text:
Talk:Surreal number

Norman Larrabee Alling, his WP entry
I thought it would be a good idea to link to his WP page from the introduction, but it turns out it doesn't exist. I woiuld think he should be notable enough... There's a biography online at http://www-history.mcs.st-andrews.ac.uk/Biographies/Alling.html and one of his colleagues published his Collected Papers in 1998: P Ribenboim (ed.), Collected papers of Norman Alling (Queen's University, Kingston, ON, 1998) -- MathSciNet review 19 13 629 912 according to an online reference -- which even has a "Vita Norman Alling" starting on page vii -- but there is no online version or structured reference that I can find, so there's about 4 lines of WP-suitable material to be gained from the online biography. It's a bit thin, all in all -- can anyone help? One last google gave me a hit on Philip Ehrlich's work page: Collected Papers of Norman Alling, edited by Paulo Ribenboim, Queen’s Papers in Pure and Applied Mathematics, Volume 107, 1998, Kingston, Ontario, Canada. Should be good enough, though an ISBN would be nice, and I'd like a peek at that Vita. — Preceding unsigned comment added by Lhmathies (talk • contribs) 23:01, 18 February 2018 (UTC)
 * Now I'm confused, the MacTutor page above says NLA "was" a mathematician which would imply he was no longer with us, but I cannot find a death date anywhere, not even on the reference at https://mathematics.library.cornell.edu/works/A which seems up to date in this regard, judging by other entries. Since the rules for biographies of living persons are stricter, it would be nice to have this confirmed... Lhmathies (talk) 04:23, 23 February 2018 (UTC)

Controversy
There should be a section here, to preserve neutrality, noting that surreal numbers are non-existent, in the finitist form of mathematics.

 * That doesn't make a lot of sense. The fraction of math articles that could have such a disclaimer is huge.  Nothing special about the surreals there.  (I wouldn't call it a "controversy" either.  Some formulations of mathematics just don't include it.)  Sniffnoy (talk) 18:53, 10 May 2018 (UTC)

 * It is clear that the surreals are intended to be developped in an infinitary set theory so there is no need to specify that. Note that although they form an infinite class, they can still be defined in finitary mathematics.Vincent Bagayoko (talk) 11:13, 13 May 2018 (UTC)

Definition of y^R and y^L
It seems that there is no definition of the expressions y^R and y^L present in the equation for division
 * My reading is that they're the same as y_R and y_L in the previous paragraph. But I'm not confident enough to make the change.  Help?DavidHobby (talk) 02:45, 7 May 2019 (UTC)
 * True, this is inconsistent notation and I changed it.Lhmathies (talk) 13:40, 8 May 2019 (UTC)

When is the birthday number / older-than relation needed?
This is mostly idle musings, a.k.a. original research -- I would have put them under this previous Talk page entry, but it has been archived.

I think the answer to that question is that you can construct the ordered class of surreal numbers without using the older-than relation, but in the process you will necessarily define it.

On the other hand, to get from forms to numbers and thus to prove existence and consistency of arithmetic operations and define a field structure, the existence of the relation is crucial.

Sketch: Given a set S of totally ordered sets, totally ordered by inclusion (order homomorphies), you can create a 'completion' by taking the union U and adding special forms { L -- there is an obvious extension of the ordering and you can extend S with the completion to get a new totally ordered set of totally ordered sets. Now start with the empty set of sets and apply transfinite induction to get the surreals as the union of all the sets in the union of the sets of sets (where the sets of numbers are the same as the generations in the Conway construction). The step of taking the union is only strictly necessary for the steps corresponding to limit ordinals, including 0; in other cases S has a maximal element.

Note that the induction step only needs a total order on the set of sets. The well-ordering is a result of using (transfinite) induction. And instead of the 'special forms' you can just take the set of upwards (or downwards) closed sets -- each 'new' number corresponds to exactly one closed set, there are no equivalence classes involved and no double or parallel induction happening. Lhmathies (talk) 13:24, 10 January 2020 (UTC)
 * Actually this came out a bit misleading -- you do need a total order relation on S to make the induction step work and that is in effect an older-than relation, but the order type of S is not important so you might not be able to express it as a number. — Preceding unsigned comment added by Lhmathies (talk • contribs) 14:29, 13 January 2020 (UTC)

Mistake in article image
Where the value of a third is shown in the diagram is in fact where two thirds should be. The tree branches right, left, right, left, r, l, r, l etc. for 2/3rds and right, left, left, right, left, right left, r, l, r, l etc. for 1/3rd. --82.30.170.253 (talk) 02:51, 20 April 2020 (UTC)

 * Yup, that's an error alright. I'd suggest commenting on the page for the image, or perhaps directly messaging the person who made it.  (You could also try editing it yourself, obviously, but that may be difficult; I tried it myself and didn't get very far...) Sniffnoy (talk) 17:13, 20 April 2020 (UTC)

I agree. It is an error. You beat me to it. I tried to comment on the page for the image, but could not find it. The answer is 2/3 as per a video from John Conway himself. — Preceding unsigned comment added by 2001:16B8:170A:4700:E942:4EC6:B9E3:1C31 (talk) 21:35, 1 June 2020 (UTC)

Does Conway need to be defended?
One of the most recent edits has "remove conflicting naming claims" as the edit summary -- but the old text did say that the name was first used by Conway. It also said that equivalent treatments had been developed earlier, and that is gone too... so what was the real reason to change it? (The previous formulation was mine from a few years ago, full disclosure).

It looks like Conway has to be the first to ever come up with anything that looks like surreal numbers -- confer the emphasis in "History of the concept" on Alling's prior treatment not being as useful, which may be true but possibly not worth so many words. Is this a cult? Lhmathies (talk) 08:59, 1 June 2020 (UTC)

Error in epsilon definition?
Am I misunderstanding or should
 * { 0 | 1, $1⁄2$, $1⁄4$, $1⁄8$, ... } = ε

actually be
 * { 1, $1⁄2$, $1⁄4$, $1⁄8$, ... | 0 } = ε

? In the former version, as it currently is in the article, 0 | 1 would be 1/2 and the subsequent infinite series of fractions seems misplaced. Kufat (talk) 02:03, 13 August 2022 (UTC)

 * Order does not matter in sets. The order used in the article is more convenient in that it describes an infinite sequence of simpler numbers that are greater than ε. The latter is not a surreal number because 0 is not less than 1. –LaundryPizza03 ( d c̄ ) 03:16, 13 August 2022 (UTC)

Exponentiation
Should the article contain a paragraph on exponentiation of surreal numbers, and perhaps even a hint of tetration, pentation, and other hyperoperations? — Q uantling (talk | contribs) 15:45, 7 November 2022 (UTC)

 * Did you see Surreal number (a subsection under the treatment of the Surreal number)? This is a general definition for any two surreal numbers -- but it seems to me that hyperoperations are only defined for integer arguments (because you need to count down to zero) and I'm at a loss as to any possible generalizations. (Of course you could allow infinite numbers as the first argument, but I'm not sure if anything more interesting happens than with integers).
 * Exponentiation with an exponent that is a dyadic fraction (of which the integers are a subset) doesn't even need the log and exp functions, but that is not treated separately.
 * Lhmathies (talk) 13:41, 21 February 2023 (UTC)

Surcomplex numbers
One passage of the article reads as follows:

"A surcomplex number is a number of the form a+bi, where a and b are surreal numbers and i is the square root of −1. The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements."

I find a great similarity between a) "(except for being a proper class)" and b) "Other than that, Mrs. Lincoln, what did you think of the play?".

My point is that there is a tremendous, indescribably large distinction between

 * a) an algebraically closed field that is a set

and

 * b) one that is not a set, that cannot be a set.

User:2601:200:c082:2ea0:48d7:7772:5ebd:f634 00:04, 29 March 2023

 * Sure. But to my naive perception, this wanders off into model theory and requires posing questions like "is there a model of the surreals in the Grothendieck universe or maybe in NBG?" or maybe this is a question about "how does the Yoneda lemma work out in this case?" or maybe the question is "how does the transfer principle work out in this case?" -- and I have no clue about this. But since the transfer principle is mostly about transferring over statements written in first-order logic, maybe something goes wonky higher logics, ascending up the sigma-pi hierarchy? Who knows? On the other hand, it is my understanding that there is one and only one model of the complex numbers. Since the paragraph closes with the statement: Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.[6]: Th.27 and so that feels very cut-n-dried and unambiguous. "Any fixed set theory" would seem to include ZFC and New Foundations and etc. I tried to rephrase your statement as a menu of plausible questions. Rephrasing statements as questions allows them to get answered. I am not sure what you might have been thinking of. 67.198.37.16 (talk) 20:51, 30 April 2023 (UTC)

"Esoteric Pun" Parenthetical.
In "Gaps and Continuity" there's a part describing gap "On" and its relation to "No". Then, in parenthesis it is described as an "Esoteric Pun", followed by a colon (indicating a link between ideas), followed by some more text.

