Talk:Split-complex number

message: This number plane shows its significance when used to begin spacetime study. Gro-Tsen may find it "somewhat boring" but this structure holds up the physicists model. The question of extension to four dimensions has led to hyperbolic quaternions, biquaternions, and coquaternions, each with its own particular utility.Rgdboer 22:19, 11 August 2006 (UTC)
replies: text: Yes, thank you for the correction, and for the reminder; you're right of course. Thanks, Jens Koeplinger 23:53, 15 August 2006 (UTC)
replies: text: For the most part, physicists have an adequate notation based on matrix representations. Physicists and mathematicians abandon/switch notations whenever the new notation seems to offer clearer insight. I see no particular advantage to using split-complex numbers at this point, although going through the algebraic exercises may be a curious diversion. linas 14:06, 18 August 2006 (UTC)

text: It it certainly possible and customary to write these numbers in matrix form, it should  be in all articles (I'll check again). Starting at split-octonions, the matrix multiplication rule must be re-defined (as detailed in that article), as they are not associative anymore, which - in my eyes - reduces the "beauty" of matrix notation, since one must now watch-out which matrix multiplication rule one must follow (the traditional one or the "split" rule). But you are certainly correct that matrix representation is the one used almost exclusively. If split-octonions were to be some kind of an endpoint in physics, then matrix formulation will likely prevail. But it's more than a diversion, I think. In Octonionic Electrodynamics (M. Gogberashvili, J. Phys. A, 39 (2006) 7099-7104) the author relates the inexistence of magnetic monopoles to algebraic properties (non-associativity), and proposes some extension of the EM field concept through using additional degrees of freedom in the split-octonion expression. Curious diversion or brilliant discovery - who knows? Thanks for writing, I'm enjoying it. Jens Koeplinger 14:32, 18 August 2006 (UTC)
thread_title: Major application

message: Is there interest in having the terminology of split-complex algebra cleaned-up? We have the following terms:
*split-complex number
*split-complex algebra
*split-quaternion (which is the coquaternion)
*split-octonion
Current use of terminology is ambiguous: Some use the term "split-compelx number" for what's described in this article, some use it for higher-dimensional constructs that can be obtained through the modified Cayley-Dickson construction as described in split-octonion. Actually, in split-octonion it refers for a more "general description" of this all to split-complex number, which is actually less general, in my opinion.
Are there any suggestions? I would ideally picture one article that features the general concept, of numbers that are built on the following bases: one real, 2^n non-real roots of +1, and (2^n - 1) roots of -1 (with n = 0, 1, 2, 3, ...). This general article would list the modified Cayley-Dickson construction, definition of j, and then refer to the first three (n = 0, 1, 2) constructs that can be obtained this way: split-complex numbers, coquaternions, and split-octonions. Then, we could remove some more general wording from the respective number systems.
The ultimate goal would be that someone who is new to the term will face a compilation of articles that doesn't seem boring or uninteresting, but actually reflects its relative importance and field of impact (as compared to other hypercomplex number types that can be defined).
At this point I'd just like to put this out for comment. Thanks, Jens Koeplinger 01:20, 15 August 2006 (UTC)
replies: text: Its pretty clear from reviewing the split-number articles what the "split-" refers to. It might not be bad to have a general article, but what you describe above is sounds like its veering off into original research. Also, having a general article should not result in the removal of "general wording" in other articles.  I would be more comfortable if you personally were more familiar with the theory of Lie groups and representation theory, which is the language favored by most modern mathematicians. linas 14:33, 18 August 2006 (UTC)
replies: text: I've updated the "See also" section accordingly, to give some account on where split-complex numbers are situated. Maybe this is all we need to do at this point. Thanks again, Jens Koeplinger 01:23, 19 August 2006 (UTC)
thread_title:  Interest in Terminology Cleanup? 

