Talk:Hypercomplex number

message: 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 
 i^2 = j^2 = -1\ \mathrm{and}\ k^2 = ijk = 1 
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)
thread_title:  Older comments 

message: 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)
replies: text: 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)
replies: text:  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 \sqrt{1} and \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)
thread_title:  "none form a field" vs. [[Tessarine]]s / conic quaternions from [[hypernumbers]] 

message: 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)
thread_title:  "External Links" section 

message: \begin{align}
  & \left\| A \right\|^{2}=\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{n}{A_{i}A_{j}g_{i,j}}} \\ 
 & g_{i,j}=\left\{ \begin{matrix}
   1 & i=j=1  \\
   -e_{i-1}e_{j-1} & i,j\ne 1  \\
   0 & else  \\
\end{matrix} \right. \\ 
\end{align}
--80.178.6.167 16:06, 27 August 2007 (UTC)
thread_title:  norms 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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,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)
replies: text: 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)
thread_title: Basis and real part

message: 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 \theta  component in 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)
replies: text: 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)

text: 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)
thread_title:  Mandelbulb stunning! References to the algebra? 

message: 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)
thread_title:  Inconsistent date? 

message: 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)
thread_title:  Two dimensions 

message: 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 \mathbb C\otimes\mathbb C, eight-dimensional biquaternions \mathbb C\otimes\mathbb H, and 16-dimensional complex octonions \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 ⊗ℝ; for example it seems ℂ ⊗ℝ ℂ ≅ ℂ ⊕ ℂ (tessarines), but ℂ ⊗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)
replies: text: 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)
replies: text: 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)

text: 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)

text: 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)
replies: text: 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 ⊗ℝ.  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)
thread_title:  Incomplete statement on producing new algebras via tensor product 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title: Break

message: No, the conventional normalization (i.e. finding a basis  satisfying  for all ) is not possible in general.
Sorry to be 3–4 years late: The French article (that I’ve discovered 10 days ago only) is based on a  (2006!) version of the English article which, as highlighted by  by asking the question, suggested that the conventional normalization is always possible.
The French article has been corrected for this point only 2 days ago, and now I’m giving a counterexample here (I don’t see how to include it nicely in the main articles).
Note: Even though we cannot in general perform conventional normalization on the whole basis, in the 3-dimensional case  it is always possible to force  then .
I hope my English grammar is not too bad… — Ethaniel (talk) 15:28, 15 February 2016 (UTC)
thread_title:  Answer 

message: Even though “it is conventional to choose the basis [] so that ”, probably due
to what is conventionally done for complex numbers, quaternions, octonions, etc. I don’t think that this conventional normalization is always the most relevant or useful one and shouldn’t be labeled as “the normalization”, only as “the conventional normalization”.
An introductory example is the diagonal basis of split-complex numbers which is in my opinion more useful than the conventional normalization (product is straightforward), but I reckon that the conventional normalization is more understandable at first.
A more striking example is the 3-dimensional hypercomplex numbers defined by basis  satisfying  and  (that algebra is associative and commutative).
As the three basis vectors are squaring to  that basis is indeed following the conventional normalization, but the strange looking inner products (which BTW is one of the few ways to achieve associativity) makes that basis not so easy to use.
Now let consider basis , we then find out that  and , so product in this new basis is straightforward and shows isomorphism with  (if I understand correctly direct sum of algebras).
In that example couldn’t  be considered as a better normalization than ? — Ethaniel (talk) 14:14, 3 March 2016 (UTC)
replies: text: In fact the matter goes back to the history of logarithms where the "natural hyperbola" xy = 1 was used to define natural logarithm as an area. That context relates to the R+R algebra where it forms a "unit sphere" since the quadratic form is xy, but in the Euclidean layout the "natural hyperbola" has radius square root of 2. It is normalization of that picture that led to the unit hyperbola as it is used for hyperbolic functions and the split-complex numbers. But note that a circle with area 2 pi has radius root 2 also, so the definitions of both circular angle and hyperbolic angle have this factor in common, making the pesky normalization unavoidable. As for the 3-algebras, read WP:NOT#FORUM. — Rgdboer (talk) 03:04, 4 March 2016 (UTC)
thread_title:  Relevance of conventional normalization? 

message: The following unreferenced text was removed:
: Finally in 1956 J. Frank Adams used topological methods to prove that there exist only four finite-dimensional real division algebras: the reals ℝ, the complexes ℂ, the quaternions ℍ, and the octonions 𝕆.
Rumors abound, but the proof with topological methods was not found. There is no mention of it on Mactutor or the Frank Adams article. Search on Mathscinet under "Adams, John Frank" produces many articles but their significance for real division algebras was not found.Rgdboer (talk) 22:35, 20 June 2015 (UTC)
replies: text: I included the paper, but it turns out the date was wrong (possibly due to confusion with the Bott periodicity theorem) so thanks for pointing this out. --Sammy1339 (talk) 00:04, 21 June 2015 (UTC)

text: I'm not sure what you're saying. The theorem necessarily involves some algebraic topology. Another source is Hatcher's book on K-theory but it's really no more accessible than the original paper. Should we not cite it because the sources are too hard to read? If you can find an additional source that might be easier you could add it to the article alongside Adams' paper. --Sammy1339 (talk) 00:32, 22 June 2015 (UTC)
thread_title:  Frank result 

message: I thought this was hilarious, and I felt sorry removing it. 
*original edit: 
*the "referenece" 
*Wrong (2005)
--dab (𒁳) 10:13, 21 December 2015 (UTC)
replies: text: The unreferenced sentence was "Review of historic particulars gives body to the generalities of modern theory". The book cited speaks to this point directly. The point is that algebra over a field is a common topic in algebra, and linear algebra is often treated in abstract n dimensions. The low-dimensional real algebras cited here, having had their day in literature, are neglected in most schools. Study of the particular algebras contributes to better understanding of the other courses. The observation of this fact in learning seemed acceptable before being backed up with a philosophical text, but you find it funny when made real with a reliable source. Thank you for your interest and constructive edits, but the observation on particulars deserves to stay. — Rgdboer (talk) 02:22, 22 December 2015 (UTC)
thread_title: Wrong (2005)

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thread_title:  External links modified 

message: --Reciprocist (talk) 16:15, 16 April 2021 (UTC)
replies: text: the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them.
Editors should also be readers. Rgdboer (talk) 16:52, 16 April 2021 (UTC)
thread_title:  An article on hypercomplex numbers that does not mention neither bicomplex numers nor tessarines, nor any commutative hypercomplex algebra at all?