Talk:Split-complex number

Definition

Are these really defined in terms of some j whose square is 1? That seems like an unfortunate definition. Defining them as 2-tuples with specified addition and multiplication (analogously to how Rudin defines complex numbers) seems like a wiser choice, and is surely equivalent. ^L_Wizard @ 03:55, 14 May 2006 (UTC)

Defining split-complex bases with non-real roots of +1. I support the definition of split-complex bases with the use of "j" as a non-real root of +1; though introducing it as two-tuples, as suggested above, is certainly possible. But the use of "counterimaginary" bases (non-real roots of +1, in contrast to the customary imaginary roots of -1 which are naturally non-real) offers a natural starting point of arithmetical investigation of such "j" bases. The most complete investigation of such kind - to my knowledge - is done in the hypernumbers program, which I had recently added to Wikipedia. It offers roots, logarithms, etc of such "j" bases (which are called $\varepsilon$ bases in hypernumbers terminology). Thanks, Jens (25 June 2006).

Yes, well, one is chained to 150 years of history. Rudin'a approach, while enlightening and expository, is just one of may ways of representing these. Different authors prefer different notations, and so there it is. linas 13:46, 18 August 2006 (UTC)

Another alternative definition: Since our practical and notational traditions have such a hold on the imagination, consideration of alternative definitions is a real aid to seeing new numbers. For students that have entered into ring theory, there is the section in the article Polynomial ring#Some uses of polynomial rings that puts all three complex planes in a broader context.Rgdboer 21:17, 18 October 2007 (UTC)

Representation and relevance of split-complex numbers

Hello - Split-complex numbers, and non-real roots of +1 in general, have come under critizism during a discussion in Wikipedia_talk:WikiProject Mathematics#Hypernumbers crackpottery. While I agree there is much work to do, the following remarks by a highly educated mathematics professional were a wake-up call to me. How can it happen that split-complex numbers are considered obsolete, by a specialist who has proven unconventional approaches (see e.g. Unlambda)?

Dr. Madore allowed me to repost his comments here (which I will do, so they stay with this page and don't get archived-away), I hope that it will lead to an inclusive and productive debate, and will at the end result in an improved public representation of split-complex numbers in Wikipedia, and ultimatively numbers in general. Thanks, Jens Koeplinger 13:31, 10 August 2006 (UTC)

[Ann.: a "power orbit" of j are all real powers of j]

"[...]

Already the article on split-complex numbers seems of dubious interest to me: most unfortunately it does not mention the (obvious) fact that, by the Chinese remainder theorem, "split-complex numbers" / "epsilon numbers" can be identified with pairs of real numbers with termwise addition and multiplication (I mean, not only are they a two-dimensional algebra over the reals, but actually they are the direct product of two copies of the real numbers), which makes them sort of boring (why bother about the product of two copies of the reals, not arbitrary tuples?); the identification takes the pair $(a,b)$ to ${\frac {a+b}{2}}+{\frac {a-b}{2}}\varepsilon$ (the number $\varepsilon$ is called $j$ in the article on split-complex numbers; and it's a trivial exercise to see that this is indeed an isomorphism). (Also, incidentally, the article is wrong in stating that split-complex numbers have nilpotents: they don't, they have divisors of zero but no nilpotents.) I'm stating all this to refute the idea that the number $\varepsilon$ is an interesting object. As to it's "power orbit", i.e., a one-parameter subgroup, once we have identified split-complex numbers with pairs of real numbers as I explained, and the number $\varepsilon$ with the pair $(1,-1)$, it is clear that one-parameter subgroups all lie in one connected component (both coordinates positive) of the multiplicative group of invertible split-complex numbers, and $\varepsilon$ is not there, so it does not have a "power orbit" (no more than -1 has in the real numbers). Similarly, trying to add both $i$ with $i^{2}=-1$ and $\varepsilon$ with $\varepsilon ^{2}=1$ just gives you pairs of complex numbers, again not very interesting. This is all basic algebra and applications of the Chinese remainder theorem. --Gro-Tsen 10:15, 10 August 2006 (UTC)

[...]

Feel free to repost my comment elsewhere if you think it wise. Personally I won't follow the "split-complex numbers" page because I don't think it's interesting in any way (but it's not really crackpot stuff either: it's just entirely boring) and I don't have time to improve it. I just find it laughable if it turns out that nobody noticed that these "split-complex numbers" are just isomorphic to pairs of real numbers (something which should be obvious from the start to anyone with a minimal background in algebra, e.g., having read Lang's book). Btw, "tessarines" / "bicomplex numbers" are similarly isomorphic to pairs of complex numbers. Any (commutative and associative) étale algebra over the real numbers is a product of copies of the real numbers and the complex numbers, anyway. --Gro-Tsen 12:38, 10 August 2006 (UTC)

One person's boredom is another person's interesting topic. The fact that no one uses these things indicates that the vast majority of mathematicians agree with you. There seems to have been a historical basis and interest in all of this; the article seems to document that. The question of what happens for the three cases of $j^{2}=-1,0,+1$ is legit, esp. given that the first is deep and fundamental, and the second is of interest to supersymmetry. So may as well discuss the third case as well, instead of pretending it doesn't exist. linas 13:55, 18 August 2006 (UTC)

Major application

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)

Just to give a few examples, on arXiv.org you find on a query for "split complex" and "split octonion" many, many hits: http://arxiv.org/find/grp_physics/1/OR+all:+AND+split+complex+all:+AND+split+octonion/0/1/0/all/0/1

Physicists are in need of algebraic explanations: How does it work, how can I use it; are my numbers commutative or not, are they normed, etc etc (all of Rgdboer's numbers have a multiplicative modulus; and I'm particularily thankful to Rgdboer for - indirectly - pointing me while back to hyperbolic quaternions). Surely, all these numbers are contained in some definition of hypercomplex numbers, like the one from from Kantor and Solodovnikov. But when one wants to actually use them for description of physical law, it is wonderful to have a step-by-step account of their algebraic properties listed - like it is in the current article. I would have loved to have this available to me 10 years ago or so.

