Talk:Hypercomplex number

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Contents 1 Older comments 2 "none form a field" vs. Tessarines / conic quaternions from hypernumbers 3 Uses of the term "hypercomplex numbers" 4 "External Links" section 5 "External Links" section 6 Recent edits 7 norms 8 Basis and real part

[edit] Older comments

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)

---

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

---

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)

---

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.

[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)

The main article should also mention that "hypercomplex number" can also refer to the nonstandard extension of complex numbers, with links to nonstandard analysis and hyperreal numbers, a usage to be distinguished from those in the rest of the article. Alan R. Fisher (talk) 00:31, 6 December 2007 (UTC)

Hello - The hyperreal numbers are mentioned and linked in the first paragraph in "Numbers with dimensionality", but only very briefly. Feel free to expand - or more likely, to add another section for "others" ... the following high-level sections were suggested to me a while back:

• Numbers with dimensionality (this is the existing section)
• Set-theoretical extensions of the reals
• Topological extensions of the reals

However, since I don't feel competent enough to do so, I've held back. ... Thanks, Koeplinger (talk) 01:46, 6 December 2007 (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)

Thank you for detailing your reasoning for the updates, and thanks for the contributions in general. I will read over them closely; as a first impression I believe that the very valid points about Clifford algebras (definition, labeling, usefulness, examples, and spinor property) is content that belong onto the Clifford algebra page (in a high-level / introductory section), and not on an overview page on the many different understandings of hypercomplex numbers that people have or had over the years. I understand that the Clifford algebra article is very content-rich and becomes technical very quickly, however, these issues should IMHO be considered on the actual article (Clifford Algebra) and not here. Until then, thanks, Jens Koeplinger 19:54, 31 March 2007 (UTC)

PS: I've been looking through the Clifford Algebra, Coquaternion, and the current hypercomplex number pages, and while I like their content as a whole I'm now planning to work on their representation over the next few weeks. Thanks for providing all the good isomorphisms. Jens Koeplinger 00:00, 1 April 2007 (UTC)

If I (a non-expert on hypercomplex numbers, whatever they may be) may be permitted to contribute. I also find the excessive focus on Clifford algebras unbalancing. There is ample space elsewhere on Wikipedia for their geometry and applications: geometric algebra, spacetime algebra, Clifford algebra, quaternions and spatial rotation, spinor, to name but a few. A reader is likely to want to know what distinguishes the general conception of a hypercomplex system from a garden-variety Clifford algebra. Nevertheless, the previous version of the article is unbalanced in the opposite direction. The Clifford algebras should be treated here as a special case of hypercomplex numbers, rather than bringing in a bunch of facts particular to them out of the blue. Anyway, the treatment also needs to be altered because the Clifford algebras don't match the definition of a hypercomplex system in a natural way. One must first identify a basis of the algebra (regarded as a real vector space): 1, i₁, ..., i_2ⁿ-1 where each element squares to 0 or ±1. Is this an exercise for the reader? Silly rabbit 17:37, 24 May 2007 (UTC)

Um... isn't that more or less what the article currently outlines (or tries to)? Start with bases e₁, ..., e_n, which are i₁, ..., i_n, where each element squares to ±1 as desired; then the remaining bases i_N+1, ..., i_2ⁿ-1, follow by closure under (Clifford) multiplication. These bases also square to ±1, but now the sign is already fixed, and cannot be chosen at will.

Yes, of course. I wasn't thinking. Maybe make it explicit. Silly rabbit 18:40, 24 May 2007 (UTC)

Maybe I'm in a minority of one, but I do see the Clifford algebras as a natural place to start in a survey of different types of hypercomplex numbers. I suppose it depends on what properties of complex numbers you view as most characteristic, but Clifford algebras seem to me to be the class of hypercomplex numbers which most faithfully extend the most characteristic features of complex numbers -- namely their association with rotations; and the complex analysis calculus results of functions of a complex variable, such as Cauchy's integral theorem, which naturally find higher dimensional analogues in Clifford analysis.

