Talk:Biquaternion

Conflict of terminology

There seems to be a conflict of terminology with regards to the name biquaternion. Hamilton appears to have used the term to mean a quaternion with complex coefficients (i.e. C⊗H), while Clifford (in Preliminary Sketch of Biquaternions, 1873) uses the term to mean an algebra isomorphic to H⊕H, which follows the quaternions in the sequence of Clifford algebras:

R → C → H → H⊕H → ...

The complexified quaternions are not isomorphic to Clifford's biquaternions. This page presently discusses Hamilton's notion, while the German version of the page discuss's Clifford's notion. Some mention should be made of the conflict. I'm not sure which term is more commonly used. -- Fropuff 21:43, 19 February 2006 (UTC)

Note that Hamilton used the term first (it appears in his 'Lectures on Quaternions', 1853, article 669, available at http://historical.library.cornell.edu/math/). Sangwine 21:50, 25 February 2007 (UTC)

Clifford biquaternion

Since the structure of Clifford biquaternions is demonstrably different than the classical twentieth century concept of biquaternions used to develop the relativity transformations, the works of the Clifford algebraists on their biquaternion need a separate space. Rgdboer 01:35, 23 February 2006 (UTC)

Well okay; I'm not sure the name is standard, but I guess we have to disambig them somehow. At any rate we should probably mention the alternate meaning somewhere in the intro to this article.

Of course, Hamilton's biquaternions also form a Clifford algebra, just with the opposite signature from Clifford's biquaternions. They fit into the sequence

R → C^~ → H^~ → C⊗H → ...

with C^~ and H^~ being the split-complex numbers and split-quaternions respectively. -- Fropuff 04:01, 23 February 2006 (UTC)

WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:44, 10 November 2007 (UTC)

Does a biquaternion have an inverse

First congratulations on this fantastic new article. I hope to watch it expand with time. These would be good things to explain in this article.

Does a biquaterion have an inverse, all the time, some of the time or none of the time?

It has to at least some of the time I think because if the imaginary part is zero, it would have an inverse, but this is trivial.

Letting h equal the imaginary scalar of ordinary algebra, since i,j and k are already taken, it would seem to me that the simple biquaterion:

hi would be its own inverse.

$hihi=(hh)(ii)=(-1)(-1)=1\,$

Your example is given in the section biquaternion#Alternative complex plane. The discriminant for biquaternions is

D = $u^{2}+v^{2}+w^{2}+x^{2}$.

When D ≠ 0, then q = u + vi + wj + xk has an inverse, otherwise it is singular.Rgdboer (talk) 20:10, 26 November 2008 (UTC)

I found were Hamilton gives the general formula for the reciprocal of a bi-quaternion:

Hamilton's general formulas for the reciprocal of a biquaternion

— Hobojaks (talk) 20:36, 28 November 2008 (UTC)

Relation to octonains

If a biquaterion does not always have an inverse, why does making the imaginary scalar anti-commutative and anti-associative thus making an octonian change the system back into a division algebra?

Hobojaks (talk) 18:19, 26 November 2008 (UTC)

The switch you describe for getting octonions results in a discriminant that is the sum of the absolute values of u, v, w, and x.

The only time this discriminant is zero is when u = v = w = x = 0, so the octonions are almost always invertable.Rgdboer (talk) 20:10, 26 November 2008 (UTC)

Trivialization

Since biquaternions and 2 x 2 complex matrices are algebras with different bases, they are not exactly the same thing, though they are isomorphic algebras. In historical or educational contexts, reference to biquaternions is natural and valid. With the sophisticated modern view of spinor representation theory, contemporary authors need make no reference to biquaternions when dealing with 2 x 2 complex matrices. But this encyclopedia is for the general public and the policy WP:NPOV explicitly prohibits pejorative remarks such as:

The term "biquaternion" is archaic and no longer used much by mathematicians, because the algebra of biquaternions is isomorphic to the algebra of 2 by 2 complex matrices.

that was contributed by a WP:User on January 14 . Therefore I am undoing the contribution.Rgdboer (talk) 22:32, 24 January 2010 (UTC)

What is wrong with that statement ? To say they are not used much is probably an understatement: they are not used at all in maths today that I am aware of. Archaic is just another word for historic, which you use - hardly pejorative.