message: According to WP:Lead_section#Provide_and_accessible_overview, "specialized terminology should be avoided in the introduction". I view Clifford algebras as specialized terminology, though I understand a proponent of systematic terminology may view Clifford algebras as an important unifying concept. In the nearly three years I've been participating in the WP project I've seen willingness and power to work co-operatively. Stating challenges to the write-up on these discussion pages frequently brings out a bold and concise solution to vague uncertainty. This page on split-complex numbers was my first contribution, and I feel gratitude to all who have given a hand at advancing this concept. Still, the "lede" needs some work; the assembled sections give some picture, but anyone interest should use the "What links here" for a more complete view. Thank you everyone for your care; now let's see if we can invite the world to our party.Rgdboer 01:23, 6 June 2007 (UTC)
replies: text: After 7 months I decided to act: moved the specialized terminology out of the lede into a later paragraph. Also contextualized this article to linear algebra, more narrow in scope than mathematics. The previous edit by RJ Chapman mentions "futility" of this article, still the link to group ring is a significant contribution. What is futile about discription of a particular linear algebra that is in constant use with special structure ? Rgdboer (talk) 21:12, 19 December 2007 (UTC)
replies: text: Hi Robert - your question answers it in itself: It's the "use", or application, that makes it a valuable article: The split-complex numbers are a primitive in many different algebras, and we're happy to learn about yet another program that they are part of. From the point of view of any wider algebra, the split-complex numbers are trivial; yet it is how they're used that makes it important to be listed here in great detail: Once you make an observation that can be modeled using split-complex numbers, you now have a wealth of algebras and programs at your disposal, by courtesy of Wikipedia's contributors, to extend your model mathematically. You can introduce additional degrees of freedom to your model, from a choice of wider algebras, and then check whether this generates predictions that can be tested. The building blocks go with the building. Thanks for the clean-up, and your attention! Jens Koeplinger (talk) 01:38, 20 December 2007 (UTC)
thread_title: Serious lede

message: In the above arguments on significance of double numbers (I prefer this terms to split-complex numbers) I am on the supportive side: double numbers nicely describe the geometry of 2D space-time. Their non-triviality becomes even more explicit if we consider Moebius transformation of them. The corresponding conformal geometries of complex, dual and double numbers are very similar, see arXiv:math/0512416. Another illustration: complex, dual and double numbers appear from the same procedure of induced representations of the group SL(2,R), see arXiv:0707.4024. V.V.Kisil 14:07, 21 August 2007 (UTC)
thread_title: Double numbers and Moebius transformations

message: The hyperbolic complex numbers are of the form e^(a)*(cosh(b)+ r e^(a)*sinh(b) union with the negative of this where r is the square root of +1.  In matrix form, they are an exponentiated 2 by 2 matrix with a on the leading diagonal and b in the other places union with the addittive inverse of this.  As such, they are an algebraic field equivalent to the euclidean complex numbers.  —Preceding unsigned comment added by 92.18.89.63 (talk) 19:01, 17 May 2008 (UTC)
replies: text: Maybe this is a terminology problem? Since the split-complexes (hyperbolic complexes) contain zero-divisor spaces, they cannot be a field (in this sense), as division is not uniquely defined for all elements other than a single 0 element: If \{1, e\} is a basis of the number system with e^2 = 1 and e \ne \pm 1, then the vectors A:=1+e and B:=1-e generate a zero-divisor space such that aA * bB = 0 for any a, b real. Does this answer your concern? Thanks, Jens Koeplinger (talk) 21:37, 17 May 2008 (UTC)
thread_title:  hyperbolic complex numbers 

message: The main article is a bit confusing in the norm. First the quadratic form  ||z|| is defined. It is shown that this is not a norm in the classical sense as it is not positive definite.
Couple of lines later I read that A split-complex number is invertible if and only if its norm is nonzero. But the norm has not been defined.  —Preceding unsigned comment added by TomyDuby (talk • contribs) 04:30, 20 July 2008 (UTC)
replies: text: Thank you very much for an insightful observation. Accordingly, I have shifted to the use of modulus to correctly steer away from use of an important term useful in the metric theory of normed spaces. On the real line and complex plane the term absolute value is frequently used. The term modulus also arises there and can be carried over into split-signature algebras. For mathematics to continue to serve all branches of application, it is important for observations such as that of TomyDuby to be noted. Such vigilance clarifies communication.Rgdboer (talk) 22:22, 20 July 2008 (UTC)
thread_title:  Norm of a split-complex number 