The "nilpotents" mishap may have been caused by sloppy use of nomenclature in physics; when physicists talk about "split-complex numbers" they may at times actually refer to "split-compelx algebra", which includes the split-complex numbers from the current article, but also constructs that can be obtained through a modified Cayley-Dickson construction (like coquaternions and split-octonions). The latter (split-octonions) are of particular interest these days in physics, and it turns-out that the outline of the more general split-complex algebra slipped into the split-octonion article (i.e. the modified Cayley-Dickson construction when chosing $\lambda =\ell ^{2}=+1$). All constructs from split-complex algebra (other than the split-complex numbers) contain nilpotents.

One suggestion would be to generate a new article split-complex algebra that introduces the extension program in general, and then trim the existing articles about split-complex number, coquaternions, and split-octonions to their actual properties.

The remaining suggestions I would have are minor; for example in the description of its idempotents remove the word "non-trivial" and write explicitely the only two 'trivial' idempotents "0, 1". Or for the zero-divisors, spell them out explicitely: Every number $~z=(a+bj)$ has zero-divisors $c~z^{*}$ (a, b, c any real, $z^{*}$ conjugate of z).

Thanks, Jens Koeplinger 01:39, 12 August 2006 (UTC)

• For z to have a zero-divisor one must have a = b or a = − b. Remember, the group of units of D covers the whole plane except for these lines.Rgdboer 21:39, 15 August 2006 (UTC)

Yes, thank you for the correction, and for the reminder; you're right of course. Thanks, Jens Koeplinger 23:53, 15 August 2006 (UTC) PS: Thanks also for the updates to the hypercomplex numbers intro. I see the "stub" notice is removed, but the article is not really complete. It would be wonderful if we had a "history" section there; but this may be something for later. Jens Koeplinger 23:59, 15 August 2006 (UTC)

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)

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)

Interest in Terminology Cleanup?

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)

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)

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)

Serious lede

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)

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)

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)

Double numbers and Moebius transformations

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)

hyperbolic complex numbers

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)

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 $e\neq \pm 1$, 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)

Norm of a split-complex number

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)

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)

"Comparison to complex numbers" section...

... would be nice? --Raijinili (talk) 19:30, 14 August 2008 (UTC)

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 e^jθ = 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)

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)

Not archaic

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)

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)

Subfield larger than R ?

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) Some edits:Rgdboer (talk) 03:24, 22 February 2010 (UTC)

Symbols for bilinear form

Symbols that look much like "<" and ">" are used to define the bilinear form Re(z w*). As recently modified, they are hard to find and must be copied and pasted. They are not on the WP:Mathsymbols page but rather on List of mathematical symbols. The use of obscure symbols does not facilitate communication. Similar symbols are used to indicate "average" in statistics and inner product when the parameters are separated by a pipe (|). Discussion on these symbols is welcome; change can be anticipated as they impede expression when so hard to obtain.Rgdboer (talk) 22:24, 15 November 2011 (UTC)

I do not think that the solution is to use the less-than and greater-than symbols. What you originally changed them from were "⟨" and "⟩" (Unicode 27E8 & 27E9: Mathematical left/right angle bracket) which are in fact readily available under "Math and logic" symbols while editing. The visually similar delimiter symbols that I used last may have been inappropriate; feel free to use the symbols from "Math and logic" toolbar at the bottom of the edit page instead. Quondum^talk_contr 06:12, 16 November 2011 (UTC)

Trial test of the symbols:≪ and ≫ work for me. On the other hand ⟨ and ⟩ show up as little squares. Otherwise there isn't an angle bracket in the "Math and logic" selection under Insert. The little squares are just before the frac template, are they the symbols you suggest? The introduction of > and < came because my screen had squares where angle brackets were expected.Rgdboer (talk) 22:30, 21 November 2011 (UTC)

Yes, the symbols ⟨ (unicode "mathematical left angle bracket") and ⟩ (unicode "mathematical right angle bracket") appear just before {{frac|}}, and just after the □ (unicode "white square") and ∠ (unicode "angle"). From this I deduce that they are intended for general use. It seems you have a font issue specific to your browser – see Template:Unicode. To change a symbol that is often required in math to an inappropriate approximation is not a solution; it will require input from the wider community as it would by implication be the start of a style creep, generally discouraged. In the absence of a general fix, I suggest (in the following approximate order):

1. using the {{Unicode|}} template if it works for you (sample: ⟨x⟩ – preferred solution),
2. raising this problem in Wikipedia_talk:WikiProject_Mathematics or MOS:MATH,
3. using a {{math|}} template if it works for you (sample: ⟨x⟩),
4. forcing PNG, but best not inline with text (sample: $\langle x\rangle$ or $\scriptstyle \langle x\rangle$),
5. somehow selecting a more suitable font on your browser.

Quondum^talk_contr 06:05, 22 November 2011 (UTC)

commutative diagram

It has been requested that the commutative diagram be redrawn, here it is...

...is this ok?