The reason they are not used is they are isomorphic, i.e. have identical properties, to 2x2 complex matrices. Mathematicians prefer to use these are they can bring their full range of tools to bear on them. E.g. they can do inverses, products, sums, etc. without having to learn a new algebra. Biquaternions is just one of a number of systems that have been created that have later turned out to be special cases of a more general and powerful algebra, so have been forgotten for good reason. It's good that they are documented but it should be made clear that they are only of historical interest. --JohnBlackburne^words_deeds 22:44, 24 January 2010 (UTC)

---

In reply to the comment;

The term "biquaternion" is archaic and no longer used much by mathematicians, because the algebra of biquaternions is isomorphic to the algebra of 2 by 2 complex matrices.

If you search the internet, then you will find this is not true. In particular, I reference you to this physics essay on biquaternions and the things they deduce in modern physics (pretty much all of it). http://arxiv.org/PS_cache/math-ph/pdf/0201/0201058v5.pdf

"To conclude this introduction, let us summarize our main point: If quaternions are used consistently in theoretical physics, we get a comprehensive and consistent description of the physical world, with relativistic and quantum effects easily taken into account. In other words, we claim that Hamilton’s conjecture, the very idea which motivated more then half of his professional life, i.e., the concept that somehow quaternions are a fundamental building block of the physical universe, appears to be essentially correct in the light of contemporary knowledge."

Secondly, I believe I now know why these equations are central to physics and physical reality (I discovered this just a few weeks ago). I have tried to explain this as simply as I am able (below) but I ask that those who better understand the mathematics of biquaternion wave functions check this to confirm it is true (I am a natural philosopher, I came to the solution not from mathematics but from the spherical standing wave structure of matter (WSM)).

If you describe reality most simply (Occam’s razor) then you must describe reality in terms of only one substance, and since we all experience existing in one common space it is reasonable (consistent with science) to take this as our foundation (Hamilton thought similarly, why he devoted his life to developing maths for real objects in real 3D space).

Since the wave properties of light and matter are well known (the particle wave duality) and since we cannot add another separate substance, matter particles, to space, but we can have space vibrating (waves flowing through space), we can then use our biquaternion wave equations and see what we get based upon this most simple conception of physical reality. i.e.

“One thing, three dimensional space exists and has planar waves flowing through it in all directions.”

If we then apply biquaternion wave functions to this we find a most stunning thing. Space can actually vibrate in two completely different ways.

1. Space vibrates in all directions (background space, quantum field, vacuum fluctuations, Tao, Akasha Prana, ...)

2. Space vibrates radially around a central point, forming a scalar spherical standing wave. In this special case the biquaternions show how the vector / transverse wave components all cancel one another, leaving a scalar spherical standing wave. The wave center is what we see as matter 'particles'

The most profound thing is that the equations give us the Dirac equation – but now we can understand the cause of spin and antimatter. The biquaternion wave equations show us that there are four different phase arrangements for the transverse waves that cause them to cancel one another – and they create two pairs of scalar spherical standing waves that have opposite phase (matter and antimatter) and for each phase there are two phase arrangements that can construct it (the two spin states of the electron and positron). This is represented by the biquaternion multiplied by its complex conjugate (I assume this represents the waves flowing in opposite directions and thus opposite vectors) and the result is a scalar spherical standing wave (the vector / transverse wave components cancel).

And from this most simple science foundation for reality – just waves flowing through space in all directions - it seems (thanks to the work of many brilliant mathematicians) that you can then perfectly explain and unite quantum physics and Einstein’s relativity and exactly deduce all the central equations of modern physics. To confirm this you just need to search biquaternions and each subject area of physics and you will find the solutions, certainly Mendel Sach’s has done a lot in the area of general relativity.

