Talk:Hypercomplex number

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I agree that "Unfortunate" is the right phrase. If the hypercomplexes formed a field, we'd be able to say a lot more interesting things about them. -- GWO

There is a lot of neat stuff connected with the quaternions, octonions and sedenions (see the external link in Octonions, for example). Saying that things would be more interesting if they were fields is missing the point. --Zundark, 2001 Dec 20

I don't think so. They'd be more interesting, (and a darn sight more useful for things like 3-dimensional versions of complex-analytic inviscid 2D fluid dynamics) and it is unfortunate that they don't.

They wouldn't be themselves if they were fields. And they wouldn't be any more useful, as they would be incorrect. This 'unfortunately' is ridiculous. --Taw

I'm sorry you feel that way, Taw, but does thiat mean you should delete someone else's words? As I mentioned in my summary when I put the word back, my math & physics professors used the word "unfortunate" to cover this type of situation, so apparently they don't share your view. -- GregLindahl

Unless I'm totally missing something, there is a type of hypercomplex number that does form a field: if one sets

it is quite easy to prove that i, j, and k follow the commutative laws. On the other hand, I could be crazy, and missing a point in my logic.Scythe33 21:46, 19 September 2005 (UTC)

Oh. They have zero divisors (1 + k)(1 - k) = 0. However, according to Mathworld (which apparently suffers from credibility attacks) these are called "the" hypercomplex numbers. Scythe33 20:27, 20 September 2005 (UTC)

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In complex numbers it is possible to represent i as

[0 -1]
[1  0]

real unit as:

[1 0]
[0 1]

Is there some equivalent representation for these other complex systems? Does it have to be 4 and 8 D, or could you pull off an arbitrary number of complex dimensions?--BlackGriffen

Yes there are. There are two for quaterion - as 2x2 complex matrix and as 4x4 real matrix. I suppose there are no for octonions because matrix multiplication is associative so it's impossible to define subgroup of matrix ring that is isomorphic with octonions. --Taw

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A description of commutative hypercomplex numbers in user-defined dimensions may be found on the web pages at www.hypercomplex.us - twjewitt@ziplink.net

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Is the term hypercomplex number really well-defined? As far as I know, it was a term used around the turn of the century, before it was clear exactly how numerous finite-dimensional algebras over the reals were. Walt Pohl 23:16, 31 Aug 2004 (UTC)

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I made up a meta-complex numbers system for multiplying is comutative. In hyper-complex its not comutative. Meta-complex numbers is sumthing like: [[[0,1],1],[-1,[2,2]]] and its comutative. Mi own wiki-web system has that page on it. --zzo38 17:25, 2004 Sep 26 (UTC)

--- In the article tt's said that hypercomplex numbers are defined on an Euclidean space. This is not always true, e. g. the numbers proposed from [User:Scythe33|Scythe33] on this discussion page. However, the problem is that the term "hypercomplex number" is not uniquely defined. I consider them according to the famous book from Kantor consisting of

• an n-dimensional vector space over a field K

AND

• a multiplication table defining products between imaginary units

SUCH THAT

• an identity element exists (unity imaginary) which commutes with every other imaginary.

Contents 1 "none form a field" vs. Tessarines / conic quaternions from hypernumbers 2 Uses of the term "hypercomplex numbers" 3 "External Links" section 4 "External Links" section 5 Recent edits

[edit] "none form a field" vs. Tessarines / conic quaternions from hypernumbers

Hello,

Like some previous concerns here, I also am not comfortable with the statement "But none of these extensions forms a field, ...". To my knowledge, Tessarines (if used with complex number coefficients) are a field, and they are isomorphic to 'conic quaternions' from the hypernumber program. They are commutative, associative, distributive, and the arithmetic is algebraically closed (contains roots and logarithms of all numbers). Is this not correct? I'll also bounce this off the "hypernumbers" Yahoo(R) discussion group, to see whether I can get some feedback.