I have a few issues with this section that I will do my best to loosely describe, and I'm hoping that either my sentiments will be reflected and something will be done, or that justification will be given and I will understand the choices that have been made in the writing of the section. In no particular order:

 * 1) The word "Esoteric" links to "Western Esotericism". Forgive my ignorance, but I see no way in which that is actually relevant here. I believe it is a different use of the word "Esoteric", possibly an informal (or even wrong) one.
 * 2) Even if it is somehow relevant, the portion after the colon following "Esoteric Pun" fails to explain how it is related to esotericism at all, or just how it is "esoteric".
 * 3) It also fails to elaborate on the pun. There is the obvious observation that "No" and "On" are English inverses of one another, but it isn't mentioned or clued into at all, in favor of simply elaborating on the mathematical relationship between the two. If this does actually serve to explain the pun a bit more, I'm just not seeing how.
 * 4) Even without the issues of the first sentence, I'm not sure what function the rest serves as a parenthetical. It's fairly long and, at least to me, just feels poorly written. I think that if there's something important that needs saying in there, it should be given it's own paragraph, rather than being stuffed into parentheses.

[Conclusive paragraph which sums up my thoughts] TheJonyMyster (talk) 03:09, 19 May 2023 (UTC)

 * Fixed it. Esoteric is one of the types of mathematical joke as explained in the lead of that article. So the first link was just a red herring and I've removed it. Though of course since the book includes its own creation myth, it's possible that there is some esoteric content, should a third-party source report on it... Skyerise (talk) 12:38, 19 May 2023 (UTC)

sup and inf
Do the surreals have the least-upper-bound property? That is, does every non-empty set of surreals with an upper bound have a least upper bound ($sup$) in the surreals? Whether yes or no, mention of this in the article would be nice. I assume the answer would be the same for a greatest-lower-bound property ($inf$), and mention of that too would be nice. Thank you — Q uantling (talk | contribs) 13:13, 7 September 2023 (UTC)

 * This question is answered in the negative in the section "Gaps and continuity". –LaundryPizza03 ( d c̄ ) 13:17, 16 September 2023 (UTC)

Dead Link
This link in the section Further reading is dead: http://www-cs-faculty.stanford.edu/~knuth/sn.html Hubert1965 (talk) 14:39, 3 November 2023 (UTC)

 * Thanks, I've fixed it. Skyerise (talk) 14:43, 3 November 2023 (UTC)

magnitudes and absolutes

 * ... infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number.

The word positive was removed, and restored with "This is necessary; an infinitisimal is not smaller than -1." But it is smaller in absolute value, unless I missed a memo.

It is not smaller than the real number zero, though. —Tamfang (talk) 05:23, 3 May 2024 (UTC)
 * I think the sentence should be reworded. "X is smaller in absolute value than Y" is unclear and could be interpreted as "the absolute value of X is smaller than Y", which is how I initially read it. Lumping infinites and infinitesimals together in one phrase also makes the sentence a little awkward. Perhaps ... but also infinite numbers, whose absolute value is larger than any real number, and infinitesimal numbers, whose absolute value is larger than zero but smaller than any positive real number. It's a little wordy but is less ambiguous than the current sentence. CodeTalker (talk) 07:06, 3 May 2024 (UTC)

Games & Applications to combinatorial game theory—merge sections?
It seems like these sections repeat some information and are in general about a similar topic. Should they be merged? Theanswertolifetheuniverseandeverything (talk) 12:44, 17 June 2024 (UTC)

Replacing an ambiguous conclusion to a paragraph
A paragraph in Surreal number closes with "These labels will also be justified by the rules for surreal addition and multiplication below". I am concerned about an ambiguity: When I reached the aforementioned "below", I expected to see a definition of addition that establishes that the labels were correct (i.e., a derivation or proof.) Instead, I found a circular definition of addition that requires a a priori knowledge of the labels. On my first few readings, I thought that I had skipped over (or forgotten) a step. I am thinking of a replacement conclusion to the paragraph with something like "These labels will used to define addition and multiplication below in a way that permits any given surreal number to be described by a bewildering array number of different forms ( such as $$\{2\tfrac 1 2|4\tfrac 1 2\}=\{2|\} = 3 \ne 3\tfrac 1 2.)\,$$" While this sentence clarifies the true nature of the temporary assignment of labels, it is a heavy sentence. Ordinarily I would place some or all of it in a footnote. But Wikipedia is not too keen on footnotes. For that reason I decided not to "Be Bold" on this edit. Guy vandegrift (talk) 16:32, 18 June 2024 (UTC)
 * It would suffice to emend that to "by many different forms". -- Elphion (talk) 03:51, 19 June 2024 (UTC)

title: Talk:Surreal number


text:
Sedenion

In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital letter S, boldface $S$ or blackboard bold $$\mathbb S$$. They are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to a subalgebra of the sedenions. Unlike the octonions, the sedenions are not an alternative algebra. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, sometimes called the 32-ions or trigintaduonions. It is possible to continue applying the Cayley–Dickson construction arbitrarily many times.

The term sedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the biquaternions, or the algebra of 4 × 4 matrices over the real numbers, or that studied by.

Arithmetic
Like octonions, multiplication of sedenions is neither commutative nor associative. But in contrast to the octonions, the sedenions do not even have the property of being alternative. They do, however, have the property of power associativity, which can be stated as that, for any element x of $$\mathbb{S}$$, the power $$x^n$$ is well defined. They are also flexible.

Every sedenion is a linear combination of the unit sedenions $$e_0$$, $$e_1$$, $$e_2$$, $$e_3$$, ..., $$e_{15}$$, which form a basis of the vector space of sedenions. Every sedenion can be represented in the form

 * $$x = x_0 e_0 + x_1 e_1 + x_2 e_2 + \cdots + x_{14} e_{14} + x_{15} e_{15}.$$

Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is distributive over addition.

Like other algebras based on the Cayley–Dickson construction, the sedenions contain the algebra they were constructed from. So, they contain the octonions (generated by $$e_0$$ to $$e_7$$ in the table below), and therefore also the quaternions (generated by $$e_0$$ to $$e_3$$), complex numbers (generated by $$e_0$$ and $$e_1$$) and real numbers (generated by $$e_0$$).

The sedenions have a multiplicative identity element $$e_0$$ and multiplicative inverses, but they are not a division algebra because they have zero divisors. This means that two nonzero sedenions can be multiplied to obtain zero: an example is $$(e_3 + e_{10})(e_6 - e_{15})$$. All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.

A sedenion multiplication table is shown below:

Sedenion properties
From the above table, we can see that:

 * $$e_0e_i = e_ie_0 = e_i \, \text{for all} \, i,$$

 * $$e_ie_i = -e_0 \,\, \text{for}\,\, i \neq 0,$$ and

 * $$e_ie_j = -e_je_i \,\, \text{for}\,\, i \neq j \,\,\text{with}\,\, i,j \neq 0.$$

Anti-associative
The sedenions are not fully anti-associative. Choose any four generators, $$i,j,k$$ and $$l$$. The following 5-cycle shows that these five relations cannot all be anti-associative.

$$(ij)(kl) = -((ij)k)l = (i(jk))l = -i((jk)l) = i(j(kl)) = -(ij)(kl) = 0$$

In particular, in the table above, using $$e_1,e_2,e_4$$ and $$e_8$$ the last expression associates. $$(e_1e_2)e_{12} = e_1(e_2e_{12}) = -e_{15}$$

Quaternionic subalgebras
The 35 triads that make up this specific sedenion multiplication table with the 7 triads of the octonions used in creating the sedenion through the Cayley–Dickson construction shown in bold:

The binary representations of the indices of these triples bitwise XOR to 0.

