# Talk:Hypercomplex number

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

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.

## "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 ${\displaystyle {\sqrt {1}}}$ and ${\displaystyle {\sqrt {-1}}}$ 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)

## 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)

## "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)

## "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)

## 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, ..., i2n-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, ..., en, which are i1, ..., in, where each element squares to ±1 as desired; then the remaining bases iN+1, ..., i2n-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 ek look too much like the ik 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 ek and ik -- they should look similar, shouldn't they? The eks are the iks. Jheald 19:12, 24 May 2007 (UTC)
No. The ek 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 ei). 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 ik 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 ik. 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: ${\displaystyle \langle A\rangle _{k}\wedge \langle B\rangle _{l}=\langle AB\rangle _{k+l}}$   (zero if k+l > n), where ${\displaystyle \langle A\rangle _{k}}$ 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 in)? 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)

## 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 ${\displaystyle \{1,i_{1},i_{2},i_{3}\}}$ 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)

## Mandelbulb stunning! References to the algebra?

Hi - what a fascinating new fractal! I see the new entry "Mandelbulb" as well. For the algebra, do I understand it correctly that addition is the three dimensional vector space addition, and multiplication is defined by adding spherical coordinate angles and multiplying radii? The ${\displaystyle \theta }$ component in ${\displaystyle xy=(r_{x}r_{y},\theta _{x}+\theta _{y},\phi _{x}+\phi _{y})}$ would then be the generalization as compared to the complexes. The fractal is very likely genuine (certainly in its stunning quality and richness!), but I bet the algebra has been looked at. This is interesting to me, as spherical coordinates can be generalized to higher dimensions (see "n-sphere"), which relates to interesting symmetries. That exactly the 8th power of the simple Mandelbrot algorithm yields the beautiful fractal with 7-fold symmetry is enigmatic, in the best of meanings. Thanks, Jens Koeplinger (talk) 14:52, 7 December 2009 (UTC)

I think this algebraic system is similar, if not identical, to the system Hamilton was investigating in the 1840's and before maybe, where a real axis is the intersection of a continuous, cyclic group of complex planes. I didn't get this from the source, though, only from an indirect reference about an article "A Commutative Multiplication of Number Triplets" (2000) by Frank R. Pfaff. I don't have access to the original article, either, so please take this reference with great caution. For algebraic properties, it should be pointed out IMHO that while multiplication in the presented system mostly does not distribute over addition, the multiplication is nonassociative (and not alternative) only at the poles of the 2-sphere. Everywhere else (after choosing proper parameter ranges for the angles) I believe multiplication is associative. Am I wrong? ... Anyway, the fractal is fantastic, but as of right now I think the algebra is rediscovered. Thanks, Jens Koeplinger (talk) 22:05, 8 December 2009 (UTC)
Nice fractal indeed, but just thought I'd add that higher and lower powers produce aesthetically good results too (perhaps increasingly less so as one approaches power two). Power eight is only very approximately the 'sweet spot' perhaps (orders seven and nine are almost identical), so there's nothing particularly special about it. See here for examples: http://images.math.cnrs.fr/Mandelbulb.html (Mandelbulb article by Jos Leys) --Skytopia (talk) 12:14, 6 April 2010 (UTC)

## Inconsistent date?

This article refers to Noether being at Bryn Mawr in 1929. but the article on Noether herself says she only went to Bryn Mawr after the Nazis expelled all Jews from German university faculties om 1933.