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)

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)

No problem! Better? Maschen (talk) 23:12, 2 December 2012 (UTC)

Yes, your diagram has simplified the text considerably. The meaning of the diagram should now be clearer. Thank you.Rgdboer (talk) 20:37, 3 December 2012 (UTC)

Interval arithmetic

I have removed the following:

In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in Bulletin de l’Academie Polanaise des Sciences (see link in References). He identified an interval [a, A] with the split-complex number

$z={\frac {A+a}{2}}+\jmath \ {\frac {A-a}{2}}\$

and called it an "approximate number". D. H. Lehmer reviewed the article in Mathematical Reviews. In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted.

The papers referred to are available online here and do not support this statement. Warmus specifically states that "Approximate numbers do not form a group under addition" (paper I, p.254). Deltahedron (talk) 20:34, 11 February 2014 (UTC)

True, on page 254 he says that for one set of operations. However, on page 256 he says that approximate numbers form a ring. It is that structure that appears isomorphic to the split-complex numbers.Rgdboer (talk) 02:20, 10 September 2014 (UTC)

Is there an independent reliable source that supports this set being isomorphic to the split-complex numbers, or is it original research? Deltahedron (talk) 06:13, 10 September 2014 (UTC)

The review by D. H. Lehmer in Mathematical Reviews, now linked at the reference, says that the system is equivalent to the so-called hyperbolic complex numbers.Rgdboer (talk) 20:45, 10 September 2014 (UTC)

• I think the confusion comes from the fact that Warmus describes not one but two systems, each of which he calls "approximate numbers". The first system is closely tied to intervals, and as he says it is "not regular enough", eg, as I quoted before, not a group (I think it is probably a semiring). The second system is more satisfactory as an algebraic system, indeed, is a ring, and is what is identified by Lehmer with the hyperbolic complexes, and here as the split-complexes. However, the connection of this second system with intervals is more obscure.

What I do question is the statement in the article He identified an interval [a, A] with the split-complex number ... and called it an "approximate number" This implies that he made the identification, but he did not, that was Lehmer. I would suggest the wording

He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra. D. H. Lehmer reviewed the article in Mathematical Reviews and observed that this second system was isomorphic to the "hyperbolic complex" numbers, ie the split complexes.

Deltahedron (talk) 19:20, 11 September 2014 (UTC)

Your suggestion has been used to modify the History section. Thank you for investigating this topic.Rgdboer (talk) 23:57, 11 September 2014 (UTC)

Clifford algebra

Until someone finds a reference, the following has been removed:

The split-complex numbers are a particular case of a Clifford algebra. Namely, they form a Clifford algebra over a one-dimensional vector space with a positive-definite quadratic form. Contrast this with the complex numbers which form a Clifford algebra over a one-dimensional vector space with a negative-definite quadratic form. (NB: some authors switch the signs in the definition of a Clifford algebra which will interchange the meaning of positive-definite and negative-definite).

In mathematics, the split-complex numbers are members of the Clifford algebra Cℓ_1,0(R) = Cℓ⁰_1,1(R) (the superscript 0 indicating the even subalgebra). This is an extension of the real numbers defined analogously to the complex numbers C = Cℓ_0,1(R) = Cℓ⁰_2,0(R).

Generally books referring to Clifford algebra are very weak in abstract algebra so it would be good to find one that acknowledges this particular structure that underlies so much of the theory of real algebras. — Rgdboer (talk) 21:02, 16 March 2016 (UTC)

Isn't Cℓ_1,0(R), by the definition of that symbol, an algebra formed by adjoining to R an element e₁ such that e²
₁ = +1, which is exactly the split-complex numbers? These would indeed form an even subalgebra of Cℓ_1,1(R), the split-quaternions, just as the standard complex numbers do. It may be difficult to find a source calling this algebra the "split-complex numbers", because they're probably not in the mood to invent another name for ²R.

BTW, I confess I don't see how these reducible algebras are anywhere near the fundamental status of the Euclidean Hurwitz algebras. The split-complexes are just ²R, and lose a lot compared to the standard complex numbers: the norm is no longer positive-definite, and you can get zero divisors along the diagonals (which infects any algebra with a "countercomplex" basis, where (1±j) will always be zero divisors.) The split-quaternions mentioned are just R(2). While it does seem to me to be somewhat interesting to see what happens when you adjoin an element i to R and set in turn i² = −1, 0, and +1, it seems pretty obvious that the first choice is by far the best. And to avoid infecting a 4D number system, then when we add j, another complex basis, we give up commutativity to avoid having k become a split-complex basis (creating the tessarines), creating H. Finally we give up associativity when we add ℓ, yet another complex basis, to create O. I don't dispute that looking at these things can offer a way into interesting results like the nine-point conic (which can be analysed using C when it is an ellipse, and ²R when it is a hyperbola). But they don't seem as fundamental, just charming things that can be thought of in a "hypercomplex" way that may shed some additional light on their more basic form. Maybe these things are more useful in physics than they are in mathematics. If that is truly the case – and even if not – it strikes me that more details should be provided on how and where it is useful (in all fields!) to consider these numbers in the "hypercomplex" way, instead of just brushing them aside as ²R. (Perhaps the analogy to the complexes and the way they illuminate the unit hyperbola.) Double sharp (talk) 21:03, 17 March 2016 (UTC)

Thank you for your thoughts on improving the article. BTW some sense of the "Clifford algebra" program can be gained from comparing bivector with bivector (complex) and their lists of references. An archived Talk is informative. — Rgdboer (talk) 20:57, 19 March 2016 (UTC)