I also have a concern with all "hypercomplex numbers" being an Euclidean-type extension. Split-complex numbers, and others that use non-real roots of 1, are an extension that is rather of hyperbolic geometry, and not on the Euclidean geometry offered through roots of -1.

And a third remark, there appears to be a small group based in Moscow that uses the term "hypercomplex number" for a different number system ( http://hypercomplex.xpsweb.com/index.php ). I'm not happy about their use of the term, but to the least, we should offer a disambiguation.

Any feedback is welcome, so we can hopefully provide some valuable (and in my eyes needed) updates to this article.

Thanks, Jens Koeplinger 01:49, 19 July 2006 (UTC)

Since det t = ww - zz can be zero for many t <> 0 there are many non-invertable tessarines and the ring cannot be a field.Rgdboer 04:25, 20 July 2006 (UTC)

Thanks, agreed. Tessarines with z = +- w are idempotent, as you show. I'm still looking for how many understandings of "hypercomplex number" are out there. So far I've found four different viewpoints: 1) Euclidean geometry extenions (using additional dimensions built on square roots of -1); 2) Euclidean and hyperbolic geometry externions (using non-real and type dimensions); 3) Extensions that use any kind on non-real dimensions; and 4) Numbers modeled for Finsler geometries. Maybe we can update the article along these lines? Thanks again, Jens Koeplinger 13:36, 20 July 2006 (UTC)

[edit] Uses of the term "hypercomplex numbers"

Hi,

I'm (still) looking for uses of the term "hypercomplex number", with first-use references. Currently, I've seen four different uses:

1) Cayley-Dickson construction type (using roots of -1)

2) Extensions using roots of -1 and +1 (Cayley-Dickson construction type and split-complex number type)

3) Numbers with dimensionality, where at least one axis is non-real

4) Use as by http://hypercomplex.xpsweb.com

Possibly after doing so, the article should be rewritten (e.g. only Cayley-Dickson construction type numbers could be considered an Euclidean extension; all others also incorporate different metrics, e.g. hyperbolic metric types from split-complex numbers). But without first-use examples, the article would remain quite fuzzy. Any ideas?

Thanks, Jens Koeplinger 21:18, 22 July 2006 (UTC)

Hi, in the absense of any response, I speculate that the term "hypercomplex number" might be used rather freely today, and that people may know by the context of their current discussion what is meant by it. Unless there's a different suggestion, I'll take the definition as in "Hypercomplex numbers : an elementary introduction to algebras", I.L. Kantor, A.S. Solodovnikov; translated by A. Shenitzer, New York: Springer-Verlag, c1989 [originally in Russian] and write something along the lines "A comprehensive modern definition of 'hypercomplex number' is given in ...". This should accomodate the fact that there were previous, older definitions of the term which (mostly?) encompass a subjset of Kantor's (et. al.) definition; and it'll then allow to group certain familiar types into categories. Their definitions also allow to add a section of number types that don't fall under their definition, but have some overlap. Thanks, Jens Koeplinger 13:39, 26 July 2006 (UTC)

Hello. I've put the first proposal for a complete rewrite out there. I tried to incorporate all statements that were there before, but put them into a better context and order. Any comments are welcome. Thanks, Jens Koeplinger 18:37, 31 July 2006 (UTC)

[edit] "External Links" section

Earlier today I reverted an edit that was adding a commercial advertisement and a link to a page that was broken and had a different description than the subject header. After reviewing the page, I've added it back now, but with a more correct description ("Clyde Davenport's Commutative Hypercomplex Math Page"). This way, I believe, the character of the referenced page is better represented, as a personal web page, which is to be taken as such. In order to add at least some more external references, I've for now added two that I deem significant, hyperjeff.com (history) and hypercomplex.ru (research group after Kantor & Solodovnikov's hypercomplex program). I guess this would be a good place to link to certain pages.

