Talk:Split-complex number

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[edit] Definition

Are these really defined in terms of some j whose square is 1? That seems like an unfortunate definition. Defining them as 2-tuples with specified addition and multiplication (analogously to how Rudin defines complex numbers) seems like a wiser choice, and is surely equivalent. LWizard @ 03:55, 14 May 2006 (UTC)


Defining split-complex bases with non-real roots of +1. I support the definition of split-complex bases with the use of "j" as a non-real root of +1; though introducing it as two-tuples, as suggested above, is certainly possible. But the use of "counterimaginary" bases (non-real roots of +1, in contrast to the customary imaginary roots of -1 which are naturally non-real) offers a natural starting point of arithmetical investigation of such "j" bases. The most complete investigation of such kind - to my knowledge - is done in the hypernumbers program, which I had recently added to Wikipedia. It offers roots, logarithms, etc of such "j" bases (which are called \varepsilon bases in hypernumbers terminology). Thanks, Jens (25 June 2006).
Yes, well, one is chained to 150 years of history. Rudin'a approach, while enlightening and expository, is just one of may ways of representing these. Different authors prefer different notations, and so there it is. linas 13:46, 18 August 2006 (UTC)


[edit] Representation and relevance of split-complex numbers

Hello - Split-complex numbers, and non-real roots of +1 in general, have come under critizism during a discussion in Wikipedia_talk:WikiProject Mathematics#Hypernumbers crackpottery. While I agree there is much work to do, the following remarks by a highly educated mathematics professional were a wake-up call to me. How can it happen that split-complex numbers are considered obsolete, by a specialist who has proven unconventional approaches (see e.g. Unlambda)?

Dr. Madore allowed me to repost his comments here (which I will do, so they stay with this page and don't get archived-away), I hope that it will lead to an inclusive and productive debate, and will at the end result in an improved public representation of split-complex numbers in Wikipedia, and ultimatively numbers in general. Thanks, Jens Koeplinger 13:31, 10 August 2006 (UTC)

[Ann.: a "power orbit" of j are all real powers of j]

"[...]
Already the article on split-complex numbers seems of dubious interest to me: most unfortunately it does not mention the (obvious) fact that, by the Chinese remainder theorem, "split-complex numbers" / "epsilon numbers" can be identified with pairs of real numbers with termwise addition and multiplication (I mean, not only are they a two-dimensional algebra over the reals, but actually they are the direct product of two copies of the real numbers), which makes them sort of boring (why bother about the product of two copies of the reals, not arbitrary tuples?); the identification takes the pair (a,b) to \frac{a+b}{2} + \frac{a-b}{2}\varepsilon (the number \varepsilon is called j in the article on split-complex numbers; and it's a trivial exercise to see that this is indeed an isomorphism). (Also, incidentally, the article is wrong in stating that split-complex numbers have nilpotents: they don't, they have divisors of zero but no nilpotents.) I'm stating all this to refute the idea that the number \varepsilon is an interesting object. As to it's "power orbit", i.e., a one-parameter subgroup, once we have identified split-complex numbers with pairs of real numbers as I explained, and the number \varepsilon with the pair (1, − 1), it is clear that one-parameter subgroups all lie in one connected component (both coordinates positive) of the multiplicative group of invertible split-complex numbers, and \varepsilon is not there, so it does not have a "power orbit" (no more than -1 has in the real numbers). Similarly, trying to add both i with i2 = − 1 and \varepsilon with \varepsilon^2=1 just gives you pairs of complex numbers, again not very interesting. This is all basic algebra and applications of the Chinese remainder theorem. --Gro-Tsen 10:15, 10 August 2006 (UTC)
[...]
Feel free to repost my comment elsewhere if you think it wise. Personally I won't follow the "split-complex numbers" page because I don't think it's interesting in any way (but it's not really crackpot stuff either: it's just entirely boring) and I don't have time to improve it. I just find it laughable if it turns out that nobody noticed that these "split-complex numbers" are just isomorphic to pairs of real numbers (something which should be obvious from the start to anyone with a minimal background in algebra, e.g., having read Lang's book). Btw, "tessarines" / "bicomplex numbers" are similarly isomorphic to pairs of complex numbers. Any (commutative and associative) étale algebra over the real numbers is a product of copies of the real numbers and the complex numbers, anyway. --Gro-Tsen 12:38, 10 August 2006 (UTC)
One person's boredom is another person's interesting topic. The fact that no one uses these things indicates that the vast majority of mathematicians agree with you. There seems to have been a historical basis and interest in all of this; the article seems to document that. The question of what happens for the three cases of j2 = − 1,0, + 1 is legit, esp. given that the first is deep and fundamental, and the second is of interest to supersymmetry. So may as well discuss the third case as well, instead of pretending it doesn't exist. linas 13:55, 18 August 2006 (UTC)

[edit] Major application

This number plane shows its significance when used to begin spacetime study. Gro-Tsen may find it "somewhat boring" but this structure holds up the physicists model. The question of extension to four dimensions has led to hyperbolic quaternions, biquaternions, and coquaternions, each with its own particular utility.Rgdboer 22:19, 11 August 2006 (UTC)

Just to give a few examples, on arXiv.org you find on a query for "split complex" and "split octonion" many, many hits: http://arxiv.org/find/grp_physics/1/OR+all:+AND+split+complex+all:+AND+split+octonion/0/1/0/all/0/1

Physicists are in need of algebraic explanations: How does it work, how can I use it; are my numbers commutative or not, are they normed, etc etc (all of Rgdboer's numbers have a multiplicative modulus; and I'm particularily thankful to Rgdboer for - indirectly - pointing me while back to hyperbolic quaternions). Surely, all these numbers are contained in some definition of hypercomplex numbers, like the one from from Kantor and Solodovnikov. But when one wants to actually use them for description of physical law, it is wonderful to have a step-by-step account of their algebraic properties listed - like it is in the current article. I would have loved to have this available to me 10 years ago or so.