OK, I've been reading through some of these. It is indeed very interesting and is slowly going into my head (not without a lot of thinking about it). I guess they certainly are more useful in physics than a simple look at their mathematical properties would seem to indicate (they're indeed reducible, but the "hypercomplex" way of looking at the split-complexes does seem to be illuminating). Perhaps some of this in the simplest cases could be included in this article, however, since the split-complexes are probably much easier to introduce to an average mathematics student than Clifford algebras in general. Double sharp (talk) 15:43, 31 March 2016 (UTC)

The removed text appears to be correct to me, but I don't have a reference. Placing the split-complex within the framework of the "Clifford algebra program" is a worthy cause. Here's why. Jurgen Jost in his book on Riemannian geometry chapter one sets up (real) clifford algebras, obtains spinors, spin manifolds and spin connections .. all before chapter two. This includes special cases of split-biquaternions or whatever, but split-complex is never mentioned. Also, he does just Riemannian, not pseudo-Riemannian. For physics, Cl(p,q) gives more insight. Thus, stating that split-complex is real Cl(1,0) is worthwhile.

After that, the literature is a mess. Apparently, the "one ring to rule them all" is (real) Cl(p,q,r) which counts how many $e_{i}^{2}=\{+1,-1,0\}$. Conventional quantum field theory can be built from Cl(3,1) which gives plenty enough for the Dirac algebra for Dirac spinors, although confusion remains about Majorana spinors because physicists are abysmally terrible at abstract algbera. What physicists call supersymmetry is built from Cl(3,1,N) (three space, one time dimension, N super-partners, often N=2) Bryce DeWitt has a book called supermanifolds which deals with the Cl(p,0,r) case, but the index never mentions Clifford (I just looked). So that's the mega-context grand-scheme of things for "why clifford".

Doublesharps mention of Euclidean Hurwitz algebra is intriguing; but it doesn't contact the conventional conversations in which clifford ideas continue to spread and dominate. 67.198.37.16 (talk) 00:07, 22 May 2024 (UTC)

OK, here's a reference. Lounesto, Pertti (2001), Clifford Algebras and Spinors (2nd ed.), Cambridge University Press, ISBN 978-0-521-00551-7, gives the following section on p.209, as part of a survey of low-dimensional Clifford Algebras and their matrix representations, following on from discussion of the elements of the Clifford Algebras of the Euclidean plane $\mathbb {R} ^{2}$, Euclidean space $\mathbb {R} ^{3}$, and 4-dimensional Euclidean space $\mathbb {R} ^{4}$:

Split complex numbers $\mathbb {R} \oplus \mathbb {R}$
The Clifford Algebra Cℓ₁ of the Euclidean line $\mathbb {R} ^{1}=\mathbb {R}$ is spanned by 1, e₁ where e₁²=1. Its multiplication table is
...
The Clifford Algebra Cℓ₁ is isomorphic, as an associative algebra, to the double field $\mathbb {R} \oplus \mathbb {R}$ of split complex numbers. The product of two elements $(\alpha _{1},\alpha _{2})$ and $(\beta _{1},\beta _{2})$ in $\mathbb {R} \oplus \mathbb {R}$ is defined component-wise
$\qquad \qquad (\alpha _{1},\alpha _{2})(\beta _{1},\beta _{2})=(\alpha _{1}\beta _{1},\alpha _{2}\beta _{2})$
The isomorphism Cℓ₁ $\simeq \mathbb {R} \oplus \mathbb {R}$ can be seen by the correspondences
$\qquad \qquad \qquad 1\leftrightarrow (1,1)$
$\qquad \qquad \qquad \mathbf {e} _{1}\leftrightarrow (1,-1)$
The Clifford Algebra Cℓ₁ $\simeq \mathbb {R} \oplus \mathbb {R}$ is a direct sum of the two ideals spanned by the idempotents ${\tfrac {1}{2}}(1+\mathbf {e} _{1})\simeq (1,0)$ and ${\tfrac {1}{2}}(1-\mathbf {e} _{1})\simeq (0,1)$

Exercise 1 on p.218 asks the reader to show that Cℓ⁺_p,q $\simeq$ Cℓ_p,q-1 and Cℓ⁺_n = Cℓ⁺_n,0 $\simeq$ Cℓ_0,n-1, where the superscripted plus is being used to indicate the even subalgebra. A consequence of this is therefore that Cℓ⁺_1,1 $\simeq$ Cℓ_1,0, the split-complex numbers, while Cℓ⁺_2,0 $\simeq$ Cℓ_0,1, the complex numbers.

Cℓ_1,1 is briefly noted as the Clifford Algebra of the hyperbolic plane $\mathbb {R} ^{1,1}$ on page 211. Complex numbers are discussed as the even Clifford subalgebra Cℓ⁺_2,0 on pages 26 to 31.

This I think broadly takes care of the most substantial aspects of what was cut.

The even Clifford subalgebras are significant because, as described at eg Geometric algebra#Rotations, or (at length) at bivector, or in Lounesto p. 57 et seq, etc., this is where elements ("rotors") of the algebra live corresponding to elements of the group $Spin(p,q)$, that effect rotations and hyperbolic rotations acting either in a double-sided way $X\mapsto RX{\tilde {R}}$ on members $X$ of the algebra, or a single-sided way $\varphi \mapsto R\varphi$ on spinors $\varphi$ (which is what enables spinors to take roles as "square roots of geometry" (M. Atiyah), etc).