{ {1, 2, 3}, {1, 4, 5}, {1, 7, 6}, {1, 8, 9}, {1, 11, 10}, {1, 13, 12}, {1, 14, 15},

{2, 4, 6}, {2, 5, 7}, {2, 8, 10}, {2, 9, 11}, {2, 14, 12}, {2, 15, 13}, {3, 4, 7},

{3, 6, 5}, {3, 8, 11}, {3, 10, 9}, {3, 13, 14}, {3, 15, 12}, {4, 8, 12}, {4, 9, 13},

{4, 10, 14}, {4, 11, 15}, {5, 8, 13}, {5, 10, 15}, {5, 12, 9}, {5, 14, 11}, {6, 8, 14},

{6, 11, 13}, {6, 12, 10}, {6, 15, 9}, {7, 8, 15}, {7, 9, 14}, {7, 12, 11}, {7, 13, 10} }

Zero divisors
The list of 84 sets of zero divisors $$\{e_a, e_b, e_c, e_d\}$$, where $$(e_a + e_b) \circ (e_c + e_d) = 0$$:

$$\begin{array}{c} \text{Sedenion Zero Divisors} \quad \{ e_a, e_b, e_c, e_d \} \\ \text{where} ~ (e_a + e_b) \circ (e_c + e_d) = 0 \\ \begin{array}{ccc} 1 \leq a \leq 6, & c > a, & 9 \leq b \leq 15 \\ 9 \leq c \leq 15 &       & -9 \geq d \geq -15 \end{array} \\ \begin{array}{ll} \{e_1, e_{10}, e_5, e_{14}\} & \{e_1, e_{10}, e_4, -e_{15}\} \\ \{e_1, e_{10}, e_7, e_{12}\} & \{e_1, e_{10}, e_6, -e_{13}\} \\ \{e_1, e_{11}, e_4, e_{14}\} & \{e_1, e_{11}, e_6, -e_{12}\} \\ \{e_1, e_{11}, e_5, e_{15}\} & \{e_1, e_{11}, e_7, -e_{13}\} \\ \{e_1, e_{12}, e_2, e_{15}\} & \{e_1, e_{12}, e_3, -e_{14}\} \\ \{e_1, e_{12}, e_6, e_{11}\} & \{e_1, e_{12}, e_7, -e_{10}\} \\ \{e_1, e_{13}, e_6, e_{10}\} & \{e_1, e_{13}, e_2, -e_{14}\} \\ \{e_1, e_{13}, e_7, e_{11}\} & \{e_1, e_{13}, e_3, -e_{15}\} \\ \{e_1, e_{14}, e_2, e_{13}\} & \{e_1, e_{14}, e_4, -e_{11}\} \\ \{e_1, e_{14}, e_3, e_{12}\} & \{e_1, e_{14}, e_5, -e_{10}\} \\ \{e_1, e_{15}, e_3, e_{13}\} & \{e_1, e_{15}, e_2, -e_{12}\} \\ \{e_1, e_{15}, e_4, e_{10}\} & \{e_1, e_{15}, e_5, -e_{11}\} \\

\{e_2, e_9,   e_4, e_{15}\} & \{e_2, e_9,    e_5, -e_{14}\} \\ \{e_2, e_9,   e_6, e_{13}\} & \{e_2, e_9,    e_7, -e_{12}\} \\ \{e_2, e_{11}, e_5, e_{12}\} & \{e_2, e_{11}, e_4, -e_{13}\} \\ \{e_2, e_{11}, e_6, e_{15}\} & \{e_2, e_{11}, e_7, -e_{14}\} \\ \{e_2, e_{12}, e_3, e_{13}\} & \{e_2, e_{12}, e_5, -e_{11}\} \\ \{e_2, e_{12}, e_7, e_9  \} & \{e_2, e_{13}, e_3, -e_{12}\} \\ \{e_2, e_{13}, e_4, e_{11}\} & \{e_2, e_{13}, e_6, -e_9  \} \\ \{e_2, e_{14}, e_5, e_9  \} & \{e_2, e_{14}, e_3, -e_{15}\} \\ \{e_2, e_{14}, e_3, e_{14}\} & \{e_2, e_{15}, e_4, -e_9  \} \\ \{e_2, e_{15}, e_3, e_{14}\} & \{e_2, e_{15}, e_6, -e_{11}\} \\

\{e_3, e_9,   e_6, e_{12}\} & \{e_3, e_9,    e_4, -e_{14}\} \\ \{e_3, e_9,   e_7, e_{13}\} & \{e_3, e_9,    e_5, -e_{15}\} \\ \{e_3, e_{10}, e_4, e_{13}\} & \{e_3, e_{10}, e_5, -e_{12}\} \\ \{e_3, e_{10}, e_7, e_{14}\} & \{e_3, e_{10}, e_6, -e_{15}\} \\ \{e_3, e_{12}, e_5, e_{10}\} & \{e_3, e_{12}, e_6, -e_9  \} \\ \{e_3, e_{14}, e_4, e_9  \} & \{e_3, e_{13}, e_4, -e_{10}\} \\ \{e_3, e_{15}, e_5, e_9  \} & \{e_3, e_{13}, e_7, -e_9   \} \\ \{e_3, e_{15}, e_6, e_{10}\} & \{e_3, e_{14}, e_7, -e_{10}\} \\

\{e_4, e_9,   e_7, e_{10}\} & \{e_4, e_9,    e_6, -e_{11}\} \\ \{e_4, e_{10}, e_5, e_{11}\} & \{e_4, e_{10}, e_7, -e_9  \} \\ \{e_4, e_{11}, e_6, e_9  \} & \{e_4, e_{11}, e_5, -e_{10}\} \\ \{e_4, e_{13}, e_6, e_{15}\} & \{e_4, e_{13}, e_7, -e_{14}\} \\ \{e_4, e_{14}, e_7, e_{13}\} & \{e_4, e_{14}, e_5, -e_{15}\} \\ \{e_4, e_{15}, e_5, e_{14}\} & \{e_4, e_{15}, e_6, -e_{13}\} \\

\{e_5, e_{10}, e_6, e_9  \} & \{e_5, e_9,    e_6, -e_{10}\} \\ \{e_5, e_{11}, e_7, e_9  \} & \{e_5, e_9,    e_7, -e_{11}\} \\ \{e_5, e_{12}, e_7, e_{14}\} & \{e_5, e_{12}, e_6, -e_{15}\} \\ \{e_5, e_{15}, e_6, e_{12}\} & \{e_5, e_{14}, e_7, -e_{12}\} \\

\{e_6, e_{11}, e_7, e_{10}\} & \{e_6, e_{10}, e_7, -e_{11}\} \\ \{e_6, e_{13}, e_7, e_{12}\} & \{e_6, e_{12}, e_7, -e_{13}\} \end{array} \end{array}$$

Applications
showed that the space of pairs of norm-one sedenions that multiply to zero is homeomorphic to the compact form of the exceptional Lie group G2. (Note that in his paper, a "zero divisor" means a pair of elements that multiply to zero.)

demonstrated that the three generations of leptons and quarks that are associated with unbroken gauge symmetry $$\mathrm {SU(3)_{c} \times U(1)_{em}}$$ can be represented using the algebra of the complexified sedenions $$\mathbb {C \otimes S}$$. Their reasoning follows that a primitive idempotent projector $$\rho_{+} = 1/2(1+ie_{15})$$ — where $$e_{15}$$ is chosen as an imaginary unit akin to $$e_{7}$$ for $$\mathbb {O}$$ in the Fano plane — that acts on the standard basis of the sedenions uniquely divides the algebra into three sets of split basis elements for $$\mathbb {C \otimes O}$$, whose adjoint left actions on themselves generate three copies of the Clifford algebra $$\mathrm Cl(6)$$ which in-turn contain minimal left ideals that describe a single generation of fermions with unbroken $$\mathrm {SU(3)_{c} \times U(1)_{em}}$$ gauge symmetry. In particular, they note that tensor products between normed division algebras generate zero divisors akin to those inside $$\mathbb {S}$$, where for $$\mathbb {C \otimes O}$$ the lack of alternativity and associativity does not affect the construction of minimal left ideals since their underlying split basis requires only two basis elements to be multiplied together, in-which associativity or alternativity are uninvolved. Still, these ideals constructed from an adjoint algebra of left actions of the algebra on itself remain associative, alternative, and isomorphic to a Clifford algebra. Altogether, this permits three copies of $$(\mathbb {C \otimes O})_{L} \cong \mathrm {Cl(6)}$$ to exist inside $$\mathbb {(C \otimes S)}_{L}$$. Furthermore, these three complexified octonion subalgebras are not independent; they share a common $$\mathrm Cl(2)$$ subalgebra, which the authors note could form a theoretical basis for CKM and PMNS matrices that, respectively, describe quark mixing and neutrino oscillations.

Sedenion neural networks provide a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Sedenion


text:
Talk:Sedenion

Multiplicative inverses and zero divisors
I wonder about the following feature of the sedenions as claimed in the article: They have multiplicative inverses and at the same time zero divisors.

In a matrix algebra, these two features cannot coexist, since a divisor of zero necessarily has determinant zero and thus it is not invertible.