message: ... would be nice? --Raijinili (talk) 19:30, 14 August 2008 (UTC)
replies: text: Hmm, this is an interesting idea. The sign flip appears literally when you try multiplying or dividing split-complexes (see the modulus), and manifests as a switch from circular to hyperbolic trigonometric functions in the equivalent of Euler's formula. Showing formulae for complexes and split-complexes together immediately illustrates these analogies. However, there are some phenomena in the split-complexes that do not have an analogue in the complexes. If you have ejθ = cosh θ + j sinh θ for a real hyperbolic angle θ, your point described is always going to be on the right branch of the unit hyperbola. To get to the left branch, you need a minus sign in front of the whole thing; this does not occur in the complexes. Further, you have a "counter-unit hyperbola", and you need to multiply by j or −j to get to these from the right branch of the unit hyperbola: not much like the complex numbers, because the complex numbers do not have a unit circle that breaks into several branches. Furthermore, the choice of bases ε = (1 + j)/2 and ε* = (1 − j)/2 identifies the split-complexes directly with pairs of real numbers; this is impossible for the complexes, which are not reducible. Finally, the split-complexes are not algebraically complete (you can only take a square root of a split-complex number with a positive norm that lies in the right half-plane, so for example j doesn't have a square root), and do not obey the fundamental theorem of algebra: an nth-degree polynomial can have anywhere between 0 (e.g. x2 + 1 = 0) and n2 (e.g. x2 − 1 = 0) roots. Double sharp (talk) 15:51, 31 March 2016 (UTC)
replies: text: Yes, you have hit on the primary distinguishing features of two planar topological rings. On the other side, the rings must match up in their roles in 2 × 2 real matrices as invariant subrings. These matrices are a first step into linear algebra for students, and it seems fair to highlight split-complex numbers early in this study so as to set aside Euclidean presumptions. — Rgdboer (talk) 22:34, 31 March 2016 (UTC)
thread_title:  "Comparison to [[complex numbers]]" section... 

message: A user has inserted the opinion that "split-complex number is an archaic term". Reference to the synonyms section will show that there are many terms that have found expression under the split-complex number label. The concept is very important for describing a symmetry of spacetime, so the unification of various authors work under this term has been important for science. If this user wishes to further discuss the issue, the Talk can serve to refine the differences in attitude to this article.Rgdboer (talk) 04:04, 16 January 2010 (UTC)
replies: text: There being no response, I undid the edit.No evidence has been given that the "direct sum of R with itself" identification is more common.Rgdboer (talk) 02:46, 22 January 2010 (UTC)
thread_title:  Not archaic 

message: A good faith edit was posted as follows
:However, the polar form exp(a+jb) together with the additive inverse of this - exp(a+jb) do form a field.
Presumably the editor in U.K. means that the half of the group of units of split-complex numbers that surrounds the real axis forms a field. While it is true that this subset forms a multiplicative group, to form a field there must be an other commutative group structure too. The addition operation (+) is associated with the "linear space" aspect of the split-complex ring. In the proposed subset take
: exp(bj) = cosh b + j sinh b , and 
:− exp(−bj) = −cosh b + j sinh b.
The sum of these elements is 2j sinh b which is not in the subset, so it is not an additive subgroup. For subalgebras generally, one requires linear subspaces as these provide a primary commutative structure for the ring.Rgdboer (talk) 02:56, 22 February 2010 (UTC)
thread_title:  Subfield larger than R ? 

message: It has been requested that the commutative diagram be redrawn, here it is...
I tried exporting to SVG and even PDF - unfortunately neither worked so in the last resort PNG has been used, which isn't ideal but maybe someone else can convert to SVG later... 
I'm not as familiar with split complex numbers as the usual complex numbers btw...
P.S. Rgdboer - did you mean Split-complex number (Matrix representations) instead of Split-complex number (Algebraic properties)? Maschen (talk) 09:29, 2 December 2012 (UTC)
replies: text: Yes, the correct section is Matrix representations. Most of the diagram you have made is right on, but the lower C and D should be R2 or R×R. Also a big dot in front of the versor provides a notation that the versor acts as a multiplicative factor. On usual versus split, you are not alone in finding these other numbers novel. Plenty of references for the dubious ! Rgdboer (talk) 22:29, 2 December 2012 (UTC)
replies: text: No problem! Better? Maschen (talk) 23:12, 2 December 2012 (UTC)
thread_title:  commutative diagram 

message: Just like multiplying by i corresponds to a rotation of π/2 anticlockwise, it seems that multiplying by j should have a geometric meaning too. It looks like it simply corresponds to a reflection across the line y = x: is that right? And is it possible to have j correspond to another geometrical meaning, like complex conjugation if we were to adjoin it to the complex numbers? (That seems to just create the split-quaternions, or 2×2 real matrices.) Double sharp (talk) 16:52, 17 March 2016 (UTC)
replies: text: Yes it is just reflection, or swapping coordinates, as stated at Unit hyperbola#Complex plane algebra. Also { i, j } is a generating set of a group where the group is the dihedral group of the square. You have noted the real algebra thus generated. — Rgdboer (talk) 20:29, 17 March 2016 (UTC)
thread_title:  Geometric interpretation of ''j'' 