IMO setting out the Clifford algebra is therefore valuable, because it establishes how the behaviour described in #Geometry fits solidly within a wider pattern, i.e. the split complex number $a+bj$ identifiable with the bivector $a+b\;\mathbf {e} _{12}$ in the even part of Cℓ_1,1 ($\simeq a+b\;\mathbf {e} _{1}$ in Cℓ_1,0) effects hyperbolic rotations in Cℓ_1,1, just as the ordinary complex number $a+bi$ identifiable with the bivector $a+b\;\mathbf {e} _{12}$ in the even part of Cℓ_2,0 ($\simeq a+b\;\mathbf {e} _{1}$ in Cℓ_0,1) effects ordinary rotations in Cℓ_2,0, or, just as, in general, rotors effect rotations in higher dimensions, such as the quaternions (elements of the even part of Cℓ_3,0) in Cℓ_3,0 -- the one difference being that for complex numbers and split-complex numbers the two rotors in the two-sided transformation can be brought together to make a single complex number or split-complex number, which is not possible in higher dimensions.

IMO it would be useful to add some or all of this back into the article (ping @Rgdboer, Double sharp, and 67.198.37.16:)

For convenience, here is the removal diff, showing the material that was previously included at the end of the Split-complex number#Algebraic properties section. But it might be useful to put it under a subhead and add a bit more content, noting how in the context of geometrical algebra the material in the #Geometry section fits into the broader pattern. Jheald (talk) 17:50, 22 May 2024 (UTC)

Geometrically, I think it's pedagogically better to think of hyperbolic numbers (a.k.a. double numbers, split-complex numbers) as the even sub-algebra of the Clifford algebra of the Lorentzian vector plane (metric signature 1,1), analogous to complex numbers as the even sub-algebra of the Clifford algebra of the Euclidean vector plane, rather than as the full Clifford algebra of the Euclidean vector line, though both perspectives are meaningful and interesting in their own right. YMMV. In this version, each hyperbolic number can be thought of as a ratio of two Lorentzian-plane vectors, or equivalently as an operator which rotates and scales one vector into another. –jacobolus (t) 18:27, 22 May 2024 (UTC)

Geometric interpretation of j

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)

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)

Para-complex numbers

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)

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)

Hi Rgdboer. On MathSciNet, "paracomplex" produces 26 matches and "para-complex" produces 11 matches, against only 9 matches for "split-complex", which seems to confirm my previous impression. I'm not sure how you did your search. At first glance, I do not see how it is used in other contexts. The majority of the uses, it seems, are by differential geometers as I suspected (along with "para-Kähler" structures, which produces 53 matches).--Seub (talk) 02:06, 27 March 2016 (UTC)

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)

The symbol D

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)

Spacio-temporal plane

The following sentence was deleted by User:Cosmia Nebula:

• In that model, the number z = x + y j 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)

"In abstract algebra"?

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)

 Fixed by removing "abstract". D.Lazard (talk) 08:49, 7 September 2021 (UTC)

"no log when b > |a|"

I tried to add the formula for logarithm $\log(a+bj)={\frac {1}{2}}j(\log(a+b)-\log(a-b))+{\frac {1}{2}}(\log(a-b)+\log(a+b))$, but user Rgdboer reverted this with comment "no log when b > |a|". Well, when b > a the logarithm indeed falls out from the split-complex plane, being represented as a tessarine (complex+split-complex) number. Still, I think, the expression for logarithm is important, at least where it is applicable. The formula can be well sourced by the web sources. For instance: https://arxiv.org/pdf/math/0008119.pdf (formula (51))--Reciprocist (talk) 16:02, 7 September 2021 (UTC)

For adding such a formula, you must first define logarithms of split complex numbers, and you need a source for this definition. AFIK, there is no common definition of such logarithms. So, I agree with the revert. D.Lazard (talk) 16:51, 7 September 2021 (UTC)

The source is linked in the comment above. Here is another one: http://3dcomplexnumbers.net/2019/03/02/the-logarithm-of-all-2d-circular-numbers-the-split-complex-numbers/ --Reciprocist (talk) 17:26, 7 September 2021 (UTC) Another source (page 3: https://www.jcscm.net/fp/126.pdf)--Reciprocist (talk) 17:32, 7 September 2021 (UTC)

None of these three links are valid sources. See WP:Reliable sources. In mathematics, a reliable source requires to be published in a journal after a peer-referring process. This confirms that I said above: there is no definition of logarithms of split complex numbers that is accepted by a siginficant part of the specialists of this mathematical structure.

By the way, it seems that the authors of your links do not know the well known and almost trivial fact that split complex numbers are isomorphic (as a ring and as a quadratic space over the complex numbers) with $\mathbb {C} ^{2}.$ D.Lazard (talk) 20:54, 7 September 2021 (UTC)

LOL. Did you look at the last source linked? https://www.jcscm.net/fp/126.pdf It is published in Journal of Computer Science & Computational Mathematics, Volume 8, Issue 1, March 2018 DOI: 10.20967/jcscm.2018.01.001. --Reciprocist (talk) 07:08, 8 September 2021 (UTC)

Thanks for providing a link to this article (the first link was buggy). It is clear that the publisher is a predatory publisher, and that this article is thus not a source for Wikipedia. D.Lazard (talk) 08:47, 8 September 2021 (UTC)

Can you provide a link that claims so? I searched over Internet and found none.--Reciprocist (talk) 10:23, 8 September 2021 (UTC)

A journal that

• publishes article that clearly do no belong to the scope delimited by the journal title,
• do not states the competences of the editorial board, and do not provides a link to their personal wen page,
• is referenced neither by mathscinet or Zentral blatt

is necessarily predatory. D.Lazard (talk) 11:49, 8 September 2021 (UTC)