Ok, I see where you're coming from (complex numbers <-> geometry). Still, I think perhaps the treatment of Clifford algebras should come later, after the combinatorial mucking-about with quaternions, Cayley algebras, etc. The way it stands now, the e_k look too much like the i_k for my comfort. Silly rabbit 18:40, 24 May 2007 (UTC)

Not just geometry, but all the results of complex analysis too. As for the e_k and i_k -- they should look similar, shouldn't they? The e_ks are the i_ks. Jheald 19:12, 24 May 2007 (UTC)

No. The e_k aren't a basis. I know it's a rather fine distinction. You say: well, obviously I mean that {e₁, ..., e₁e₂, ..., etc} is the basis. But I think that the *first* definition of a hypercomplex system really should fit the model given in the definition. Example:

The quaternions are the hypercomplex system with i₁² = i₂² = i₃² = -1, and i₁i₂ = i₃, i₂i₃ = i₁, i₃i₁ = i₂. It's a direct example of the definition. No need to introduce quadratic forms, distinguished elements (your e_i). It obviously and unobjectionably fits the bill of a hypercomplex system. All I'm saying is that the Clifford algebra is more subtle. Silly rabbit 19:36, 24 May 2007 (UTC)

So that's why I see the Clifford algebras as the most straightforward and feature-preserving of the various possible higher dimensional analogues of complex numbers presented in this article. But I'm happy that other people might foreground the preserving other properties of C by other families of algebras, as being more characteristic. YMMV.

A reader is likely to want to know what distinguishes the general conception of a hypercomplex system from a garden-variety Clifford algebra. I don't necessarily disagree, but this page may quite likely be the first time a reader has ever heard of Clifford algebra. So it makes sense to give some presentation of these "garden-variety" hypercomplex numbers first, I would argue. Jheald 18:25, 24 May 2007 (UTC)

Ok, see my comment above. The definition of a Clifford algebra doesn't fit naturally in with the general treatment of hypercomplex numbers. If you want to see how the definition works in examples, then writing down the relations among the i_k should be a priority rather than implicitly relying on the closure of the Clifford algebra to give you these relations. Think quaternions first. Maybe then Clifford algebras. Also, its probably better to start with something familiar to all readers. Chances are, anyone coming across this page has heard of quaternions at least.

What part of the definition of hypercomplex numbers presented in this article are you suggesting a Clifford algebra doesn't fit in with? The point, surely, is that different families of sorts of hypercomplex numbers have different family relations among the i_k. The Clifford algebra codifies one particular (somewhat prescriptive) set of relations, which lead to systems of numbers which preserve certain aspects of the properties of complex numbers.

As for quaternions, I would see them firstly as a paradigmatic example of a Clifford algebra/geometric algebra. That's surely the easiest way in to their geometic properties and general physical usefulness; and it's the Clifford algebras which preserve those properties into algebras defined for other spaces.

And, if we're going to be putting Clifford algebras in detail, why not exterior algebras as well? Surely these are more easily described than Clifford algebras. Silly rabbit 18:40, 24 May 2007 (UTC)

Well, because we're assuming a metric, and that addition is defined between arbitrary elements of the algebra. The latter in particular tends to be part of what we look for in a "number". Jheald 19:12, 24 May 2007 (UTC)

You can assume a metric if you want to. But there is nothing in the article that makes it a requirement. Silly rabbit 19:36, 24 May 2007 (UTC)

And, by the way, you can add elements of different homogeneous degrees in the exterior algebra. Actually, it's done quite frequently in some circles. Silly rabbit 19:46, 24 May 2007 (UTC)

But you can't add elements of different grades -- eg a scalar to a bivector -- so that's where it fails to fit the brief. Of course, you could always start with a Clifford algebra, and simply regard the exterior algebra as encapsulating its metric-independent properties.  :-) Jheald 20:31, 24 May 2007 (UTC)