I"d like to see an example of two sedenions with inverses and a product of zero. --J"org Knappen
 * What the article says is:"The sedenions have a multiplicative identity element 1 and multiplicative inverses, but they are not a division algebra. This is because they have zero divisors."Take a look at zero divisors which demonstrates the principle with matrices. A given sedenion might be a zero divisor when paired with one particular other, but nothing stops either having an inverse as far as I can see. HTH --Phil | Talk 11:56, Jan 11, 2005 (UTC)

 * Seen that -- nice demostration of what is possible in infinitely many dimensions. My argument above is for finite matrices (what"s the determinant of an infinite matrix, anyway?) I just like to see an example for sedenions, too --J"org Knappen

 * J"org - Matrices form an associative algebra. Sedenions are non-associative. You are not able to map a sedenion algebra on matrices (as opposed to e.g. octonions or quaternions). For examples of calculations with sedenions see e.g. the two references to articles from K. Carmody. Please note, however, that the sedenions in there are of a different type as the ones displayed in the multiplication table (I'll try to have this corrected). --65.185.222.50 13:19, 12 September 2005 (UTC)

 * Within the current article it says  $$(e_3+e_{10})(e_6-e_{15}) = 0$$, and the inverse of any $$s$$ can be calculated by $$s^*/|s|^2$$, so $$(e_3+e_{10})^{-1} = -1/2 (e_3+e_{10}),  (e_6-e_{15})^{-1}=-1/2 (e_6-e_{15})$$. Generally, the inverses of (nontrivial) zero divisors are again zero divisors. Summsumm2 (talk) 15:19, 5 July 2019 (UTC)

Corrigendum

The multiplication table given in the article applies to sedenions of the type discussed by Imaeda/Imaeda, however, the sedenions discussed by Carmody are of a different type, first proposed by C. Musès. The following reference should be added to establish the correct originatorship of the latter sedenion type: C. Musès, Appl. Math. and Computation, 4, pp 45-66 (1978) --65.185.222.50 13:19, 12 September 2005 (UTC)

History
when were they discovered? are there larger algebras with the same properties? 83.79.181.211 19:05, 29 September 2005 (UTC)

 * About the term "sedenion" and the discovery; as far as I know, the cited articles are the earliest publications I could find that go deeper into arithmetic laws of the sedenions, and thereby solidify the term (so it's quite recent). Surely, as part of the Cayley-Dickson construction, their existence was discovered earlier within that program. Any earlier known uses of the term "sedenion"? Thanks, Jens Koeplinger 12:40, 15 September 2006 (UTC)

There are larger algebras with the same properties, in fact an infinite number of them. One can perform the Cayley-Dickson construction on sedenions to get a 32-element algebra, and again to get a 64-element algebra - in fact, one can get an algebra of 2n for any non-negative n. n=0 gives the reals, n=1 gives complex numbers, n=2 quaternions, n=3 octonions, n=4 sedenions and so forth. --Frank Lofaro Jr. 22:33, 2 March 2006 (UTC)

https://hsm.stackexchange.com/questions/11411/who-pioneered-the-study-of-the-sedenions

Who pioneered the study of the sedenions?

On Bibliography of Quaternions and Allied Mathematics by Alexander Macfarlane I found this:

On page 72; James Byrnie Shaw 1896 Sedenions (title). American Assoc. Proc., 45, 26.

I couldn't find this reference, but the same author wrote this book:

Synopsis of Linear Associative Algebra: A Report on its Natural Development and Results Reached up to the Present Time. 1907.

From its Table of Contents; Part II: Particular Algebras. Section XVIII: Triquaternions and Quadriquaternions. Page 91.

Early on Sedenions were also known as "quadriquaternions".

Section XIX: Sylvester Algebras. Page 93. Covers "Nonions" (9-ions), and "Sedenions" (16-ions). Here the Sedenions are attributed to James Joseph Sylvester.

On page 76; James Joseph Sylvester 1883-4 On quaternions, nonions, sedenions, etc. Johns Hopkins Univ. Circ., 3. Nos. 7 and 9. 4, No. 28.

This second reference can be found among The Collected Mathematical Papers of James Joseph Sylvester, [Volume IV(1882—1897)]:

Sylvester, James Joseph (1973) [1904], Baker, Henry Frederick (ed.), The collected mathematical papers of James Joseph Sylvester, IV, New York: AMS Chelsea Publishing, ISBN 978-0-8218-4238-6

Then, as far as I understood, James Joseph Sylvester appears on the literature as the proponent of two Algebras; the one from the Nonions and the Sedenions. I will leave as an open question for the community to confirm, correct, or debunk that J.J. Sylvester "discovered" or "was the pioneer" of their study in 1883. — Preceding unsigned comment added by 178.48.57.121 (talk) 04:10, 9 February 2020 (UTC)

Is Moreno's result incorrect?
A few days ago I reverted an unsourced edit to section Sedenion, but the issue that was brought up may have to be taken into account in some way, either by citing the added content or removing the current sentence in this section (see the discussion on my talk page for further details). Regards, Gap9551 (talk) 15:42, 20 April 2016 (UTC)
 * It is small mistake in Moreno work, not in the proof but in final statement. I saw this statement is repeated in other works. The correct statement should be: the set $$\{(a,b): a*b=0; a,b \in S^{15}\}$$ is homeomorphic to Lie group $$G_2$$. Thus it is not set of "norm 1 sedenions zero-divisors". It is set of pairs of norm 1 sedenions which product is zero.
 * The proof is following. The zero divisors in sedenion sphere is $$a+b\iota$$ where $$a, b$$ are perpendicular imaginary octonions and $$\iota$$ is extra element in Cayley-Dickson definition. Thus this is 11-dimensional subset of $$S^{15}$$ of shape $$S^6 \times \times S^5$$. The $$G_2$$ we obtain by considering unit octonion $$c$$ perpendicular to quaternion space generated by $$a, b$$. Then we have $$(a+b\iota)(c+a(bc)\iota)=0$$. Regards, Marek Mitros

— Preceding unsigned comment added by 193.41.170.245 (talk) 08:33, 21 April 2016 (UTC)

 * The correction by Marek Mitros is indeed right; I looked at Moreno's paper and verified this. So, I've edited the article to make this correction.  John Baez (talk) 03:48, 28 July 2016 (UTC)

Double simetry in Sedenion
For a double simetry in a sedenion, resulting in a Quaternion, resultant wave function in the Quantum mechanics, is a matrix that results in the Elementary particle of the standard model (without Higgs).

Please, add the simplification.

--188.171.58.12 (talk) 22:17, 26 December 2019 (UTC)

Multiplication Table
Could the multiplication table be fixed such that it shows -e correctly for all of the elements? In my browser (Chrome) it shows the negative sign above the e for e values with two-digit subscripts, and it took me a moment to figure out what this was. TricksterWolf (talk) 18:55, 29 February 2020 (UTC)

Incorrect number of primitive zero divisors
There are actually 168 primitive zero divisors. The Quaternion triplets mentioned are those created by the Cayley-Dickson doubling process with sequential new basis elements but is not unique, others exist. Please consult my paper: https://vixra.org/abs/2010.0086?ref=11771562 where all 168 primitive zero divisors for the standard Sedenion form mentioned are itemized, the 15 Octonion subalgebra candidates are itemized by Sedenion basis element subsets, an algebraic proof all 15 cannot be oriented as proper Octonion algebras, a method to determine if a set of seven oriented Quaternion triplets otherwise suitable for an Octonion Algebra actually describes one, a method to find all sets of 5 Octonion subalgebra candidates that cannot all be properly oriented and how this shows the maximal number of Octonion Algebra candidates that can be oriented as proper Octonion is eight; any seven sharing a common basis element plus any one of the other eight.

The definition variation for all Cayley-Dickson algebras through Sedenions is discussed in detail in my paper describing all valid Quaternion triplet orientations. A doubling process by variation is provided. There is much information provided that could improve the wiki for Octonion Algebra.

EightOnions (talk) 23:43, 14 January 2021 (UTC)

Flexible alternativity
In it is demonstrated that all Cayley - Dickson algebras are flexible alternative. I feel that only saying 'sedenions are not an alternative algebra', thought it is true, does not describe the essence of sedenions. — Preceding unsigned comment added by Crodrigue1 (talk • contribs) 01:44, 30 January 2021 (UTC)

Zero Divisor correction
I think there are 2 mistakes in the Sedenion Zero divisor list:

"(e1+e13)×(e7+e-14) = 0" is False, but "(e1+e13)×(e2+e-14) == 0" is True. "(e6+e10)×(e7+e-13) = 0" is False, but "(e6+e12)×(e7+e-13) == 0" is True.