Have you a reference for that categorization? From a published source.--Reciprocist (talk) 12:51, 8 September 2021 (UTC) Here is another source: https://dergipark.org.tr/en/download/article-file/869566 p. 190 --Reciprocist (talk) 13:03, 8 September 2021 (UTC)

The exponential of the j-axis is the unit hyperbola. Thus exp(a+bj) lies in the quadrant |b|<a. Since logarithm is the bijective inverse of exp, it is undefined outside of that quadrant. Rgdboer (talk) 02:53, 10 December 2021 (UTC)

Yes, except if we allow tessarines. But it is defined inside that quadrant.--Reciprocist (talk) 21:45, 20 February 2022 (UTC)

Diagonal basis symbols

Can we use names $e_{-}={\tfrac {1}{2}}(1-j)$ and $e_{+}={\tfrac {1}{2}}(1+j)$ or similar instead of $e={\tfrac {1}{2}}(1-j)$ and $e^{*}={\tfrac {1}{2}}(1+j)$? 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)

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)

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)

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

Should this article be moved to double number?

Both the term "double number" and the term "hyperbolic number" seem to be in much wider use than "split-complex number" in academic literature (if anything could be called a "standard" name, I would say double number comes the closest). The oldest example of the name "split-complex number" I can see is from the mid 1980s, by Rosenfeld, and "split-complex" remains a relatively uncommon name. Much of the recent adoption of the term is probably driven by this Wikipedia article (cf. citogenesis): Google Scholar turns up only 34 sources containing "split-complex numbers" from before 2004 when this article was created, compared to 1540 sources containing "double numbers" [not all of these refer to this concept, but I would guess at least hundreds do] and 107 containing "hyperbolic numbers". Edit: ping user:Rgdboer. –jacobolus (t) 16:50, 8 April 2023 (UTC)

Double number is easily confused with even number. The prefix "split" occurs with other composition algebras. Note the many synonyms. "Complex" is not very descriptive, but entrenched. The symbol D acknowledges the alternative name and adjacency to C. One of the ways the concept of this article has been suppressed is by re-naming. Rgdboer (talk) 20:32, 11 April 2023 (UTC)

Whoops, I missed the reply here, sorry. I don't really understand your points. Why would "double number" be confused with "even number"? I have never in my life encountered someone calling even numbers "double number": do you have any example of that ever occurring? (I can find examples of people using the name "doubly even numbers" to refer to multiples of 4. I guess I could imagine someone calling powers of 2 something like "doubling numbers", but I don't think "double numbers" is really in danger of any serious confusion. It's not going to show up in contexts where these other kinds of numbers would make any sense as interpretations for the name.)

The prefix "split" does not especially often occur with the particular system we are talking about, and is a minority name by a significant margin. Using it in Wikipedia seems to me more like agenda pushing than neutrally describing the current state of the literature. Whether it occurs in a related system seems like a secondary concern.

The name "split quaternion" also seems like a terrible name to me, but has been picked up by most of the recent sources discussing it, so we're unfortunately stuck with it.

If we are worried about some naming consistency, having “complex numbers”, “split-complex numbers”, “dual numbers” seems incredibly inconsistent. Something along the lines of “elliptic numbers”, “hyperbolic numbers”, “parabolic numbers” would be more logically coherent, but “complex”/“double”/“dual” at least has the advantage of all three being different single-word names.

The symbol D acknowledges the alternative name – does't that seem pretty confusing? Like "we use name X for the name but name Y for the symbol, just to mess with readers".

One of the ways the concept of this article has been suppressed – the main reason the concept has been rare is because mathematicians learn about complex numbers in school and not double numbers; secondarily, mathematicians really love the field axioms and have been loath to work with systems where they don't hold if they can possibly avoid it. But I agree the concept has also been hampered by the lack of a concise, widely adopted name. Swapping out the most common short name for a much more cumbersome and less common alternative amplifies that problem. –jacobolus (t) 17:27, 28 April 2023 (UTC)

The concept of this article and split quaternions are composition algebras not included in Hurwitz's_theorem_(composition_algebras). Applications in science are evident. Thank you for the point by point response. Every even number is double another number, so the doubles might be just the evens. A general reader might think so. These low dimensional algebras that complement the division rings fit into the structure theory of algebras. — Rgdboer (talk) 02:18, 29 April 2023 (UTC)

You could still call this "split-complex numbers" in an article about composition algebra, if you think it makes that article clearer. My main point here is still unanswered though: wouldn’t you agree that the name "split-complex number" is significantly less common than either "hyperbolic number" or especially than "double number"? The nature of split quaternions is not really that relevant to this article in my opinion; these double numbers (or hyperbolic, perplex, ... numbers) are of independent interest, with a whole range of their own properties, used in a wide variety of applications that have nothing to do with split quaternions. It seems to me like the name "split-complex number" is preferred by a small group with particular niche interests, and using it in Wikipedia seems like pushing an agenda more than describing the state of the broader literature. ––jacobolus (t) 02:41, 29 April 2023 (UTC)

Please see Cayley–Dickson construction#Modified Cayley–Dickson construction for the context of this mathematical structure. Mathematics is mainly about structure so highlighting "independent interest" and "particular niche interests" or "pushing an agenda" are not in the cards. Perhaps it took a century to find the context of this structure, but now it is known, home is found, and eviction will be resisted. — Rgdboer (talk) 21:32, 29 April 2023 (UTC)

The context of this structure is that it is the basic setting for understanding conformal maps between 2-dimensional pseudo-Riemannian manifolds (if you like, "Lorentzian surfaces") differentially/locally, the same way the complex numbers are the basic setting for conformal maps between 2-dimensional Riemannian manifolds. We could also I guess mention Lorentz surfaces, analogous to Riemann surfaces. The double numbers (or split-complex numbers) are the the even-subalgebra of the algebra of vectors in the Minkowski plane (a pseudo-Euclidean plane of metric signature (1, 1)), in the same way that the complex numbers are the even subalgebra of the algebra of vectors in the Euclidean plane.

