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I agree that "Unfortunate" is the right phrase. If the hypercomplexes formed a field, we'd be able to say a lot more interesting things about them. -- GWO
There is a lot of neat stuff connected with the quaternions, octonions and sedenions (see the external link in Octonions, for example). Saying that things would be more interesting if they were fields is missing the point. -- Zundark, 2001 Dec 20
I don't think so. They'd be more interesting, (and a darn sight more useful for things like 3-dimensional versions of complex-analytic inviscid 2D fluid dynamics) and it is unfortunate that they don't.
They wouldn't be themselves if they were fields. And they wouldn't be any more useful, as they would be incorrect. This 'unfortunately' is ridiculous. -- Taw
I'm sorry you feel that way, Taw, but does thiat mean you should delete someone else's words? As I mentioned in my summary when I put the word back, my math & physics professors used the word "unfortunate" to cover this type of situation, so apparently they don't share your view. -- GregLindahl
Unless I'm totally missing something, there is a type of hypercomplex number that does form a field: if one sets
it is quite easy to prove that i, j, and k follow the commutative laws. On the other hand, I could be crazy, and missing a point in my logic. Scythe33 21:46, 19 September 2005 (UTC)
Oh. They have zero divisors (1 + k)(1 - k) = 0. However, according to Mathworld (which apparently suffers from credibility attacks) these are called "the" hypercomplex numbers. Scythe33 20:27, 20 September 2005 (UTC)
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
AND
SUCH THAT
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)
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)
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)
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)
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
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)
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)
-- 80.178.6.167 16:06, 27 August 2007 (UTC)
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)
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)
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 component in 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)
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)
My edit from January 14th was removed as an alleged original research. I disagree for 2 reasons:
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)
It is actually a problem the of detecting the first source. For example in https://archive.org/details/hyper_number_4_0 , you may catch some, but to find the first one is a a full-time job. Best Regards — Preceding unsigned comment added by 190.114.50.75 ( talk) 21:19, 2 November 2022 (UTC)
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)
The following statement in Hypercomplex number#Tensor products strikes me as inadequately defined:
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)
In the test I read the sentence
It is conventional to normalize the basis so that .
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)
Of course, but we have to distinguish between a scalar product (or a more general symmetric, eventually indefinite bilinear form) (for an vector space ) and the multiplication wich makes be an algebra. For example, is always true whereas 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 , because it has a non-real ideal:
Consider a (commutative but not associative and not even alternative) hypercomplex algebra with basis where
Obviously, the --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 for some "new" imaginary basis elements, we have to find 2 linearly independent elements and with purely real squares:
This should be real, so both parentheses should equal zero; this immediately yields or . In the first case, for , of course(otherwise it were too trivial because the element were already a real number), this yields . Indeed,
so this works. In the second case, it yields , and
In this case, it seems that 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)
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)
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)
No, the conventional normalization (i.e. finding a basis {1,i1,…,in} satisfying ik²∈{-1,0,+1} for all k) 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 very old (2006!) version of the English article which, as highlighted by Slow Phil 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).
Let’s take 3-dimensional hypercomplex numbers with basis {1,i1,i2} satisfying i1² = 0, i2² = 2+i1 and {i1,i2} = 0 ( i.e. i1i2 = -i2i1 = whatever).
We’ll search for a new basis {1,i1,j2}, so we can write in all generality j2 = α+βi1+γi2 with α, β and γ being real coefficients and γ≠0 (otherwise it would not be a basis anymore). In the new basis we immediately find that j2² = (2γ²-α²)+γ²i1+2αj2, so the conventional normalization is impossible.
Note: Even though we cannot in general perform conventional normalization on the whole basis, in the 3-dimensional case {1,i1,i2} it is always possible to force i1²∈{-1,0,+1} then i2²∈{-1,0,+1}∪{α+i1|α∈ℝ}.