Personally, I would continue to object having a link to hypercomplex.us here, because it's more an advertisement than an information. But I would pull back if some would suggest otherwise. There are elaborate reviews of commercial software here in Wikipedia, so maybe the "external links" section would be appropriate.

There's one concern, though: If we're adding personal web pages here, then we might have to add a whole bunch of pages: A simple internet search for "hypercomplex" reveals all kinds of pages, and I'm not sure that Wikipedia ought to be displaying results that one could just as well obtain from an internet search. I'm entertaining a Yahoo discussion group, and participate in another, and I don't think they need to be listed here; people will find them anyway, through simple searches.

Anyway, there's a fine line what ought and ought not to be referenced, so for now I only suggest to leave-out the hypercomplex.ru link, keep the Clyde Davenport link (in the new and more up-front version now proposed), and add some more to it over time. But it's more thinking out loud than suggesting a plan.

Thanks, Jens Koeplinger 02:36, 22 September 2006 (UTC)

[edit] "External Links" section

Jens,

Would you consider a link to http://www.hypercomplex.us/docs/generalized_number_system.pdf and/or http://www.hypercomplex.us/docs/hypercomplex_signal_processing.pdf in either the section entitled "References" or "External Links"?

Tom Jewitt

Hello Tom,

Thanks for asking. With no other responses here, it seems that it'll be your choice. Since it is a product home page, maybe we could accomodate with making this clear? Here's a suggestion:

Hypercomplex Numerical Computing and Algorithmic Trading Software

Thanks again, Jens Koeplinger 14:11, 7 October 2006 (UTC)

PS: I searched the US PTO database but cannot find the patent 60/352660 which you have referenced as pending. Could you give me a reference? I would be interested in what exactly you are attempting to patent (simply because you are referencing the patent on your paper).

I read over one of your papers, and would like to give you a few other points of reference, if you are interested. The commutative hypercomplex numbers after Kantor and Solodovnikov are also at times called "polynumbers" (in particular in the hypercomplex.ru group). The 3-dimensional numbers which are part of your "N+" program have also recently been evaluated here, together with some higher-dimensional counterparts. In addition, hypercomplex numbers with commutative mutliplication are also currently being investigated in the hypercomplex Yahoo group (public; the "polynary" #s after Armahedi and the "polyplex" after Marek; the forum pretty much started with looking at these programs). I find your "N+" numbers contained in some of these programs, but I may be wrong. Hope you find these references helpful! (And let me know if you know anything else that may be going on in this direction). Thanks, Jens Koeplinger 14:39, 7 October 2006 (UTC)

[edit] Recent edits

I am very concerned about the recent edits, which appear to be changing the overview article into an article that focuses on Clifford algebras. Also, the section on Clifford algebras contanis much detail that is not needed in an overview article. I also disagree with the grouping of Clifford algebras as having to have more than one non-real axis, which is not correct. I will wait until the recent edits are completed, but will most likely object against most of these. Thanks, Koeplinger 19:33, 30 March 2007 (UTC)

Seems reasonably in balance to me. There's about as much material on Clifford algebras as there is on Cayley-Dickson derived algebras — and by and large there's a lot more to say about Clifford algebras, because their spinor properties make them so useful.

I'd probably accept that the sentence about the quadratic form property in the first two lines could be done more smoothly - it does jar a bit at the moment; but it's important to at least try to establish what is the property of complex numbers and quaternions which Clifford algebras preserve.

After that, I can't see anything that anyone would want to cut. If you look at what's there, to me it all seems to earn its space:

• Defining anticommutation property
• Labelling scheme
• Usefulness in physics
• Examples
• Spinor property (which is what most of the individual algebra pages actually concentrate on)

That seems pretty bare-bones to me.

It also gives a great feed in to the Cayley-Dickson section. If Clifford algebras are the dull but dependable "meat and potatoes" extension of complex numbers, what are the features of complex numbers they don't capture? ... Cue the octonions.

I think that's quite a good way to structure the article. Jheald 22:48, 30 March 2007 (UTC)