The most obvious and intuitive relevant feature is that they are strongly related to hyperbolas, hyperbolic functions, distances and isometries in hyperbolic space, and so on (hence "hyperbolic numbers"). Names like "Lorentz numbers" and "spacetime numbers" are also pretty reasonable, since the Minkowski plane is a slice through Minkowski space (flat spacetime). "Split-complex number" is in my opinion a terrible, confusing name that obscures the relevant features and relationships.

But Wikipedia is not about making such decisions. Wikipedia's policy guideline is to reflect the status of the existing literature, not try to promote one or another point of view. Existing literature has for better or worse predominantly adopted the name “double number” (presumably so-called because both components square to +1).

See WP:COMMONNAME––jacobolus (t) 21:50, 29 April 2023 (UTC)

It's actually quite a bit more dramatic a difference than I suggested previously. Most of the sources I can find mentioning "split-complex numbers" says something like "the hyperbolic numbers, sometimes called split-complex numbers ..." or "the double numbers, also known as split-complex numbers ...". There are very few sources that I can find using the name "split-complex numbers" as their canonical name for this concept. Both "hyperbolic numbers" and "double numbers" are more than an order of magnitude more common, even today and despite any influence from Wikipedia. ––jacobolus (t) 00:04, 30 April 2023 (UTC)

I've started collecting sources at User:Jacobolus/Double (roughly in chronological order by section). Feel free to take a look or add others. –jacobolus (t) 01:16, 30 April 2023 (UTC)

Underdrawing

Looking at the work of Isaak Yaglom and Walter Benz one sees mathematics as art, where this number system is an underdrawing of their later geometries. Yaglom's Geometry of Complex Numbers (1968) developed the dual numbers and double numbers. His A Simple Non-Euclidean Geometry and it Physical Basis (1979) does not refer to the planar algebras. Similarly, Benz's Vorlesungen uber Geometrie der Algebren (1973) developed anormal-complex numbers to illustrate projective geometry over a ring. Later proponents of Benz planes describe the Minkowski plane without reference to anormal-complex numbers. These Russian and German mathematicians, in polishing their geometric theses, painted over the numbers as a mere underdrawing. — Rgdboer (talk) 20:29, 2 May 2023 (UTC)

I don't understand what point you are trying to make. Yaglom's book puts this number system to use in the context of transformations in Cayley-Klein geometry, specifically the inversive geometry of the "pseudo-Euclidean plane of Minkowski" (and other planes with locally hyperbolic angles) and the Laguerre geometry of the "Lobachevskii plane" (and other planes with hyperbolic distances), analogous to Möbius transformations in the Euclidean plane (and sphere and hyperbolic plane, which have circular angles) or Laguerre transformations in the sphere (or other planes with circular distances), which use complex numbers. The same geometries are described in a more synthetic manner in his earlier book (translated later) Yaglom, Isaak Moiseevich (2009) [1956]. Geometric Transformations IV: Circular Transformations. Translated by Shenitzer, Abe. Mathematical Association of America. Originally published as Геометрические преобразования, Vol. 2 (in Russian). Moscow: GITTL. 1956.

But Yaglom's original goals don't really have much bearing on what this system is most commonly named in current literature. I think both "double number" and "hyperbolic number" are defensible names per WP:COMMONNAME (they are both in very common use in different but overlapping mathematical sub-communities, and are within maybe a factor of 2 of each-other in commonness), but "split complex number" really is not (it is at least 5–10x less common than either of the other names), and "split-complex number" with a hyphen is significantly rarer than that. –jacobolus (t) 21:17, 2 May 2023 (UTC)

As an aside the article Minkowski plane would be more helpful to readers if it provided more context. And the pseudo-Euclidean plane (a.k.a. Lorentzian plane or Minkowski plane) should be given a dedicated article that talks about its use generally and relation to spacetime, Clifford algebra, etc., rather than just focusing on Benz's incidence geometry stuff, which seems like a much more niche topic. –jacobolus (t) 21:33, 2 May 2023 (UTC)

WP:PRECISE says that a name should be an exact designation. "Double" shows up as #1091 Baumann's revised General Service List of the most common English words, too common for precise description. The association with "complex number" should be retained, with understanding that the algebraic elements described are not in a division algebra but a split algebra, one with a null vector. Thank you for assembling the list of references since it expands the topic, though some archive.org texts are not WP:RL. As the complex numbers have the unit circle, this algebra has the unit hyperbola. The comparison is strong enough to perpetuate the suggestion of Benz and include "complex" in the name. — Rgdboer (talk) 02:48, 5 May 2023 (UTC)

Note that the list I have gathered so far are made up of papers that either (a) use one of the very obscure terms, (b) are within the last couple years and published in a high-impact journal, or (c) have been cited at least several times (I think you mean "reliable sources" rather than "rugby league" and arxiv rather than archive.org: the arxiv papers are those with at least 5+ citations, including some written by experts with dozens of citations), though I was just eyeballing these categories rather than applying a super consistent methodology. I went through essentially every more-than-trivially-cited paper in the Google Scholar corpus mentioning "split complex numbers" or "split-complex numbers", of which I would guess at least half primarily used the name "double number" or "hyperbolic number" and only mentioned the "split" name as a synonym. I didn't yet finish going through the full list for the other names because as I said they are >5x longer. I'm not sure I will ultimately bother going through all of them. But it gives at least some idea so far, and can be a start for finding sources for specific claims.