Is it possible that Noether had summered at Bryn Mawr in the 1920s? Or is this a simple mistake? Floozybackloves (talk) 06:55, 25 June 2010 (UTC)

## Not an original research

My edit from January 14th was removed as an alleged original research. I disagree for 2 reasons:

1. It is just a mathematical transformation and therefore it is not a question of personal opinion whether it's correct or not.
2. Kantor and Solodovnikov showed for the 2D-case how to normalize the non-real unit to a normalized non-real unit j ∈ {0,1,-1}, so I'm not the first to do so.--Slow Phil (talk) 11:56, 17 January 2011 (UTC)
But it makes no sense. I can see how ũ arises from completing the square but what is j and where does it come from? As it's defined as ũ / a. and a is a square root with the same value as ũ, then trivially j is 1 or -1 (because of the square root). But this proves nothing. More generally it's hard to follow with 'clearly' and 'obviously' used when things aren't clear and obvious and with an unclear connection to dual numbers made in the middle. "It's just a mathematical transformation" is not a good reason to include it – such things should not be included unless reliably sourced. There is an exception for routine arthmetic calculations but this is neither routine nor arithmetical.--JohnBlackburnewordsdeeds 12:16, 17 January 2011 (UTC)
The square root a does not have the same value as ũ because the former is necessarily a (positive) real number whereas the latter is not. The intermediate result ũ is rather a purely non-real number whos square - if non-zero - is either a positive or a negative real number but in general still different from ±1. So, a is the square root of the absolute value of the square of ũ, and not j but =±1.
If, on the other hand, ũ² is zero, there is nothing left to normalize and thus, ũ is already the non-real unit of the dual numbers.--Slow Phil (talk) 13:12, 17 January 2011 (UTC)
But again it proves nothing. You define ũ as something that squares to a0 + Template:Frac, so ũ is the square root of that expression. You then define "a positive real number a" to be the square root of that same expression. Then define j to be ũ / a. Trivially it is 1 because you are dividing the same thing by itself, except it's ±1 because one (but not the other for some reason) square root can take positive and negative values. All you've shown is two things that have the same magnitude when divided give ±1, though arguably you've not even shown that, just stated it.--JohnBlackburnewordsdeeds 14:24, 17 January 2011 (UTC)
I'm sure you mean a0+Template:Frac which is a real number. And, yes, ũ squares to this real number but, if a0+Template:Frac is non-negative, there is also a non-negative real number a which does so. Again, there are 3 cases:
1. a0+Template:Frac = 0: We can define j=ũ and are ready.
2. a0+Template:Frac > 0: In this case, a is one square root of the expression whereas ũ = aj with the non-real unit j, is another one. It's not as trivial as you think because 1, as a split-complex number, has the 4 (!) square-roots +1,-1, +j and -j.
3. a0+Template:Frac < 0: We can define a as the positive real number which squares to |a0+Template:Frac| = -a0-Template:Frac. In this case, ũ = ai with the usual imaginary unit i.--Slow Phil (talk) 16:02, 17 January 2011 (UTC)
I side with John. The first section, "Catalogue", begins with the definition of 'hypercomplex numbers' after Kantor and Solodovnikov (K&S). The new edit by Slow Phil begins with: "This is easy ..." ( link ). I don't know what "this" refers to, or what would be "easy" as opposed to seemingly not easy. The prior paragraph is just reference to a definition and stands as such. I believe Slow Phil attempts to discuss why a full understanding of K&S 'hypercomplex numbers' in two dimensions can be written in few paragraphs. Maybe a simple statement would do, that in two dimensions K&S must fall into one of three cases: complex, split, and dual (with cite into the K&S chapter). Then as for style, I don't like "easy" or "obviously" (2x). Thanks, Jens Koeplinger (talk) 14:34, 17 January 2011 (UTC)
PS: I just notice that the very next section already states the two dimensional case, and gives a reference. Therefore, I'm afraid I don't see the value of what Slow Phil added. At the risk of causing a WP:3RR, I'll revert it again. My apologies if there is something that should still be written here that I overlooked. As an overview page, 'hypercomplex number' now points to the two most prominent programs that use the term: K&S, and Clifford algebras. Maybe algebraic properties of the respective algebras are better discussed in the topic pages? Thanks, Jens Koeplinger (talk) 15:36, 17 January 2011 (UTC)