What seems truly remarkable is that we have solved all the mathematics first, without ever understanding the amazingly simple physical reality behind it all that caused it. However, it should be acknowledged that Clifford was partly correct with his work "On the Space-Theory of Matter". He wrote;

"I hold: 1) That small portions of space are in fact analogous to little hills on a surface which is on the average flat, namely that the ordinary laws of geometry are not valid in them. 2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave. 3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal. 4) That in this physical world nothing else takes place but this variation subject to the law of continuity." http://en.wikisource.org/wiki/On_the_Space-Theory_of_Matter

Finally, with respect to the argument that quaternions are archaic because we now have more advanced maths that are more flexible and can work in infinite dimensions. What they forget is that reality determines the mathematics we must use to describe it (thus the current confusions of higher dimensions), and the reality we all experience is three dimensional space (where we can now deduce that the further 4th dimension of time is cause by the wave motion of this space). This is why biquaternions are so useful and important to mathematical physics (and to humanity), it seems their structure correctly represents the structure of physical reality. Geoffrey Haselhurst (14/05/2011) Haselhurst (talk) 06:09, 14 May 2011 (UTC) —Preceding unsigned comment added by Haselhurst (talk • contribs) 05:19, 14 May 2011 (UTC)

More footnotes

The tag asking for more inline citations has been removed. This is a mathematical article and assertions are verifiable. If a reader or editor requires further edification, please use this Talk space to make comments or raise questions.Rgdboer (talk) 23:24, 20 January 2013 (UTC)

Confusion in "Linear representation"

Without reading this entire article line by line, I was confused by the claim about the Linear representation that "each of these three arrays has a square equal to the negative of the identity matrix." Let's compute:

When I square the first array, I get h² times the identity matrix:

${\begin{pmatrix}h&0\\0&-h\end{pmatrix}}{\begin{pmatrix}h&0\\0&-h\end{pmatrix}}={\begin{pmatrix}h^{2}&0\\0&h^{2}\end{pmatrix}}$

When I first looked, I didn't see the Definition that h is "the square root of minus one".

By the way: Is "the square root of minus one" in this context the same as the imaginary unit? I assume so -- and if so, it might be nice to link to that Wikipedia on imaginary unit.

I"m making a change of this nature.

Thanks, DavidMCEddy (talk) 09:10, 14 December 2015 (UTC)

CA

The following was removed as unreferenced:

They can be classified as the Clifford algebra Cℓ₂(C) = Cℓ⁰₃(C). This is also isomorphic to the Pauli algebra Cℓ_3,0(R), and the even part of the spacetime algebra Cℓ⁰_1,3(R).

CA texts don't highlight biquaternions so why should this article direct to CA ? Compared to composition algebras, CA,s have weak foundation. — Rgdboer (talk) 02:43, 1 February 2017 (UTC)

I've added references and replaced the CA stuff.

CA texts don't highlight biquaternions as such because CA provides a uniform and consistent means of understanding such geometric structures in arbitrary dimensions and signatures, so the name "biquaternion" is an unhelpful classification for those of us who don't enjoy the botanist's approach to mathematics. On the other hand, I saw a reference to biquaternions and wondered what they were, so I came to this page hoping to find out. Since I'm used to CA notation, I would have understood immediately, and could have stopped reading and gone on to more useful endeavors if I had seen those classifications. As it is I had to scratch my head, figure it out for myself, wonder if I was mistaken because surely wikipedia would have said if there was a simple CA classification of this algebra, then look through the revision history, and finally make the edits myself to regain the information you removed from this page. More generally, the entire idea behind hypertext is that links between concepts are simple to make and useful to have; ignore them if you want, and let other people use them if they want. So "this article [should] direct to CA" to take advantage of one of the most important innovations of the past century.

I am also befuddled by your claim that CAs have a "weak foundation". What could that possibly mean? CA has an enormous literature by now, and they are absolutely vital in modern treatments of geometry and theoretical physics. I expect that you will find many people like myself who think and speak almost entirely in terms of CA. In any case, they are clearly part of the academic mainstream, so insisting on their removal looks like tendentious POV, especially in light of the fact that several other claims about the algebra do not have references. MOBle (talk) 16:15, 29 August 2017 (UTC)

The issue of coinage is pertinent to this article since proponents of Geometric Algebra assert use of namespace bivector for a 2-vector in multilinear algebra, diminishing the claim to the namespace by those attentive to literature in mathematics using the term who are now compelled to revert to bivector (complex) to express the vector part of a biquaternion. You are encouraged to remove the script notations of your team, as the tenure at Bivector is but one piece of kompromat that can be displayed. Any editor, understanding what is at stake, can do it for you. — Rgdboer (talk) 21:47, 30 August 2017 (UTC)

"Introduction to the complex quaternions in particle physics"

I have removed the above reference again as I was unable to find it or where it was published, so am not persuaded it is a reliable source. If it’s online then there should be a link. If it’s in a journal there should be a DOI. Then anyone can check it’s reliability or otherwise. The Youtube videos are certainly not a reliable source, and links to Youtube do not belong in articles.--JohnBlackburne^words_deeds 07:58, 18 August 2017 (UTC)

In reply: Of course, the videos are fully referenced. If you watch the films, you will find these references listed clearly at the end of every video. The relevant publications are also listed right in the description of each video.