Correction would then be:

{e1,e13,e7,e-14} -> changed to -> {e1,e13,e2,e-14} {e6,e10,e7,e-13} -> changed to -> {e6,e12,e7,e-13}

I am not math certified. I wrote a program to locate zero divisors and found these 2 differences. Maybe someone could verify. Peawormsworth (talk) 19:36, 6 August 2022 (UTC)

Names of further algebras
Names have been given to various further algebras, but this appears inconsistent. The 32 unit algebra is referred to in one section as the "pathions" but elsewhere as the "trigintaduonions" (this name presumably courtesy of Raoul E. Cawagas). Both of these are inconsistent with the established naming scheme given by the names of the quaternions, octonions and sedenions. The name of the sedenions shows that the 32 unit algebra following it should be the duotriginions, as the tens prefix (den-) is preceeded by the ones prefix (se-), and since the "ta" of "triginta" would be omitted to fit the established nomenclaturalogical pattern ("...-nions"). This would give:

Real numbers (1)

Complex numbers (2)

Quaternions (4)

Octonions (8)

Sedenions (16)

Duotriginions (32)

Quatersexaginions (64) *

Cenoctoviginions (128)

Duocensequinquaginions (256) *

and as the article says, ad infinitum.

*: The established prefixes "se" and "quater" might be considered only due to ease of pronunciation, and "sex" and "quattuor" (the correct latin prefixes) may in fact be more appropriate. HYPERIAPATH (talk) 22:03, 25 May 2024 (UTC)

Fixed Zero Divisors
I changed...

$$\{e_1,e_{13},e_7,-e_{14}\}$$

$$\{e_6,e_{10},e_7,-e_{13}\}$$

to...

$$\{e_1,e_{13},e_2,-e_{14}\}>$$

$$\{e_6,e_{12},e_7,-e_{13}\}$$

Because this is False...

$$(e_1+e_{13})\times (e_7-e_{14}) = 0$$

$$(e_6+e_{10})\times (e_7-e_{13}) = 0$$

Proof...

$$(e_1+e_{13})\times (e_7-e_{14}) = (e_1\times e_7)+(e_{13}\times e_7)-(e_1\times e_{14})-(e_{13}\times e_{14})$$

$$= (-e_6)+(e_{10})-(e_{15})-(-e_3) = e_3-e_6+e_{10}-e_{15}$$

$$(e_6+e_{10})\times (e_7-e_{13}) = (e_6\times e_7)+(e_{10}\times e_7)-(e_6\times e_{13})-(e_{10}\times e_{13})$$

$$= (e_1)+(-e_{13})-(e_{11})-(e_7) = e_1-e_7-e_{11}-e_{13}$$

And this is True...

$$(e_1+e_{13})\times (e_2-e_{14}) = 0$$

$$(e_6+e_{12})\times (e_7-e_{13}) = 0$$

Proof...

$$(e_1+e_{13})\times (e_2-e_{14}) = (e_1\times e_2)+(e_{13}\times e_2)-(e_1\times e_{14})-(e_{13}\times e_{14})$$

$$= (-e_3)+(e_{15})-(e_{15})-(-e_3) = 0$$

$$(e_6+e_{12})\times (e7-e{13}) = (e_6\times e_7)+(e_{12}\times e_7)-(e_6\times e_{13})-(e_{12}\times e_{13})$$

$$= (e_1)+(e_{11})-(e_{11})-(e_1) = 0$$

I made the change live, because I believe it is true and the last comment was deprecated.

Citation: the multiplication table of this wiki page. Peawormsworth (talk) 11:27, 23 June 2024 (UTC)

title: Talk:Sedenion


text:
Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface $O$ or blackboard bold $$\mathbb O$$. Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.

Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and quantum logic. Applying the Cayley–Dickson construction to the octonions produces the sedenions.

History
The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Rowan Hamilton's discovery of quaternions. Graves called his discovery "octaves", and mentioned them in a letter to Hamilton dated 26 December 1843. He first published his result slightly later than Arthur Cayley's article. The octonions were discovered independently by Cayley and are sometimes referred to as "Cayley numbers" or the "Cayley algebra". Hamilton described the early history of Graves's discovery.

Definition
The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the unit octonions:

 * $$\{e_0, e_1, e_2, e_3, e_4, e_5, e_6, e_7\},$$

where $e_{0}$ is the scalar or real element; it may be identified with the real number 1. That is, every octonion $x$ can be written in the form
 * $$x = x_0e_0 + x_1e_1 + x_2e_2 + x_3e_3 + x_4e_4 + x_5e_5 + x_6e_6 + x_7e_7,\,$$

with real coefficients $x_{i}$.

Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the products of all the terms, again like quaternions. The product of each pair of terms can be given by multiplication of the coefficients and a multiplication table of the unit octonions, like this one (due to Cayley, 1845, and Graves, 1843):

Most off-diagonal elements of the table are antisymmetric, making it almost a skew-symmetric matrix except for the elements on the main diagonal, as well as the row and column for which $e_{0}$ is an operand.

The table can be summarized as follows:

e_i e_j = \begin{cases} e_j, & \text{if }i = 0 \\ e_i, & \text{if }j = 0 \\ - \delta_{ij}e_0 + \varepsilon _{ijk} e_k, & \text{otherwise} \end{cases} $$

where $δ_{ij}$ is the Kronecker delta (equal to 1 if and only if $i = j$), and $ε_{ijk}$ is a completely antisymmetric tensor with value 1 when $ijk = 123, 145, 176, 246, 257, 347, 365$.

The above definition is not unique, however; it is only one of 480 possible definitions for octonion multiplication with $e_{0} = 1$. The others can be obtained by permuting and changing the signs of the non-scalar basis elements ${e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7}} |undefined$. The 480 different algebras are isomorphic, and there is rarely a need to consider which particular multiplication rule is used.

Each of these 480 definitions is invariant up to signs under some 7-cycle of the points (1234567), and for each 7-cycle there are four definitions, differing by signs and reversal of order. A common choice is to use the definition invariant under the 7-cycle (1234567) with $e_{1}e_{2} = e_{4}$ by using the triangular multiplication diagram, or Fano plane below that also shows the sorted list of 124 based 7-cycle triads and its associated multiplication matrices in both $e_{n}$ and $$\mathrm{IJKL}$$ format.
 * Octonion-124-137-156-235-267-346-457.svg

A variation of this sometimes used is to label the elements of the basis by the elements $∞$, 0, 1, 2, ..., 6, of the projective line over the finite field of order 7. The multiplication is then given by $e_{∞} = 1$ and $e_{1}e_{2} = e_{4}$, and all expressions obtained from this by adding a constant (modulo 7) to all subscripts: in other words using the seven triples (124) (235) (346) (450) (561) (602) (013). These are the nonzero codewords of the quadratic residue code of length 7 over the Galois field of two elements, $GF(2)$. There is a symmetry of order 7 given by adding a constant mod 7 to all subscripts, and also a symmetry of order 3 given by multiplying all subscripts by one of the quadratic residues 1, 2, 4 mod 7. These seven triples can also be considered as the seven translates of the set {1,2,4} of non-zero squares forming a cyclic (7,3,1)-difference set in the finite field $GF(7)$ of seven elements.

The multiplication table for a geometric algebra of signature $(−−−−)$ can be given in terms of the following 7 quaternionic triples (omitting the identity element):

in which the lowercase items are vectors and the uppercase ones are bivectors and $(I, j, k), (i, J, k), (i, j, K), (I, J, K), (∗I, i, m), (∗J, j, m), (∗K, k, m)$ (which is the Hodge star operator). If the $∗ = mijk$ is forced to be equal to the identity then the multiplication ceases to be associative, but the $∗$ may be removed from the multiplication table resulting in an octonion multiplication table.

In keeping $∗$ associative and thus not reducing the 4-dimensional geometric algebra to an octonion one, the whole multiplication table can be derived from the equation for $∗ = mijk$. Consider the gamma matrices. The formula defining the fifth gamma matrix shows that it is the $∗$ of a four-dimensional geometric algebra of the gamma matrices.

Cayley–Dickson construction
A more systematic way of defining the octonions is via the Cayley–Dickson construction. Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions. Addition is defined pairwise. The product of two pairs of quaternions $∗$ and $(a, b)$ is defined by
 * $$(a,b)(c,d)=(ac-d^{*}b,da+bc^{*}),$$

where $(c, d)$ denotes the conjugate of the quaternion $z$. This definition is equivalent to the one given above when the eight unit octonions are identified with the pairs

Fano plane mnemonic
A convenient mnemonic for remembering the products of unit octonions is given by the diagram, which represents the multiplication table of Cayley and Graves. This diagram with seven points and seven lines (the circle through 1, 2, and 3 is considered a line) is called the Fano plane. The lines are directional. The seven points correspond to the seven standard basis elements of $z*$ (see definition below). Each pair of distinct points lies on a unique line and each line runs through exactly three points.

Let $(1, 0), (i, 0), (j, 0), (k, 0), (0, 1), (0, i), (0, j), (0, k)$ be an ordered triple of points lying on a given line with the order specified by the direction of the arrow. Then multiplication is given by

 * $e_{0}$ and $Im(O)$

together with cyclic permutations. These rules together with

 * 1 is the multiplicative identity,
 * $${e_i}^2=-1$$ for each point in the diagram

completely defines the multiplicative structure of the octonions. Each of the seven lines generates a subalgebra of $(a, b, c)$ isomorphic to the quaternions $ab = c$.