I know that the next section states that there are just 3 kinds of 2D hypercomplex algebras and describes two of them briefly but I wasn't satisfied with that. Aside from the statement that the non-real units can be normalized, I additionally wanted to show how these 3 cases emanate from the general case where u² is an arbitrary linear combination of 1 and u.
I still don't think this is unimportant, let alone uninteresting, and I feel brought to some criticism: I spent some time and energy on adding it, and I did not anticipate the deprecatory reaction to it. I find it o.k. when other editors alter it, ask me to alter it or ask me for a source, but simply reverting it is a rather destructive kind of handling it, making an editors - admittedly not always perfect - effort a Sisyphos work.
I will try to bring it back in another manner, and I beg you not to use the sledgehammer in handling statements you aren't content with.--Slow Phil (talk) 18:07, 17 January 2011 (UTC)

Sounds good, thank you for taking the time. Sorry for the drastic reverts, your edits were at a prominent place in an overview article. You don't need to convince me about it being interesting. To the contrary, completeness of whatever 'hypercomplex' number program may be envisioned is quite interesting to me (therefore my edits and involvement here). Thanks, Jens Koeplinger (talk) 18:51, 17 January 2011 (UTC)

## Two dimensions

The tricotomy in the 2D case has been formulated as a theorem. I have trimmed the exposition given by linking in a pair of articles on relevant algebra. The three cases have their own articles, note that complex plane mentions the lesser known cases in notes. Further edits on this section can focus on getting the proof communicated for interested readers and need not expand on the cases. Thanks are in order to editors that came out of the shell to try to improve this article.Rgdboer (talk) 03:08, 26 January 2011 (UTC)

## Incomplete statement on producing new algebras via tensor product

The following statement in Hypercomplex number#Tensor products strikes me as inadequately defined:

In particular taking tensor products with the complex numbers leads to four-dimensional tessarines ${\displaystyle \mathbb {C} \otimes \mathbb {C} }$, eight-dimensional biquaternions ${\displaystyle \mathbb {C} \otimes \mathbb {H} }$, and 16-dimensional complex octonions ${\displaystyle \mathbb {C} \otimes \mathbb {O} }$.

In particular, it seems to be necessary to qualify the tensor product of the vector spaces tensor product of algebras as being over the field ℝ, either by stating this or by using the symbol ⊗; for example it seems ℂ ⊗ ℂ ≅ ℂ ⊕ ℂ (tessarines), but ℂ ⊗ ℂ ≅ ℂ (complex numbers). I'd appreciate someone who has more familiarity of the math and notation than I have making this change. — Quondum 05:14, 2 February 2012 (UTC)

The article tensor product of algebras applies since the subject is algebra over a field rather than vector space.Rgdboer (talk) 21:22, 2 February 2012 (UTC)
I stand corrected with regard to the referenced article/subject. Nevertheless, the ambiguity seems to remain: in forming the tensor product ℂ ⊗ ℂ, is ℂ to be regarded as a two-dimensional algebra over the field/ring ℝ, or a one-dimensional algebra over the field/ring ℂ? (It seems the former is intended, but this is only by inference.) The dimensionality of the result seems to depend upon this. Forgive me if I'm missing something "obvious"; this type of tensor product is new to me. — Quondum 22:40, 2 February 2012 (UTC)
The article complexification explains the process as an example of extension of scalars. Thank you for finding that jump.Rgdboer (talk) 02:20, 5 February 2012 (UTC)
As suggested at the outset of this article, the term is traditional in modern algebra, and refers to the algebras related to quaternions that were found in the nineteenth century. Emil Artin was one of the last to use the term in 1927. Now in 1930 and 32 Richard Brauer devised his Brauer groups for central simple algebras. The group product is the tensor operation. The more modern context of algebra over a field might be more suitable for developing tensor products of algebras.Rgdboer (talk) 02:43, 5 February 2012 (UTC)
I note that articles such as complexification are careful to define the base field of the algebra: The complexification of V is defined by taking the tensor product of V with the complex numbers (thought of as a two-dimensional vector space over the reals), and proceed to use the symbol ⊗. Some things (especially assumptions/premises) may seem obvious to you, but are not to me. I'll edit the article and you can fix it if I'm wrong. — Quondum 13:43, 5 February 2012 (UTC)