This list of papers should obviously be dramatically culled before trying to actually make citations in this article, and we should try to find the most cited, most topical, best written source for every particular claim.

The association with "complex number" should be retained – This is just you pushing your personal preferences, and does not align with Wikipedia guidelines.

too common for precise description – there are thousands of precise terms that are phrases consisting of only common English words. The terms "hyperbolic number" and "double number" are both quite precise, and used primarily (if not exclusively) for this purpose. (You can find other uses of the word pair "double number" in the context of miscellaneous sentences but rarely if ever in a context where it is used as a dedicated term.)

As the complex numbers have the unit circle, this algebra has the unit hyperbola. Considering your comments, I think we should compromise on the term "hyperbolic number". In actually doing some literature search, this term is more common than I had previously realized (roughly comparable to "double number", within at worst about a factor of 2 in how often it is used, if not closer), is pretty descriptive, is not really confusable for any other meaning, and is used often by sources dedicated specifically to this topic rather than just mentioning it. From what I can tell the name "hyperbolic complex number" was sometimes used before the 1990s, but Sobczyk's paper employing the name "hyperbolic number" was then very influential (hundreds of direct citations), especially by authors working on Clifford algebras and related topics, and pretty much crowded that one out, as well as taking "mindshare" from other names, for example Kocik apparently decided at some point that "hyperbolic numbers" is preferable to "duplex numbers". –jacobolus (t) 03:50, 5 May 2023 (UTC)

See isotropic quadratic form#Hyperbolic plane for some of the ambiguity associated with "hyperbolic". The split-complex numbers do not pertain to hyperbolic geometry in the main. "Hyperbolic" is overused as can be seen at hyperbolic (a disambiguation page). And a double number could be two digits, a pair of numbers (3,4), or a doubly indicated number VII, 7. The popularizers Yaglom and Sobzyk do not give the composition algebra context, nor the Dickson construction. The connection with "complex" is essential since that planar field is a counterpoint to this split algebra, also in the plane. References from the Clifford camp are less reliable than most math sources. — Rgdboer (talk) 02:51, 8 May 2023 (UTC)

Hyperbolic numbers are the basic context for representing transformations in the Minkowski plane (a.k.a. Lorentz plane) where angles are "hyperbolic", as well as the de Sitter and anti-de Sitter planes, in just the same way that complex numbers are the basic context for representing transformations of the Euclidean plane where angles are "circular", as well as the hyperbolic plane and sphere. They are also the basic context for representing distances in the hyperbolic plane. Every aspect of them is irrevocably bound together with hyperbolic geometry, hyperbolic functions, and hyperbolas.

But that's all besides the main point, which is that "split-complex number" and "split complex number" are minority names which will not stand up to challenge under the WP:COMMONNAME guideline. –jacobolus (t) 03:18, 8 May 2023 (UTC)

P.S. Your link to Isotropic quadratic form § Hyperbolic plane is directly relevant here, and a good example of why "hyperbolic" is an appropriate name. –jacobolus (t) 03:20, 8 May 2023 (UTC)

See Coordinate systems for the hyperbolic plane to realize that is article is not appropriately associated with that topic. Furthermore, in science the use of binomial nomenclature makes for precision: this article is about the genre of complex planes, of the split species. The common terms "double" and "hyperbolic", though commonly used in popular sources, are not specific enough. The redirects are in place for those terms, why have the scientific article title changed to something that loses accurate direction? — Rgdboer (talk) 23:33, 8 May 2023 (UTC)

Distances and translations in the hyperbolic plane are "hyperbolic angles", naturally best represented by unit-magnitude hyperbolic numbers, in just the same way that circular rotations are naturally represented by unit-magnitude complex numbers.

The use of double names to refer to particular species in phylogeny/biological taxonomy is completely irrelevant to this discussion. Notice that WP:COMMONNAME doesn't say anything remotely similar to: "all scientific concepts must be named using a convention similar to the one Linnaeus set up for biologists to use when naming types of organisms in the mid 18th century". ––jacobolus (t) –jacobolus (t) 00:10, 9 May 2023 (UTC)

Consistency is a parameter for evaluating a name. The redirect Split algebra was posted no note split-complex number is consistent with names of related articles. — Rgdboer (talk) 02:47, 11 May 2023 (UTC)

That's fine if you are choosing a name for your own academic paper, monograph, or blog post. It just doesn't match Wikipedia's policy guidelines. –jacobolus (t) 07:28, 11 May 2023 (UTC)

See WP:CRITERIA for precision and consistency, criteria appropriate to this discussion. — Rgdboer (talk) 22:42, 11 May 2023 (UTC)

Great. Notice that these criteria are roughly ordered from most to least important, with the key being that "Article titles are based on how reliable English-language sources refer to the article's subject." The title "Split-complex number" trails badly in "Recognizability", "Naturalness", and "Concision", is marginally worse at "Consistency" (consistency with Complex number and Dual number is the most relevant here, and "Split-complex" is clearly not in any consistent scheme with those), and any of these names is more or less fine for "Precision". I should probably make a formal proposal and take it to more eyeballs. –jacobolus (t) 23:45, 11 May 2023 (UTC)

Not a metric space

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)