Conjugate, norm, and inverse
The conjugate of an octonion

 * $$x = x_0\,e_0 + x_1\,e_1 + x_2\,e_2 + x_3\,e_3 + x_4\,e_4 + x_5\,e_5 + x_6\,e_6 + x_7\,e_7$$

is given by

 * $$x^* = x_0\,e_0 - x_1\,e_1 - x_2\,e_2 - x_3\,e_3 - x_4\,e_4 - x_5\,e_5 - x_6\,e_6 - x_7\,e_7.$$

Conjugation is an involution of $ba = −c$ and satisfies $O$ (note the change in order).

The real part of $x$ is given by

 * $$\frac{x + x^*}{2} = x_0\,e_0$$

and the imaginary part by

 * $$\frac{x - x^*}{2} = x_1\,e_1 + x_2\,e_2 + x_3\,e_3 + x_4\,e_4 + x_5\,e_5 + x_6\,e_6 + x_7\,e_7.$$

The set of all purely imaginary octonions spans a 7-dimensional subspace of $H$, denoted $O$.

Conjugation of octonions satisfies the equation

 * $$-6 x^* = x+(e_1x)e_1+(e_2x)e_2+(e_3x)e_3+(e_4x)e_4+(e_5x)e_5+(e_6x)e_6+(e_7x)e_7.$$

The product of an octonion with its conjugate, $(xy)* = y*x*$, is always a nonnegative real number:

 * $$x^*x = x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2.$$

Using this, the norm of an octonion can be defined as

 * $$\|x\| = \sqrt{x^*x}.$$

This norm agrees with the standard 8-dimensional Euclidean norm on $O$.

The existence of a norm on $Im(O)$ implies the existence of inverses for every nonzero element of $x*x = xx*$. The inverse of $R^{8}$, which is the unique octonion $O$ satisfying $O$, is given by

 * $$x^{-1} = \frac {x^*}{\|x\|^2}.$$

Properties
Octonionic multiplication is neither commutative:

 * $x ≠ 0$ if $i$, $j$ are distinct and non-zero,

nor associative:

 * $x^{−1}$ if $i$, $j$, $k$ are distinct, non-zero and $xx^{−1} = x^{−1}x = 1$.

The octonions do satisfy a weaker form of associativity: they are alternative. This means that the subalgebra generated by any two elements is associative. Actually, one can show that the subalgebra generated by any two elements of $e_{i}e_{j} = −e_{j}e_{i} ≠ e_{j}e_{i}$ is isomorphic to $(e_{i}e_{j})e_{k} = −e_{i}(e_{j}e_{k}) ≠ e_{i}(e_{j}e_{k})$, $e_{i}e_{j} ≠ ±e_{k}$, or $O$, all of which are associative. Because of their non-associativity, octonions cannot be represented by a subalgebra of a matrix ring over $$\mathbb R$$, unlike the real numbers, complex numbers and quaternions.

The octonions do retain one important property shared by $R$, $C$, and $H$: the norm on $R$ satisfies

 * $$\|xy\| = \|x\|\|y\|.$$

This equation means that the octonions form a composition algebra. The higher-dimensional algebras defined by the Cayley–Dickson construction (starting with the sedenions) all fail to satisfy this property. They all have zero divisors.

Wider number systems exist which have a multiplicative modulus (for example, 16-dimensional conic sedenions). Their modulus is defined differently from their norm, and they also contain zero divisors.

As shown by Hurwitz, $C$, $H$, $O$, and $R$ are the only normed division algebras over the real numbers. These four algebras also form the only alternative, finite-dimensional division algebras over the real numbers (up to isomorphism).

Not being associative, the nonzero elements of $C$ do not form a group. They do, however, form a loop, specifically a Moufang loop.

Commutator and cross product
The commutator of two octonions $x$ and $y$ is given by

 * $$[x, y] = xy - yx.$$

This is antisymmetric and imaginary. If it is considered only as a product on the imaginary subspace $H$ it defines a product on that space, the seven-dimensional cross product, given by

 * $$x \times y = \tfrac{1}{2}(xy - yx).$$

Like the cross product in three dimensions this is a vector orthogonal to $x$ and $y$ with magnitude

 * $$\|x \times y\| = \|x\| \|y\| \sin \theta.$$

But like the octonion product it is not uniquely defined. Instead there are many different cross products, each one dependent on the choice of octonion product.

Automorphisms
An automorphism, $A$, of the octonions is an invertible linear transformation of $O$ which satisfies

 * $$A(xy) = A(x)A(y).$$

The set of all automorphisms of $O$ forms a group called $Im(O)$. The group $O$ is a simply connected, compact, real Lie group of dimension 14. This group is the smallest of the exceptional Lie groups and is isomorphic to the subgroup of $O$ that preserves any chosen particular vector in its 8-dimensional real spinor representation. The group $G_{2}$ is in turn a subgroup of the group of isotopies described below.

See also: $G_{2}$ – the automorphism group of the Fano plane.

Isotopies
An isotopy of an algebra is a triple of bijective linear maps $a$, $b$, $c$ such that if $Spin(7)$ then $Spin(7)$. For $PSL(2,7)$ this is the same as an automorphism. The isotopy group of an algebra is the group of all isotopies, which contains the group of automorphisms as a subgroup.

The isotopy group of the octonions is the group $xy = z$, with $a$, $b$, $c$ acting as the three 8-dimensional representations. The subgroup of elements where $c$ fixes the identity is the subgroup $a(x)b(y) = c(z)$, and the subgroup where $a$, $b$, $c$ all fix the identity is the automorphism group $a = b = c$.

Applications
The octonions play a significant role in the classification and construction of other mathematical entities. For example, the exceptional Lie group $Spin_{8}(R)$ is the automorphism group of the octonions, and the other exceptional Lie groups $Spin_{7}(R)$, $G_{2}$, $G_{2}$ and $F_{4}$ can be understood as the isometries of certain projective planes defined using the octonions. The set of self-adjoint 3 × 3 octonionic matrices, equipped with a symmetrized matrix product, defines the Albert algebra. In discrete mathematics, the octonions provide an elementary derivation of the Leech lattice, and thus they are closely related to the sporadic simple groups.

Applications of the octonions to physics have largely been conjectural. For example, in the 1970s, attempts were made to understand quarks by way of an octonionic Hilbert space. It is known that the octonions, and the fact that only four normed division algebras can exist, relates to the spacetime dimensions in which supersymmetric quantum field theories can be constructed. Also, attempts have been made to obtain the Standard Model of elementary particle physics from octonionic constructions, for example using the "Dixon algebra" $E_{6}$.

Octonions have also arisen in the study of black hole entropy, quantum information science,  string theory, and image processing.

Octonions have been used in solutions to the hand eye calibration problem in robotics.

Deep octonion networks provide a means of efficient and compact expression in machine learning applications.

Integral octonions
There are several natural ways to choose an integral form of the octonions. The simplest is just to take the octonions whose coordinates are integers. This gives a nonassociative algebra over the integers called the Gravesian octonions. However it is not a maximal order (in the sense of ring theory); there are exactly seven maximal orders containing it. These seven maximal orders are all equivalent under automorphisms. The phrase "integral octonions" usually refers to a fixed choice of one of these seven orders.

These maximal orders were constructed by, Dickson and Bruck as follows. Label the eight basis vectors by the points of the projective line over the field with seven elements. First form the "Kirmse integers" : these consist of octonions whose coordinates are integers or half integers, and that are half integers (that is, halves of odd integers) on one of the 16 sets

of the extended quadratic residue code of length 8 over the field of two elements, given by $E_{7}$, $E_{8}$ and its images under adding a constant modulo 7, and the complements of these eight sets. Then switch infinity and any one other coordinate; this operation creates a bijection of the Kirmse integers onto a different set, which is a maximal order. There are seven ways to do this, giving seven maximal orders, which are all equivalent under cyclic permutations of the seven coordinates 0123456. (Kirmse incorrectly claimed that the Kirmse integers also form a maximal order, so he thought there were eight maximal orders rather than seven, but as pointed out they are not closed under multiplication; this mistake occurs in several published papers.)

The Kirmse integers and the seven maximal orders are all isometric to the $C ⊗ H ⊗ O$ lattice rescaled by a factor of $1/\sqrt{2}$. In particular there are 240 elements of minimum nonzero norm 1 in each of these orders, forming a Moufang loop of order 240.

The integral octonions have a "division with remainder" property: given integral octonions $a$ and $∅ (∞124) (∞235) (∞346) (∞450) (∞561) (∞602) (∞013) (∞0123456) (0356) (1460) (2501) (3612) (4023) (5134) (6245)$, we can find $q$ and $r$ with $∅$, where the remainder $r$ has norm less than that of $b$.