I hope my English grammar is not too bad… — Ethaniel ( talk) 15:28, 15 February 2016 (UTC)
Even though “it is conventional to choose the basis [{1,i1,…,in}] so that ik²∈{-1,0,+1}”, 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 {1,i1,i2} satisfying i1² = i2² = 1 and i1i2 = i2i1 = -1-i1-i2 (that algebra is associative and commutative). As the three basis vectors are squaring to 1 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 {j0,j1,j2} = {-i1+i2/2,1+i1/2,1+i2/2}, we then find out that ja² = ja and jajb|a≠b = 0, so product in this new basis is straightforward and shows isomorphism with ℝ ⊕ ℝ ⊕ ℝ (if I understand correctly direct sum of algebras). In that example couldn’t {j0,j1,j2} be considered as a better normalization than {1,i1,i2}? — Ethaniel ( talk) 14:14, 3 March 2016 (UTC)
The following unreferenced text was removed:
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)
J.F. Adams' important result in homotopy is related to the Hurwitz theorem cited but that relation is not found in M.A. Kervaire's review of Adam's paper: MR 0141119. There are articles that make the connection: Ignacio Megia "Which spheres admit topological structure" MR 2397665 and C.T Yang (1982) "Division algebras and fibrations of spheres by great spheres" MR 661656. Your observation placed here in a historical article has not be observed as a application of Homotopy or in the article on the Hurwitz theorem. Since algebraic topology is a challenging study not easily appreciated by a general reader, attention to J.F. Adams result might be better placed in an article where readers are prepared for the challenge. Rgdboer ( talk) 21:40, 21 June 2015 (UTC)
Source material should more or less transparently support the claims they are citing. I spent quite a while reviewing the original source to figure out if it actually supported the claim, and concluded that it was not obvious. It is, however, obvious what the paper is trying to prove (it is stated very clearly in the intro and in Theorem 1.1.1) so I decided to replace the passage with something more similar to that. While the original statement might be true under some interpretation of the text, that would seem to overstep the boundaries of what an encyclopedic reference is supposed to do. I hope this compromise I've put in place is a better alternative. Rschwieb ( talk) 13:33, 10 September 2018 (UTC)
I thought this was hilarious, and I felt sorry removing it.
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I agree that "Unfortunate" is the right phrase. If the hypercomplexes formed a field, we'd be able to say a lot more interesting things about them. -- GWO
There is a lot of neat stuff connected with the quaternions, octonions and sedenions (see the external link in Octonions, for example). Saying that things would be more interesting if they were fields is missing the point. -- Zundark, 2001 Dec 20
I don't think so. They'd be more interesting, (and a darn sight more useful for things like 3-dimensional versions of complex-analytic inviscid 2D fluid dynamics) and it is unfortunate that they don't.
They wouldn't be themselves if they were fields. And they wouldn't be any more useful, as they would be incorrect. This 'unfortunately' is ridiculous. -- Taw
I'm sorry you feel that way, Taw, but does thiat mean you should delete someone else's words? As I mentioned in my summary when I put the word back, my math & physics professors used the word "unfortunate" to cover this type of situation, so apparently they don't share your view. -- GregLindahl
Unless I'm totally missing something, there is a type of hypercomplex number that does form a field: if one sets
it is quite easy to prove that i, j, and k follow the commutative laws. On the other hand, I could be crazy, and missing a point in my logic. Scythe33 21:46, 19 September 2005 (UTC)
Oh. They have zero divisors (1 + k)(1 - k) = 0. However, according to Mathworld (which apparently suffers from credibility attacks) these are called "the" hypercomplex numbers. Scythe33 20:27, 20 September 2005 (UTC)
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
AND
SUCH THAT
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)
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)
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)
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)
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
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)
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)
-- 80.178.6.167 16:06, 27 August 2007 (UTC)
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)
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)
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 component in 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)
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)
My edit from January 14th was removed as an alleged original research. I disagree for 2 reasons:
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)
It is actually a problem the of detecting the first source. For example in https://archive.org/details/hyper_number_4_0 , you may catch some, but to find the first one is a a full-time job. Best Regards — Preceding unsigned comment added by 190.114.50.75 ( talk) 21:19, 2 November 2022 (UTC)
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)
The following statement in Hypercomplex number#Tensor products strikes me as inadequately defined:
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)
In the test I read the sentence
It is conventional to normalize the basis so that .