## Is the normalization possible in general?

In the test I read the sentence
It is conventional to normalize the basis so that ${\displaystyle i_{k}^{2}\in \{-1,0,+1\}}$.
Maybe, but is it also always possible, if there are more than one 'imaginary' unit the sqare of each is dependent on others?--Slow Phil (talk) 11:20, 4 April 2012 (UTC)

This is a good point. It is probably not possible in general. Even when it is possible, the term normalization does not apply; it would be better phrased as It is conventional to find a basis so that.... It is in any event not in general conventional to do so, unless to do so is straightforward/natural. For example, there are what are the 3D hypercomplex numbers sometimes referred to as "tricomplex numbers", with multiplication conventionally defined by i2=j, j2=i, ij=ji=1. I can identify a basis (1,α,β) where α2=+1 (with α=Template:FracTemplate:FraciTemplate:Fracj), but it appears β2 cannot simultaneously be a real scalar. I would suggest that the offending sentence should be modified or removed. — Quondum 13:41, 4 April 2012 (UTC)
I've thaught about it. It is certainly impossible to make ${\displaystyle i_{k}^{2}}$ to be ${\displaystyle \in \{-1,0,+1\}}$ - if the algebra contains a purely imaginary subalgebra whos basis elements don't all sqare to zero. Nevertheless, it should be possible to do a kind of "orthonormalization" which means that a product of two basis elements should be a single basis element, its negative or zero rather than a general linear combination of several basis elements.--Slow Phil (talk) 11:15, 21 May 2013 (UTC)
I *think* it's always possible in real-closed fields. In other fields there will probably be more options. If 1, 0 and -1 are "representatives of signatures" over the reals, (and 1 and 0 are representative over the complex numbers) I wonder what can be said about representative sets of signatures over other fields. Rschwieb (talk) 19:31, 21 May 2013 (UTC)
Hypercomplex algebrae are seldom fields at all: the only ones are (trivially) the real numbers themselves and the complex numbers. These both and the quaternions are the only division rings (or skew-fields, as they are also called) among the hypercomplex algebrae, as Frobenius' theorem also says. A hypercomplex algebra where, for example, ${\displaystyle i_{k}^{2}=+1}$ for some k is clearly not a division algebra, let alone a field because ${\displaystyle (1+i_{k})(1-i_{k})=1+i_{k}-i_{k}-i_{k}^{2}=1-1=0}$ in this case. There are also hypercomplex algebrae that have purely imaginary ideals, and in such cases, it will not be generally possible.--Slow Phil (talk) 00:33, 28 May 2013 (UTC)
Fortunately, normalizing those vectors has nothing to do with the algebra: only the inner product space. Go read Sylvester's law of inertia to see that every real symmetric bilinear form is cogredient to one with {-1,0,1} on the diagonal, in varying numbers. This is a basic result appearing in any textbook covering symmetric bilinear forms. Rschwieb (talk) 02:56, 28 May 2013 (UTC)

Of course, but we have to distinguish between a scalar product (or a more general symmetric, eventually indefinite bilinear form) ${\displaystyle s({\mathfrak {u}},{\mathfrak {v}})\in \mathbb {R} }$ (for an ${\displaystyle \mathbb {R} }$ vector space ${\displaystyle V}$) and the multiplication ${\displaystyle {\mathfrak {uv}}\in V}$ wich makes ${\displaystyle V}$ be an ${\displaystyle \mathbb {R} }$ algebra. For example, ${\displaystyle s({\mathfrak {u}},{\mathfrak {v}})=s({\mathfrak {v}},{\mathfrak {u}})}$ is always true whereas ${\displaystyle {\mathfrak {uv}}={\mathfrak {vu}}}$ might be not. This depends on the internal multiplication rules, and I didn't yet find a way of constructing a scalar product from these rules in general.