In the integral octonions, all left ideals and right ideals are 2-sided ideals, and the only 2-sided ideals are the principal ideals $nO$ where $n$ is a non-negative integer.

The integral octonions have a version of factorization into primes, though it is not straightforward to state because the octonions are not associative so the product of octonions depends on the order in which one does the products. The irreducible integral octonions are exactly those of prime norm, and every integral octonion can be written as a product of irreducible octonions. More precisely an integral octonion of norm $mn$ can be written as a product of integral octonions of norms $m$ and $n$.

The automorphism group of the integral octonions is the group $(∞124)$ of order 12,096, which has a simple subgroup of index 2 isomorphic to the unitary group $E_{8}$. The isotopy group of the integral octonions is the perfect double cover of the group of rotations of the $b ≠ 0$ lattice.

title: Octonion


text:
Talk:Octonion

Anticommutativity
The lead says that they are noncommutative and nonassociative. They are actually anticommutative and antiassociative, right? Bubba73 You talkin' to me? 03:45, 16 November 2012 (UTC)
 * No: ℝ and ℂ are both subspaces which are commutative, while any two ei and ej for i ≠ j, i, j > 1 anticommute. So like quaternions they are neither commutative nor anticommutative. I don't even know what antiassociative means; there's not a simple negative form of associativity, as there is of commutativity.-- JohnBlackburne wordsdeeds 04:25, 16 November 2012 (UTC)

 * From the top of the properties section:
 * $$e_ie_j = -e_je_i \neq e_je_i\,$$
 * Isn't that what anticommutative means? Bubba73 You talkin' to me? 04:36, 16 November 2012 (UTC)
 * It is not true if i or j =0 ($$e_0e_i=e_ie_0$$) It corrected the affirmation in the article.--Cbigorgne (talk) 07:20, 16 November 2012 (UTC)

Thank you, that helps clear it up. Bubba73 You talkin' to me? 15:51, 16 November 2012 (UTC) Hello. I do not think this is resolved. Actually, it is true that $$e_ie_j=-e_je_i$$ if $$i,j\neq 0$$ and $$i\neq j$$. For the non-associative part, the conditions are even harder to write...2A01:E35:8A15:73A0:6134:310F:F279:7C8E (talk) 22:20, 21 March 2013 (UTC)

I assume "antiassociative" would mean (ab)c = −a(bc). This isn't always true for the octonions: it must be so since the octonions contain copies of the quaternions within them. But it will be true if {a,b} generate an algebra isomorphic to the quaternions that is orthogonal to c (essentially what the article says). Double sharp (talk) 17:00, 12 March 2016 (UTC)

Multiplication table
What do you guys think about slightly increasing the width of each column in the table? What about replacing $e _{5}$ with e 5 for the duration of the table?

Both of these changes would only be for aesthetic purposes. It would match the corresponding table for sedenions. BradBentz44 (talk) 06:33, 28 May 2014 (UTC)

 * I'd suggest replacing the var tags with italics quotes ( →  ), since the semantics are not of variables.  Controlling column width (and making them a little wider) in the table would be good. Ordinarily, I would not agree with removing the   template, but here the subscripts end up too large on my browser.  An alternative is removing the subscript from the font modification: $e$5 – does this achieve what you're after aesthetically? —Quondum 13:37, 28 May 2014 (UTC)

 * I do think that $e$5 looks better than $e _{5}$. There isn't much difference between $e$5 and e 5. Do you think we should go ahead with the change? BradBentz44 (talk) 20:30, 28 May 2014 (UTC)
 * I don't think it's needed. The formatting matches that of the inline math above and below; changing it would make it far less consistent and so visually jarring. And a subjective reason such as 'for aesthetics purposes' is not a good reason to change formatting from one style to another; an article should not be changed just for such reasons. See e.g. MOS:STABILITY.-- JohnBlackburne wordsdeeds 20:49, 28 May 2014 (UTC)
 * I see the redirect at the link MOS:STABILITY that you provided does not go anywhere but WP:MOS – any idea which section it should go to?
 * The formatting is not consistent: the inline math does not in general use math, whereas the table does. This is sufficient reason to change the article to a consistent format.  Perhaps your browser setting result in very similar rendering, but with mine the subscript size gets really messed up with one.  But I agree that this is not a reason for any choice.  So really, should we go with math throughout except for the standalone <math>? —Quondum
 * There's no heading but MOS:STABILITY goes to the third para of the lead: "Style and formatting..." with links to relevant arbitration cases. MOS:MATH has a similar guideline at MOS:MATH which refers back to the main manual of style.-- JohnBlackburne wordsdeeds 11:49, 29 May 2014 (UTC)
 * Thanks, I missed the anchor. Anyhow, I'm not trying to dispute the principle that style should not be changed without reason, I only thought it was a link that needed fixing. That still leaves us with my observation that this article mixes HTML with and without : the table uses it, but the only place outside the table where it is used is in the sentence immediately after the table. We should change the article to use and   uniformly, or remove the  usage. —Quondum 13:25, 29 May 2014 (UTC)

Invention of "associative"
I removed a remark: "Hamilton invented the word associative so that he could say that octonions were not associative."

In this was credited to Cayley, then  and moved before the citation, which does not involve the term "associative" at all. This makes me suspect it wasn't meant very seriously.

Graves wrote to Hamilton about his "octaves" on December 26, 1843 (according to Hamilton's account: but Hamilton was already using the term "associative character" to talk about quaternions in his paper presented November 13, 1843:

Hopefully this is sufficient explanation to remove the remark rather than sticking a citation request on it. Agashlin (talk) 12:16, 2 May 2017 (UTC)

 * Just wanted to add that I regret having written an uncharitable summary for that edit, I don't think there was malicious intent and I should have assumed good faith anyway. Agashlin (talk) 12:44, 2 May 2017 (UTC)
 * Since this ia a Talk page, references stay in sections. — Rgdboer (talk) 21:16, 11 October 2017 (UTC)

Matrix representation
first it was stated: Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions

and next time you wrote: Because of their non-associativity, octonions do not have matrix representations, unlike quaternions.

but if a quaternion can be represented by a matrix, and octonions can be defined as 4 quaternions then it could be represented uniquely by 4x4x4 (3D) matrix.. No? — Preceding unsigned comment added by Xakepp35 (talk • contribs)
 * You should look at the Cayley–Dickson construction. It is how each of the algebras is derived from the previous one, so how octonions is derived from quaternions. And because of how it works each time the new algebra loses certain properties. As you move from quaternions to octonions you lose associativity, so there is no matrix representation (as matrices are associative).-- JohnBlackburne wordsdeeds 06:28, 11 October 2017 (UTC)

 * I guess the question is "what do you mean by matrix representation?" Obviously it is not a subalgebra of a matrix ring because any such algebra is associative. But the elements of any finite dimensional algebra can be viewed as a linear transformation and therefore a square matrix.  This is, at least, the spirit of Jacobson's structural description of general finite dimensional algebras.  That being the case, I'm rewording to be more explicit about the assumption that was previously made: that a 'matrix representation' meant "subalglebra of a matrix ring". Rschwieb (talk) 15:01, 8 January 2021 (UTC)

Contemporary Octonion Research
Snark aside, I fail to see why this isn't serious contemporary research worthy of note on Wikipedia. Flesh it out for me.

Than again: https://www.math.columbia.edu/~woit/wordpress/?p=3665 Still not even wrong or no? kencf0618 (talk) 19:41, 28 July 2018 (UTC)

 * Sorry, my mastery of the English language may not be sufficient to reign any snark, apologies in advance.
 * Well, I am not scared away by pentagrams, not even with candles in their corners, so why should I shy away from a minimal projective plane. Associating selected objects from the physicists' particle zoo with distinguished elements of an algebra is free, at least for a while. Projections of predictive power of these associations are nevertheless missing. Promising is all there is that I am aware of, for the time being. I am, however, really scared away by the alleged multiple tensor(!) product of groups, or is it rather vector spaces, or division algebrae, being myself really unsure about a valid outcome.
 * The redactrice of the Quanta Magazine evidently knows a lot about puffery and qualification of her readership, too (see the comments section there!), but since I am neither female, nor playing an accordeon (not even a bandoneon while doing a tango), and not at all a martial artist, I won't comment on her authority in estimating the scientific weight of her starring protegee, nor on that of the latter's article. The article in Quanta Magazine has no scientific weight to my measures.
 * The linked remarks from 2011 by J. Baez at al., meanwhile ripened during a period of seven years and however prophetic, may also be applicable, but is there any reliable source (not talking about secondary), that is beyond speculative expectations, since then?
 * Please, feel free to edit according to your estimation what is best for the encyclopedicity of this article. Purgy (talk) 09:32, 29 July 2018 (UTC)

 * We'll just have to keep an eye on it. kencf0618 (talk) 00:18, 31 July 2018 (UTC)

A problematic reference to the multiplication table and mnemonics used based on $$e_1e_2 =e_4$$ mod-7 construction
It seems beginning on July 10, 2019 User:MatthewDougherty correctly added a tag that some text related to a 7-cycle construction based on e1e2 = e4 was confusing and the reference did not match the multiplication table above, i.e. $$e_1e_2 = e_3$$. On Apr 8, 2020 User:Pianostar9 deleted some of that text, but not the tag.