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)
Of course, but we have to distinguish between a scalar product (or a more general symmetric, eventually indefinite bilinear form) (for an vector space ) and the multiplication wich makes be an algebra. For example, is always true whereas 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 , because it has a non-real ideal:
Consider a (commutative but not associative and not even alternative) hypercomplex algebra with basis where
Obviously, the --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 for some "new" imaginary basis elements, we have to find 2 linearly independent elements and with purely real squares:
This should be real, so both parentheses should equal zero; this immediately yields or . In the first case, for , of course(otherwise it were too trivial because the element were already a real number), this yields . Indeed,
so this works. In the second case, it yields , and
In this case, it seems that 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)
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)
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)
No, the conventional normalization (i.e. finding a basis {1,i1,…,in} satisfying ik²∈{-1,0,+1} for all k) 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 very old (2006!) version of the English article which, as highlighted by Slow Phil 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).
Let’s take 3-dimensional hypercomplex numbers with basis {1,i1,i2} satisfying i1² = 0, i2² = 2+i1 and {i1,i2} = 0 ( i.e. i1i2 = -i2i1 = whatever).
We’ll search for a new basis {1,i1,j2}, so we can write in all generality j2 = α+βi1+γi2 with α, β and γ being real coefficients and γ≠0 (otherwise it would not be a basis anymore). In the new basis we immediately find that j2² = (2γ²-α²)+γ²i1+2αj2, so the conventional normalization is impossible.
Note: Even though we cannot in general perform conventional normalization on the whole basis, in the 3-dimensional case {1,i1,i2} it is always possible to force i1²∈{-1,0,+1} then i2²∈{-1,0,+1}∪{α+i1|α∈ℝ}.
I hope my English grammar is not too bad… — Ethaniel ( talk) 15:28, 15 February 2016 (UTC)
Even though “it is conventional to choose the basis [{1,i1,…,in}] so that ik²∈{-1,0,+1}”, 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 {1,i1,i2} satisfying i1² = i2² = 1 and i1i2 = i2i1 = -1-i1-i2 (that algebra is associative and commutative). As the three basis vectors are squaring to 1 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 {j0,j1,j2} = {-i1+i2/2,1+i1/2,1+i2/2}, we then find out that ja² = ja and jajb|a≠b = 0, so product in this new basis is straightforward and shows isomorphism with ℝ ⊕ ℝ ⊕ ℝ (if I understand correctly direct sum of algebras). In that example couldn’t {j0,j1,j2} be considered as a better normalization than {1,i1,i2}? — Ethaniel ( talk) 14:14, 3 March 2016 (UTC)
The following unreferenced text was removed:
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)
J.F. Adams' important result in homotopy is related to the Hurwitz theorem cited but that relation is not found in M.A. Kervaire's review of Adam's paper: MR 0141119. There are articles that make the connection: Ignacio Megia "Which spheres admit topological structure" MR 2397665 and C.T Yang (1982) "Division algebras and fibrations of spheres by great spheres" MR 661656. Your observation placed here in a historical article has not be observed as a application of Homotopy or in the article on the Hurwitz theorem. Since algebraic topology is a challenging study not easily appreciated by a general reader, attention to J.F. Adams result might be better placed in an article where readers are prepared for the challenge. Rgdboer ( talk) 21:40, 21 June 2015 (UTC)
Source material should more or less transparently support the claims they are citing. I spent quite a while reviewing the original source to figure out if it actually supported the claim, and concluded that it was not obvious. It is, however, obvious what the paper is trying to prove (it is stated very clearly in the intro and in Theorem 1.1.1) so I decided to replace the passage with something more similar to that. While the original statement might be true under some interpretation of the text, that would seem to overstep the boundaries of what an encyclopedic reference is supposed to do. I hope this compromise I've put in place is a better alternative. Rschwieb ( talk) 13:33, 10 September 2018 (UTC)
I thought this was hilarious, and I felt sorry removing it.
-- dab (𒁳) 10:13, 21 December 2015 (UTC)
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From first section (History):
Editors should also be readers. Rgdboer ( talk) 16:52, 16 April 2021 (UTC)