I was already on the verve to revert your revert because I was convinced I had found an example where it's impossible to make ${\displaystyle i_{k}^{2}\in \{-1,0,+1\}}$, because it has a non-real ideal:

Consider a (commutative but not associative and not even alternative) hypercomplex algebra with basis ${\displaystyle \{1,e_{1},e_{2}\}}$ where

${\displaystyle e_{1}^{2}=-e_{2}^{2}=e_{1},e_{1}e_{2}=e_{2}e_{1}=-e_{2}}$.

Obviously, the ${\displaystyle e_{1}}$-${\displaystyle e_{2}}$-plane is a subalgebra (and within itself even a division algebra, although it has no unity element) and moreover an ideal, so I thougt it's impossible to leave the plane once you have multiplied into it. With just one non-real basis element, it's not just possible but easy to find another basis element whos square is real (see the article), but with 2 or several such basis elements, I wasn't sure whether this remains so. I just found out it is - in this special case: For getting ${\displaystyle i_{k}^{2}\in \{-1,0,+1\}}$ for some "new" imaginary basis elements, we have to find 2 linearly independent elements ${\displaystyle a_{0}+a_{1}e_{1}+a_{2}e_{2}}$ and ${\displaystyle b_{0}+b_{1}e_{1}+b_{2}e_{2}}$ with purely real squares:

${\displaystyle (a_{0}+a_{1}e_{1}+a_{2}e_{2})^{2}=a_{0}^{2}+2a_{0}a_{1}e_{1}+2a_{0}a_{2}e_{2}+a_{1}^{2}e_{1}^{2}+2a_{1}a_{2}e_{1}e_{2}+a_{2}^{2}e_{2}^{2}}$
${\displaystyle =a_{0}^{2}+2a_{0}a_{1}e_{1}+2a_{0}a_{2}e_{2}+a_{1}^{2}e_{1}-2a_{1}a_{2}e_{2}-a_{2}^{2}e_{1}}$
${\displaystyle =a_{0}^{2}+(2a_{0}a_{1}+a_{1}^{2}-a_{2}^{2})e_{1}+(2a_{0}a_{2}-2a_{1}a_{2})e_{2}}$
${\displaystyle =a_{0}^{2}+(2a_{0}a_{1}+a_{1}^{2}-a_{2}^{2})e_{1}+2a_{2}(a_{0}-a_{1})e_{2}}$

This should be real, so both parentheses should equal zero; this immediately yields ${\displaystyle a_{2}=0}$ or ${\displaystyle a_{1}=a_{0}}$. In the first case, for ${\displaystyle a_{1}\neq 0}$, of course(otherwise it were too trivial because the element were already a real number), this yields ${\displaystyle 2a_{0}=-a_{1}}$. Indeed,

${\displaystyle (1-2e_{1})^{2}=1-4e_{1}+4e_{1}^{2}=1-4e_{1}+4e_{1}=1}$,

so this works. In the second case, it yields ${\displaystyle 3a_{0}^{2}=3a_{1}^{2}=a_{2}^{2}}$, and

${\displaystyle (1+e_{1}+{\sqrt {3}}e_{2})^{2}=1+2e_{1}+2{\sqrt {3}}e_{2}+e_{1}^{2}+2{\sqrt {3}}e_{1}e_{2}+3e_{2}^{2}}$
${\displaystyle =1+2e_{1}+2{\sqrt {3}}e_{2}+e_{1}-2{\sqrt {3}}e_{2}-3e_{1}=1}$

In this case, it seems that ${\displaystyle e_{1},e_{2}}$ may just have been a "clumsy" choice, the choise of a kind of "skew" plane which isn't really purely imaginary. Of course, this is not a general proof. Do you know a general algorithm for this kind of 'normalization'?--Slow Phil (talk) 14:09, 28 May 2013 (UTC)