Yes you can. For an example (albeit over the complex numbers with the bigrading), see Dolbeault complex. In fact, some treatments of the Dolbeault operator are strictly logically dependent on the ability to add such things (e.g., for almost complex manifolds and CR manifolds). Moreover, the even/odd decomposition of the exterior algebra is used in Hodge theory and index theory more generally and here you really do need to be able to add different grades. (Now your immediate response is going to be that they're somehow invoking a version of the Clifford algebra "in disguise." No, they aren't.) I know that geometric algebraists love to regard the Clifford algebra as fundamental to everything, but there are other algebras out there which bear absolutely no relation to it. And the exterior algebra is not a "metric independent" version of the Clifford algebra. The product is different. Period. They are different algebras. They are not isomorphic in any way, unless you are willing to break the usual rules of what is meant by isomorphism. Silly rabbit 20:45, 24 May 2007 (UTC)

An even more clear-cut example is the Chern character. If we aren't allowed to add different grades, then according to you this should not even be a well-defined object. Silly rabbit 20:53, 24 May 2007 (UTC)

The exterior product is simply the maximal grade part of the Clifford product:   (zero if k+l > n), where denotes the k-grade part of the multivector A. It's the only grade of the Clifford product which doesn't involve a contraction, and is thus independent of the metric. Considering the contributions to different grades of Clifford operations is absolutely central to their geometric significance - it is right at the heart of understanding Clifford algebra as geometric algebra. Jheald 21:11, 24 May 2007 (UTC)

The Clifford algebra isn't graded by degree. It's filtered, but not graded. You could make the argument that the exterior algebra is just the associated graded algebra (which is what you seem to be suggesting), but it needs to be stated properly. Silly rabbit 21:22, 24 May 2007 (UTC)

That's right. The Clifford algebra is not a graded algebra, it's an algebra that operates on a graded linear space. And that's what makes it possible to talk about the Clifford product in terms of its contribution to different grades. Jheald 21:46, 24 May 2007 (UTC)

On a related note, I'm apparently not alone in being confused as to what a hypercomplex system actually is. For instance, van der Waerden defines them to be unital associative finite-dimensional algebras over a field (classically this field is R, and I have no objections to keeping it this way). In the associative case, is this equivalent to the definition provided (via the special basis i_n)? If you can point to a theorem in this direction I'd be much obliged. Silly rabbit 17:37, 24 May 2007 (UTC)

Well, if the algegra is defined over the reals and closed under addition, then one can presumably find a plurality of sets of n basis elements which span it; and if a norm is defined, one can scale the basis elements appropriately, no? Jheald 19:12, 24 May 2007 (UTC)

But they have to square to 0 or ±1. It's non-trivial if even true. Silly rabbit 19:29, 24 May 2007 (UTC)

That's true. My blunder. Squaring is not the same as taking a norm. The existence of n linearly independent elements that square to scalars does appear to be an additional requirement beyond the statement of van der Waerden you quote. But it also seems to me an absolutely central characteristic of what one would seek in anything called a hypercomplex number. Jheald 20:31, 24 May 2007 (UTC)

Thanks for that lengthy discussion above, which is very helpful in putting Clifford algebra into context. Looking at today's section about Clifford algebra here on the hypercomplex number article, I find this section much, much more suitable for an introductory paragraph on the actual Clifford algebra article. For example, the Clifford algebra article mentions already in the introduction that the reader should have prerequisites in multilinear algebra (which most don't). I am in support of a notion that approaches a subject in a way that requires the least amount of prerequisites at the beginning, and the more the article progresses, the more detailed it becomes, and the more pre-existing knowledge on reader's behalf may be assumed.