These films were created at Cambridge University. If you check the references which are contained within the videos, you will see that this work has been reviewed both as a PhD thesis at the University of Waterloo / Perimeter Institute for theoretical physics, and in Phys.Rev.D cited therein.

Again, JohnBlackburne, could you please describe your training which would qualify you to evaluate research in particle physics? Do you have a PhD in particle theory?

I will add a link to the work on which these videos are based, for clarity.

Theor-phys (talk) 11:33, 18 August 2017 (UTC)

You misunderstand what is meant by a reliable source. In general a reliable source is one that is published by a publisher with a reputation. This might be a reputable academic journal, or a recognised publisher of textbooks, or an established news organisation with a reputation for fact checking. What references they contain does not matter – every academic paper contains references, even ones of limited value that are never published in a journal, such as most PhD theses.

YouTube is not such a publisher, as anyone can upload a video to it, and much of the content on there is of very low quality. YouTube is further problematic as it not accessible to everyone, and due to the heavy and intrusive advertising it carries. So videos are generally not used as sources. The only exceptions are videos published by reliable sources, but that is not the case here.

If there is a CU course on Biquaternions then it might be worth linking to that – the University is a reliable publisher. But I see no evidence of this, it just looks like someone’s personal project, based on a PhD.--JohnBlackburne^words_deeds 12:05, 18 August 2017 (UTC)

---

Dear JohnBlackburne,

I have checked the regulations on videos, and there is no restriction against chalkboard presentations from youtube. Could you please demonstrate otherwise?

Also, as I have already mentioned above, the material in these videos has been published in PRD, a well known peer reviewed journal in particle physics. The content is the same in the video as what has already been published, but the videos provide a visual way to present the content.

Throughout our discussion, you have offered many reasons for why you feel this link should be removed. This includes stating that the links are not references (they are), stating that they are not good links (they are), or stating that youtube is not acceptable due to advertising, etc (however I have not been able to find anything banning youtube in the regulations). This makes it difficult to understand your motivation.

Theor-phys (talk) 12:49, 18 August 2017 (UTC)

Historically YouTube videos have not been regarded as reliable sources. Perhaps the relevant section of the rules is this?

If the videos contain the same content as certain articles in PRD, then why not just cite those articles? Mgnbar (talk) 15:36, 18 August 2017 (UTC)

I think it would be best to provide the peer-reviewed references first; conveniently, the arXiv postings have the DOIs for the journal versions ([1] [2] [3]). If, having done that, one wanted to say, "The author has also covered this material in lectures, for which recordings are available", I wouldn't object. Videos can be helpful supplements, because people sometimes explain their motivations and emphasize the key highlights more in lectures than they do in writing. But we should really have our articles point to the material that has gone through the peer-review process. XOR'easter (talk) 16:27, 18 August 2017 (UTC)

I agree with XOR'easter. Boris Tsirelson (talk) 17:07, 18 August 2017 (UTC)

I had a look at those papers earlier and none of them concern biquaternions. They don’t even mention biquaternions, so anyone looking at them for some insight into the topic will be confused. They do mention C ⊗ H, but just as one of many algebraic structures used, and offer no particular insights into the topic of the article that I can see.JohnBlackburne^words_deeds 17:19, 18 August 2017 (UTC)

Again, this is not true. This is another angle which seems to be unjustified. The content is covered in both ([4]) and in the PRD article ([5]). Here, all of the Lorentz representations of the standard model: scalars, spinors, 4-vectors, and the field strength tensor are demonstrated to be given by invariant subspaces of the biquaternions. Theor-phys (talk) 17:34, 18 August 2017 (UTC)

With respect to the argument that videos are not acceptable content: On a quick search, I have found wikipedia articles ([6]), ([7]), ([8]) which all contain videos, some on youtube. So there is indeed precedence for including videos as instructional tools.Theor-phys (talk) 17:41, 18 August 2017 (UTC)