Yes, such an algorithm can be deduced from the proofs of the theorem I am referencing. See Jacobson's Basic algebra I Theorem 6.6 on page 358, or Bishop's Tensor analysis of manifolds Theorem 2.21.1 pgae 104. Both were visible through googlebooks when I checked. Rschwieb (talk) 15:19, 28 May 2013 (UTC)
I'm enthousiastic. It reduces the otherwise vast number of different possible structures and multiplication rules greatly without simply leaving the most of them out of sight. So, a "really purely imaginary" basis element of a hypercomplex algebra always squares to a real number and can be normalized, if necessary, and any "imaginary unit" that squares to some linear combination of several or even all "imaginary units" is actually a mixture with a real and an imaginary part.
What about the products of two different imaginary units? Is there also an algorithm which assures that such products are always either another single imaginary unit or its negative or zero?--Slow Phil (talk) 16:57, 28 May 2013 (UTC)
Hold on here, I mistook the talkpage I am editing in :) I had thought we were in the realm of Clifford algebras, but now I see we are not. Everything I have said has been in the context of symmetric bilinear forms on vector spaces, and I do not know that it applies in general to real/complex algebras. I have no idea what it means to "normalize" in such an algebra. Sorry not to have caught myself earlier, but I am frequently in the Clifford algebra wikis :) You may need to revert my edit after all. Rschwieb (talk) 17:13, 28 May 2013 (UTC)
I've gone ahead and made adjustments. Hope I didn't cause any disappointment with my earlier claims :) As I said, I don't know what is meant by "normalize" in a generic algebra, and that I can only guess the original author meant that "it's convenient to have algebras with bases whose elements square to -1,0 or 1." I don't see any reason at all why an algebra must contain a basis of elements squaring to -1, 0 or 1. Rschwieb (talk) 17:24, 28 May 2013 (UTC)

### Break

AFAIK, hypercomplex algebrae have something to do with Clifford algebrae. Unfortunately, I'm not very experienced with them. Anyway, you showed me that having a "purely imaginary" ideal doesn't necessarily mean that there were no way to 'normalize' these algebrae in the way we have described it, and thanks for this. We should keep talking about it.--Slow Phil (talk) 17:37, 28 May 2013 (UTC)

The real and complex Clifford algebras are special cases of hypercomplex algebras. There are not really that many real and complex Clifford algebras at all. While all real algebras together form a proper class, the collection of the real (or complex) Clifford algebras form a set. In real Clifford algebras, v2 is real for any v in the original vector space V, and picking an orthonormalized basis is always possible. Rschwieb (talk) 19:45, 28 May 2013 (UTC)
Another thing: real Clifford algebras either have exactly two or exactly four two-sided ideals. I'm actually at a loss at the moment on how the four-ideal ones behave, but the two-ideal ones don't have any proper ideals, much less "purely imaginary" ideals. I haven't seen any use of "purely imaginary" ideals until now, but I would be interested if you know some practical uses. Rschwieb (talk) 19:59, 28 May 2013 (UTC)

I've noticed that the definition (in section Definition) indicates that a hypercomplex number is unital (but this is unfortunately not in the lead). Since an algebra appears to be distributive, the hypercomplex numbers appear to be nothing but the class of unital R-algebras (perhaps this should be in the lead?). The restriction to being unital severely restricts the possible algebras of a given dimension – dramatically so for two dimensions (from infinity to 3 algebras, up to isomorphism). But given that any non-unital algebra can be made unital by adjoining 1, and that a non-unital 2-dimensional algebra can be constructed that is quite resistant in terms of "normalization" (in the sense that no finite nontrivial group under multiplication is contained in the algebra), for higher dimensions one could probably construct algebras that are somewhat pathological. This is all off the top of my head and taking a few intuitive leaps of faith from ages ago when I tried classifying all 2d R-algebras, so take it with a pinch of salt, but also as a warning of "here be monsters". — Quondum 23:56, 28 May 2013 (UTC)