Since the term "hypercomplex number" has been overloaded so many times by different programs, I lobby for trying to shape the "hypercomplex number" article into somewhat an extended disambiguation page: The programs should be mentioned, with high-level definition, uses, and applicability, and then fairly quickly link to the topic article. At first I thought we should do this straight-away with the Clifford algebra section, but then quickly realized that we can't do this at this point, since the Clifford algebra article is not easily accessible.

James - you have wide knowledge on the foundations of Clifford algebra and its uses; could you picture youself trying to add a new section to Clifford algebra, like "A Basic Introduction into Clifford Algebra" that is close to what's currently in the "hypercomplex number" section? Then, the currently existing sections in Clifford algebra would become a more detailed description of what it is. Maybe that would be a start?

I realize that pretty much all other sections in the "hypercomplex number" article would need to be worked on similarily. I see two primary uses of the term "hypercomplex numbers", with Clifford algebra the dominating use in the U.S., and Kantor / Solodovnikov in the Russian speaking part of the world. The programs overlap to a degree. Thanks, Jens Koeplinger 14:42, 26 May 2007 (UTC)

[edit] norms

--80.178.6.167 16:06, 27 August 2007 (UTC)

[edit] Basis and real part

Hypercomplex systems evolved before modern linear algebra had standardized notions and terminology. This article is an opportunity to help students by leaning on the learning experience that evolved into linear algebra.

To study vector spaces one needs the notion of a basis (linear algebra). So far this article uses the term "bases" instead of standard usage: "elements of a basis". Editing for consistency with standard linear algebra would reassure students.

The tradition of the real part of a complex number was carried forward by the quaternionists to the real and vector parts of a quaternion. Today we say that even the real part is a vector in the 4-space of quaternions; this attitude reflects the homogeneity of vector space elements. Yet for hypercomplex numbers we are learning about multiplicative structure as in associative algebras. There is some advantage of transparency when a real part is identified to an element of a hypercomplex number system. The term "scalar part" has been applied, say in the quaternion article and this is the original terminology. For Hamilton, the tensor of a quaternion was what we now call its norm or modulus. Though the article real part has not been prepared for application here, one might consider that, in the interests of education and historical note, steps may be taken. Comments?Rgdboer (talk) 22:41, 17 January 2008 (UTC)

You're correct, of course; a linear algebra has one basis (and not two or more). The basis may have one or more elements, as you write. Thanks for your diligence. Jens Koeplinger (talk) 23:39, 17 January 2008 (UTC)

Bases is being used there as the plural of "base", not the plural of "basis". This appears to be drawn from the primary literature and may be intentional: a "base" is more special than just any old basis element. For example, a linear combinations of them does not give a new "base". Jheald (talk) 10:07, 18 January 2008 (UTC)

Ok - well, all I really wanted to say is that - for the parts that I wrote - I may not have exercised proper caution regarding terminology ("bases" vs. "elements of the basis"). It seems cleaner, e.g. for quaternions, to speak of a basis {1,i₁,i₂,i₃} that has four elements. Jheald, you've expanded the section on Clifford algebras, so I trust your judgment there. For the remainder, we might not even find consistent terminology throughout literature, I'm not sure. I don't have the Kantor & Solodovnikov book at hand, but I'll check (at least, the English translation thereof, yet another possible point of terminology mixup). Thanks, Jens Koeplinger (talk) 21:46, 20 January 2008 (UTC)

Checked my main reference: In the English translation of K&S, chapter 5.1 "Definition of a Hypercomplex Number System", it does not mention the term basis at all; rather, it is translated to "units". So, I thank Jheald for the call for caution regarding terminology. Thanks, Jens Koeplinger (talk) 03:10, 1 February 2008 (UTC)

Quick response! Hamilton based his delineation on projections S and V for scalar and vector parts. He used T for tensor, our modulus, taking real number values (as acknowledged at tensor#History). My comments above and here merely aspire to help clarify some of that evolving field:linear algebra.Rgdboer (talk) 01:34, 18 January 2008 (UTC)