message: How strange that in the (very long) list of synonyms, I don't see "para-complex numbers", when I thought that was the most "canonical" name (that's just my impression, coming from differential geometry). Am I missing something?--Seub (talk) 20:59, 26 March 2016 (UTC)
replies: text: The usual procedure is to present a citation using "paracomplex number". A venture into MathSciNet shows that F. Reese Harvey and H. Blaine Lawson inject this name in their 2012 article "Split Special Lagrangian Geometry" in Progress in Mathematics #297 (Birkhauser). But they do not provide a reference. The term paracomplex is used in other contexts. What sources in differential geometry do you know ?  — Rgdboer (talk) 01:47, 27 March 2016 (UTC)

text: A reference to Paracomplex numbers in Cruceanu, Fortuny & Gadea (1996) has been added. Thank you for calling attention to this line of work. — Rgdboer (talk) 20:15, 29 March 2016 (UTC)
thread_title:  Para-complex numbers 

message: I notice on the talk page that D has been used for the split-complexes. Is it standard? (If so it probably ought to be mentioned in the article.) And if so, what does it stand for? "Double numbers"? Double sharp (talk) 15:44, 31 March 2016 (UTC)
thread_title:  The symbol '''D''' 

message: Substituted at 02:36, 5 May 2016 (UTC)
thread_title: Assessment comment

message: The following sentence was deleted by User:Cosmia Nebula:
* In that model, the number  represents an event in a spacio-temporal plane, where x is measured in nanoseconds and y in Mermin’s feet.
Objections to content of an article are to be discussed on the Talk: this wiki that sits behind "Split-complex number". 
This topic is mentioned at Unit hyperbola#Minkowski diagram and Minkowski diagram, with the six references given in #History, just previous to the sentence given above. Discussion can proceed in this section of the Talk. Rgdboer (talk) 02:34, 7 December 2016 (UTC)
thread_title:  Spacio-temporal plane 

message: The article starts with words "in abstract algebra...", but split-complex numbers have nothing to do with abstract algebra. At least, no more than usual complex numbers.--Reciprocist (talk) 07:00, 7 September 2021 (UTC)
replies: text: by removing "abstract". D.Lazard (talk) 08:49, 7 September 2021 (UTC)
thread_title:  "In abstract algebra"? 

message: Can we use names e_- = \tfrac12(1 - j) and e_+= \tfrac12(1 + j) or similar instead of e = \tfrac12(1 - j) and e^* = \tfrac12(1 + j)? Using + and – seems like it would be more mnemonic, and having only one of these with a special symbol seems like an artificial distinction between the two. –jacobolus (t) 03:25, 5 November 2022 (UTC)
replies: text: The two idempotents are conjugates of each other so the * is used. Some notations are traditional such as e for an identity element or idempotent of a code. Rgdboer (talk) 04:42, 5 November 2022 (UTC)
replies: text: I get that they are conjugates. I think Sobczyk has pretty clear notation though, in https://garretstar.com/secciones/publications/docs/HYP2.PDF – he calls standard basis elements \{1, u\}, uses a superscript - for conjugation, and then names the idempotent basis elements u_- and u_+, so he can write an arbitrary split-complex number as w = x + uy = w_-u_- + w_+u_+, with scalar ("real") coordinates w_- = x - y and w_+ = x + y. Because u_+u_- = 0 we have wu_- = w_-u_- and wu_+ = w_+u_+. It seems to me that this choice of notation makes the algebra more legible than the version in this article which instead uses  z = x + jy = ae + be^*, etc. –jacobolus (t) 05:57, 5 November 2022 (UTC)

text: For instance at Rng (algebra)#Properties weaker than having an identity Rgdboer (talk) 04:58, 5 November 2022 (UTC)
thread_title:  Diagonal basis symbols 

message: The plane of split-complex numbers is not a metric space while the hyperbolic plane is a metric space. This article should not be added to the WP:Category of hyperbolic geometry. To highlight association with hyperbolic geometry with a name-change should be avoided. For consistency the name should continue to be associated with split algebra and complex plane. There is no mention in the guide of priority of one criteria over another, and assertion of such is misdirection. The current name is not overly long and is precise for mathematics, though popular expositions have used other, more ambiguous, names. Rgdboer (talk) 21:54, 14 May 2023 (UTC)
thread_title:  Not a metric space