I've inspected only Polarization (waves). In that article, videos appear in the External links section, not in the Notes and references. The latter consist of books with ISBNs, papers from arXiv, etc. Even if that article did cite YouTube, it would demonstrate only that that article needed work on its citations. Mgnbar (talk) 19:10, 18 August 2017 (UTC)

I don't know whether you're really new to Wikipedia, but let me add: When I first started editing here, many of the policies seemed strange, restrictive, or even misguided. But over time I have come to see why they're here. In fact, most of them I now recognize as essential to the success of the project. And the reliable sources policy, even if it's not perfect, is essential. Mgnbar (talk) 19:14, 18 August 2017 (UTC)

If you take a candid look at a range of wikipedia articles, such as ([9]), ([10]), ([11]), etc., it may become apparent that videos (youtube or otherwise) are abundant within wikipedia, usually without much referencing. It therefore might strike someone as odd that including videos here should suddenly be problematic. Having said that, I believe you have a point that the content may often be found in the 'External links' section. if this content is better suited in an “External links” section, then I can move the videos there.Theor-phys (talk) 19:34, 18 August 2017 (UTC)

It’s not suddenly. It’s part of the policies here. That’s why a bot reverted your first addition, as the prohibition on YouTube is so strong that it’s treated as disruptive editing and reverted automatically by a bot. As you discovered it’s easy to overcome the bot action: just revert it and it won’t remove them a second time. Maybe this happened on other pages, or maybe the links were added before this policy was being enforced this way. The three articles you link to does not make them 'abundant'. There are millions of articles on Wikipedia. More than those three have links to YouTube, but it’s still a tiny fraction of the totality of articles.--JohnBlackburne^words_deeds 20:08, 18 August 2017 (UTC)

Theor-phys, yes, there is a large distinction between Notes and references and External links. The former are reliable sources required to support the article's claims. The latter are tangential enrichment material. So please do sort your edits accordingly. Thanks. :) Mgnbar (talk) 20:56, 18 August 2017 (UTC)

Ok, thanks Mgnbar. Will do. Theor-phys (talk) 21:17, 18 August 2017 (UTC)

Video based on a PhD thesis?

That's just pushing it too far. I'll remove it. YohanN7 (talk) 10:08, 28 August 2017 (UTC)

I raised the question (about possible self-promotion) at Wikipedia talk:WikiProject Mathematics#Talk:Biquaternion. YohanN7 (talk) 10:46, 29 August 2017 (UTC)

A large fraction of wikipedia pages have videos on them with no DOIs, and no referencing at all for that matter. There have now been many different explanations given by a couple people for why these videos should be removed. Frankly, I think the real issue here is that I posted a video of a woman teaching physics, and that makes some people feel uncomfortable.Theor-phys (talk) 12:08, 29 August 2017 (UTC)

Are you C. Furey? YohanN7 (talk) 12:39, 29 August 2017 (UTC)

Theor-phys, I've never viewed the videos. (I didn't need to, to know that they aren't reliable sources suitable for citation.) And the links do not hint that the presenter is a woman. So I haven't known that the presenter is a woman until now.

You have to understand that active Wikipedia editors encounter link spam on a daily basis. When a new editor, who is active on just one topic, fights doggedly to add external links to Wikipedia, alarm bells go off in my mind (whether or not the external links turn out to have merit). I'm just trying to help you understand what might be going through the heads of your fellow editors. Mgnbar (talk) 14:15, 29 August 2017 (UTC)

I think I found a solution everybody can live with. YohanN7 (talk) 13:07, 1 September 2017 (UTC)

What is this "complex light cone" method that has superseded the biquaternions?

At the end of the section https://en.wikipedia.org/wiki/Biquaternion#Associated_terminology, there is a mention of a "complex light cone" method that has superseded the biquaternions. Can someone explain what this is? Cheers. — Preceding unsigned comment added by 82.13.14.165 (talk) 05:42, 27 May 2019 (UTC)

The move to general relativity brought in differential geometry and different linear algebra. See Newman–Penrose formalism. — Rgdboer (talk) 22:49, 27 May 2019 (UTC)

Bad conjugation notation

The notation of $q^{*}$ and $q^{\star }$ to denote the 2 different conjugations is bad; they can't even be distinguished if the font is small. As a replacement, I would suggest these ones:

$q^{*}\to {\bar {q}}\quad =\gamma ^{0}q^{\dagger }\gamma _{0}$,

$q^{\star }\to q^{c}\quad =\gamma ^{0}q\gamma _{0}$,

also let $(q^{\star })^{*}\to q^{\dagger }$.

