Musès' hypernumbers. They are also contained in Clifford algebras. Following the URL link and the references listed therein, there appears to be another extension program which I am not familiar with (copied: G. B. Price, An Introduction to Multicomplex Spaces and Functions, Marcel Dekker Inc., New York, 1991). It appears to me that it would be of an advantage to refer to complex number extension programs, and list names of isomorphic select number system types within each program. From the other side, it appears to be of advantage when referring to individual number systems to refer to their first use, and disambiguate from there. Nevertheless, if there's enough momentum or reason as to why the term "bicomplex number" in its current form should have its own article (e.g. a significant discovery was made using this term, or widespread use), then of course the article should stay. All I'd like to do is clean-up all these different terms for number systems, which have been used criss-cross throughout the past 160 years or so, and properly describe and reference them. Thanks, Jens Koeplinger 15:33, 8 August 2006 (UTC)[reply]

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)[reply]

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)[reply]

There should also be a link to the matrix representation of quaternions. Septentrionalis 14:33, 8 August 2006 (UTC)[reply]
- What about adding a sentence instead that refers to the quaternion article: "Quaternions, in contrast, are not commutative under multiplication." There are so many different number types that can be represented in matrix form, but the commutativity of bicomplex numbers ( $\equiv$ $\equiv$ tessarines $\equiv$ $\equiv$ conic quaternions $\equiv$ $\equiv$ Cℓ₂(C) = M₂(C), whatever term one may prefer) is an outstanding algebraic property for which readers might find interest in a cross reference. Thanks, Jens Koeplinger 15:33, 8 August 2006 (UTC)[reply]
  - I meant a specific link; but I see that should be from Tessarines. The idea of a dab here is probably sound. Septentrionalis 17:52, 8 August 2006 (UTC)[reply]
    - Thanks for the additions there; please have a look at the wording modifications I made. This way we have them nicely contrasted. As for the disambiguation, right now I'm leaning towards keeping the current article and add sections for each use; just to be able to write a few sentences about them. Thanks again, Jens Koeplinger 18:22, 8 August 2006 (UTC)[reply]

Hello, I've read the first chapter in [G. B. Price, An Introduction to Multicomplex Spaces and Functions, Marcel Dekker Inc., New York, 1991], which defines "bicomplex numbers" as described in the current article, as the first expansion step in a more general multicomplex number program. Banach space is mentioned. The reference to the URL in the current article points to an author who has several (newer) publications about such-constructed bicomplex numbers. But he is referring back to G. B. Price's definition. While I'll wait some more time for feedback on "bicomplex number", I'll go ahead and dig some more into what exactly "multicomplex numbers" are (they are not in Wikipedia yet), and will to the least update the hypercomplex number article accordingly. Jens Koeplinger 18:37, 8 August 2006 (UTC)[reply]

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)[reply]

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)[reply]