The "nilpotents" mishap may have been caused by sloppy use of nomenclature in physics; when physicists talk about "split-complex numbers" they may at times actually refer to "split-compelx algebra", which includes the split-complex numbers from the current article, but also constructs that can be obtained through a modified Cayley-Dickson construction (like coquaternions and split-octonions). The latter (split-octonions) are of particular interest these days in physics, and it turns-out that the outline of the more general split-complex algebra slipped into the split-octonion article (i.e. the modified Cayley-Dickson construction when chosing \lambda = \ell^2 = +1). All constructs from split-complex algebra (other than the split-complex numbers) contain nilpotents.

One suggestion would be to generate a new article split-complex algebra that introduces the extension program in general, and then trim the existing articles about split-complex number, coquaternions, and split-octonions to their actual properties.

The remaining suggestions I would have are minor; for example in the description of its idempotents remove the word "non-trivial" and write explicitely the only two 'trivial' idempotents "0, 1". Or for the zero-divisors, spell them out explicitely: Every number ~z = (a + bj) has zero-divisors c~z^* (a, b, c any real, z * conjugate of z).

Thanks, Jens Koeplinger 01:39, 12 August 2006 (UTC)

  • For z to have a zero-divisor one must have a = b or a = − b. Remember, the group of units of D covers the whole plane except for these lines.Rgdboer 21:39, 15 August 2006 (UTC)
Yes, thank you for the correction, and for the reminder; you're right of course. Thanks, Jens Koeplinger 23:53, 15 August 2006 (UTC) PS: Thanks also for the updates to the hypercomplex numbers intro. I see the "stub" notice is removed, but the article is not really complete. It would be wonderful if we had a "history" section there; but this may be something for later. Jens Koeplinger 23:59, 15 August 2006 (UTC)
For the most part, physicists have an adequate notation based on matrix representations. Physicists and mathematicians abandon/switch notations whenever the new notation seems to offer clearer insight. I see no particular advantage to using split-complex numbers at this point, although going through the algebraic exercises may be a curious diversion. linas 14:06, 18 August 2006 (UTC)
It it certainly possible and customary to write these numbers in matrix form, it should be in all articles (I'll check again). Starting at split-octonions, the matrix multiplication rule must be re-defined (as detailed in that article), as they are not associative anymore, which - in my eyes - reduces the "beauty" of matrix notation, since one must now watch-out which matrix multiplication rule one must follow (the traditional one or the "split" rule). But you are certainly correct that matrix representation is the one used almost exclusively. If split-octonions were to be some kind of an endpoint in physics, then matrix formulation will likely prevail. But it's more than a diversion, I think. In Octonionic Electrodynamics (M. Gogberashvili, J. Phys. A, 39 (2006) 7099-7104) the author relates the inexistence of magnetic monopoles to algebraic properties (non-associativity), and proposes some extension of the EM field concept through using additional degrees of freedom in the split-octonion expression. Curious diversion or brilliant discovery - who knows? Thanks for writing, I'm enjoying it. Jens Koeplinger 14:32, 18 August 2006 (UTC)

[edit] Interest in Terminology Cleanup?

Is there interest in having the terminology of split-complex algebra cleaned-up? We have the following terms:

Current use of terminology is ambiguous: Some use the term "split-compelx number" for what's described in this article, some use it for higher-dimensional constructs that can be obtained through the modified Cayley-Dickson construction as described in split-octonion. Actually, in split-octonion it refers for a more "general description" of this all to split-complex number, which is actually less general, in my opinion.

Are there any suggestions? I would ideally picture one article that features the general concept, of numbers that are built on the following bases: one real, 2n non-real roots of +1, and (2n − 1) roots of -1 (with n = 0, 1, 2, 3, ...). This general article would list the modified Cayley-Dickson construction, definition of j, and then refer to the first three (n = 0, 1, 2) constructs that can be obtained this way: split-complex numbers, coquaternions, and split-octonions. Then, we could remove some more general wording from the respective number systems.

The ultimate goal would be that someone who is new to the term will face a compilation of articles that doesn't seem boring or uninteresting, but actually reflects its relative importance and field of impact (as compared to other hypercomplex number types that can be defined).

At this point I'd just like to put this out for comment. Thanks, Jens Koeplinger 01:20, 15 August 2006 (UTC)

Its pretty clear from reviewing the split-number articles what the "split-" refers to. It might not be bad to have a general article, but what you describe above is sounds like its veering off into original research. Also, having a general article should not result in the removal of "general wording" in other articles. I would be more comfortable if you personally were more familiar with the theory of Lie groups and representation theory, which is the language favored by most modern mathematicians. linas 14:33, 18 August 2006 (UTC)
I've updated the "See also" section accordingly, to give some account on where split-complex numbers are situated. Maybe this is all we need to do at this point. Thanks again, Jens Koeplinger 01:23, 19 August 2006 (UTC)
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