These ones will match operations on biquanternions represented as commutators of gamma matrices (shown to the very right above):

$h=i\gamma ^{5}$,

$\mathbf {i} =\sigma ^{jk}$,

$h\mathbf {i} =\sigma ^{0i}$;

where $\sigma ^{\mu \nu }={\frac {1}{2}}\left[\gamma ^{\mu },\gamma ^{\nu }\right]$ and $\epsilon _{ijk}=1$. 150.135.165.6 (talk) 06:45, 15 April 2023 (UTC)

Agreed that the two conjugations needed clearer distinction. \star has been replaced with \bar in the math markup. \overline can be used with longer expressions. The asterisk has been kept for the biconjugate; note that it is the conjugate expressed with the Cayley–Dickson construction for the general case. Thank you for bringing this lack of clarity to notice. — Rgdboer (talk) 23:19, 23 April 2023 (UTC)

The BiQuaternion multiplication table references

I was looking into the reference article on the table addition made on February 20 added by an IP (not a WP user). The link https://www.naturalspublishing.com/files/published/e71f3zs34zg62q.pdf has this table, but it simply defers to a reference to another article [12] S. J. Sangwine, T. A. Ell, N. Le Bihan, Adv. Appl. Clifford Algebras 21, 607-636 (2011).

I couldn't find that specific published article online, but I did find a well referenced article by the same authors from the prior year 2010 arXiv:1001.0240v1 [math.RA] 1 Jan 2010 Fundamental representations and algebraic properties of biquaternions. The table in that is much different (no way to get from that to the table in this WP Biquaternion), but looks right. Other references I find to biquaternion tables seem closer to this and I am concerned the one on the WP page might be wrong (or unsubstantiated) without further published source detail.

::Jgmoxness (talk) 20:05, 8 May 2024 (UTC)

Multiplicaton is clear from commutivity of h with the three imaginary units, which multiply as in the quaternion group. The table does not contribute to the article, IMO, and is moved here for comment:

A Biquaternion multiplication table Naturalspublishing is shown below:

$e_{i}e_{j}$		$e_{j}$
		$e_{0}$	$e_{1}$	$e_{2}$	$e_{3}$	$e_{4}$	$e_{5}$	$e_{6}$	$e_{7}$
$e_{i}$	$e_{0}$	$e_{0}$	$e_{1}$	$e_{2}$	$e_{3}$	$e_{4}$	$e_{5}$	$e_{6}$	$e_{7}$
	$e_{1}$	$e_{1}$	$-e_{0}$	$e_{3}$	$-e_{2}$	$e_{5}$	$-e_{4}$	$e_{7}$	$-e_{6}$
	$e_{2}$	$e_{2}$	$-e_{3}$	$-e_{0}$	$e_{1}$	$e_{6}$	$-e_{7}$	$-e_{4}$	$e_{5}$
	$e_{3}$	$e_{3}$	$e_{2}$	$-e_{1}$	$-e_{0}$	$e_{7}$	$e_{6}$	$-e_{5}$	$-e_{4}$
	$e_{4}$	$e_{4}$	$e_{5}$	$e_{6}$	$e_{7}$	$-e_{0}$	$-e_{1}$	$-e_{2}$	$-e_{3}$
	$e_{5}$	$e_{5}$	$-e_{4}$	$e_{7}$	$-e_{6}$	$-e_{1}$	$e_{0}$	$-e_{3}$	$e_{2}$
	$e_{6}$	$e_{6}$	$-e_{7}$	$-e_{4}$	$e_{5}$	$-e_{2}$	$e_{3}$	$e_{0}$	$-e_{1}$
	$e_{7}$	$e_{7}$	$e_{6}$	$-e_{5}$	$-e_{4}$	$-e_{3}$	$-e_{2}$	$e_{1}$	$e_{0}$

Opinions requested. — Rgdboer (talk) 01:57, 9 May 2024 (UTC)