Then on May 3 User:CFjohnny1955, with some clarification by User:XOR'easter, deleted the tag by adding text that inferred the multiplication table and the mnemonic were different octonions: "...by using the triangular multiplication diagram, or Fano plane, below (and instead of the multiplication table, above)."

To be clear, the table and both 2D and 3D Fano mnemonics are (and always have been) consistently based on the triads 123-145-176-246-257-347-365. The reference to the 124-137-156-235-267-346-457 octonion Fano plane is not present in this article (yet). FYI - my triad lists are generated algorithmically and sorted in ascending order.

So this incorrectly implies there is a difference in the octonion triads used in the table and the mnemonics. While I understand the motivation to clarify and explain how to perform the interesting 7-cycle math based on the triangular rotation of the Fano plane $$e_1e_2 = e_4$$, the current comment either needs to be removed, changed, or a new Fano plane mnemonic needs to be added for that purpose. I guess we could replace the table and the mnemonics to all be based on the 124 triads, but that would be disruptive to the historical use of, and references to, the 123 triad representation in the article. Also, that representation putting the IJK quaternions in the upper left quadrant is also very important for understanding Cayley-Dickson constructions, so changing it would not be a good idea.

Since I was the one who generated these mnemonics from my Mathematica codebase using any of the 480 possible triad combinations, I created a reference to the different sets (yes, I can create just the 124 Fano plane for reference w/o the triads and tables, if you wish):

124

123

What is the best approach to correcting the issue?

Jgmoxness (talk) 21:38, 20 July 2021 (UTC)
 * I believe my edit had simply moved and reformatted text. Pianostar9 (talk) 14:17, 23 July 2021 (UTC)

I think the current diagram and the right table are incorrect. The yellow picture in Octonion-124-137-156-235-267-346-457.svg suggests e3 = ij, while the table and text suggest the convention e4 = ij. The yellow picture also doesn't match the convention that the product of two points on a line is the third point (up to sign). For example, look at the line 137. One should have e1*e3 = e7 (which in the picture is lk), but instead, e1*e3 = ik. This seems to be incorrect in both picture and right table. I think the fix should be to remove the ij in the middle point of the picture, leaving it as e3 = k, changing the bottom right point to e7 = ik (instead of e7 = lk), changing the top point to e5 = jk (instead of li), and changing the bottom left point to e6 = kl (instead of lj). The right table should be adjusted accordingly. Sadly, this is not an edit I can make; User:Jgmoxness is the one who generated this picture. 2A02:A454:920E:1:8141:EF58:A3B3:4002 (talk) 13:10, 6 October 2021 (UTC)
 * I now see you moved the text (vs. deleting it), but this was not the crux of my point - which was just trying to outline the complex history of the changes related to the confusing set of points being made in the article.
 * You are correct about a IJKL notation issue between the Fano diagrams' association of e# format to the header rows in the IJKL multiplication tables and your suggested fix for e3=k is the right approach. I should also remove the =lij on e7=lij=lk. These two changes remove the underlying problem with trying to link IJKL headers to the e# on all 480 diagrams.
 * The problem with the IJKL notation is that in trying to have the headers be consistent with the table math sequences for all 480 representations is very confusing and no longer value added. I tried to make it simple with a consistent header structure linking to e#, but included the mistake of adding the math elements in the e3 and e7 Fano plane nodes. My apologies.
 * So I have corrected those WP Fano diagrams.
 * IJKL is all just a holdover from extending imaginary to quaternions with only 2 possible representations where ij=k (up to sign). In extending it to octonions, the IJKL convention was largely dropped for readability and the practicality since there are now 480 possible representations (many where ij≠k in all but the 123 based tables).
 * Having said this, maybe dropping the all of the IJKL references in the Fano diagrams and deleting the IJKL tables would be better.
 * Thoughts?
 * Jgmoxness (talk)

Huh???
English next time please??? Carlimited (talk) 03:29, 16 December 2022 (UTC)

 * Could you be specific, please? It would be easier to address your issue if you made it clear.—Anita5192 (talk) 06:34, 16 December 2022 (UTC)

UI problem
One of the graphs stick out past the Tools bar. I don't know how to fix this, could someone else do it? UnqaidIntern (talk) 17:13, 16 May 2023 (UTC)

 * They look fine to me. Which graph are you referring to?—Anita5192 (talk) 17:21, 16 May 2023 (UTC)
 * It's the IJKL graph, I'm on a Chromebook so that might be the reason. UnqaidIntern (talk) 17:23, 16 May 2023 (UTC)

 * I see what you mean. The diagram should be replaced by separate, smaller diagrams, but I don't have the facilities for doing that. Hopefully someone else will.—Anita5192 (talk) 18:10, 16 May 2023 (UTC)
 * Much easier to simply reduce the image size - fixed.Jgmoxness (talk) 15:44, 7 August 2023 (UTC)

Consistent representation of the Fano plane please
The Fano plane in this image and don't match. For instance, e3 is the center of one, but the other has e4. Also the inner circle is counterclockwise in one while the other is clockwise.

It's likely these are identical under relabeling, but it would be a lot easier if a consistent definition was used. I'm just learning this for the first time and got a little confused! Pstetz (talk) 20:08, 4 August 2023 (UTC)

 * These are not the same diagram because they are not the same octonion (which is addressed in the article text). There are 480 possible and different octonion multiplication matrices. These are typically identified through a set of triads 123-145-176-246-257-347-365 or 124-137-156-235-267-346-457.
 * Consistency in diagrams means forcing a selection of a single unique triad set. This would provide clarity in the visual representation, but would limit the understanding of what octonions really are. These two diagrams are likely the most commonly used among about 1/2 dozen across the literature and their differences are interesting (IMO). For the complete set of 480 diagrams and matrices (which includes split octonions by triad), see splitfano.pdf
 * Jgmoxness (talk) 15:35, 7 August 2023 (UTC)

Alternativity implies power associativity
At the top of the page, it says of the octonions "they are alternative. They are also power associative." This is technically correct, but the language suggests that alternativity and power associativity are independent. They aren't - power associativity is a weaker form of (and thus prerequisite to) alternativity. This should probably be rephrased to something like "they are alternative, and thus power associative", but I can't find the best way to phrase that sentence. Cyversch (talk) 19:04, 13 May 2024 (UTC)

title: Talk:Octonion


text:
Hypercomplex number

In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.

History
In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them.

The cataloguing project began in 1872 when Benjamin Peirce first published his Linear Associative Algebra, and was carried forward by his son Charles Sanders Peirce. Most significantly, they identified the nilpotent and the idempotent elements as useful hypercomplex numbers for classifications. The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals $$\mathbb{R}$$, the complexes $$\mathbb{C}$$, the quaternions $$\mathbb{H}$$, and the octonions $$\mathbb{O}$$, and the Frobenius theorem says the only real associative division algebras are $$\mathbb{R}$$, $$\mathbb{C}$$, and $$\mathbb{H}$$. In 1958 J. Frank Adams published a further generalization in terms of Hopf invariants on H-spaces which still limits the dimension to 1, 2, 4, or 8.

It was matrix algebra that harnessed the hypercomplex systems. For instance, 2 x 2 real matrices were found isomorphic to coquaternions. Soon the matrix paradigm began to explain several others as they were represented by matrices and their operations. In 1907 Joseph Wedderburn showed that associative hypercomplex systems could be represented by square matrices, or direct products of algebras of square matrices. From that date the preferred term for a hypercomplex system became associative algebra, as seen in the title of Wedderburn's thesis at University of Edinburgh. Note however, that non-associative systems like octonions and hyperbolic quaternions represent another type of hypercomplex number.

As Thomas Hawkins explains, the hypercomplex numbers are stepping stones to learning about Lie groups and group representation theory. For instance, in 1929 Emmy Noether wrote on "hypercomplex quantities and representation theory". In 1973 Kantor and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989.

Karen Parshall has written a detailed exposition of the heyday of hypercomplex numbers, including the role of mathematicians including Theodor Molien and Eduard Study. For the transition to modern algebra, Bartel van der Waerden devotes thirty pages to hypercomplex numbers in his History of Algebra.

Definition
A definition of a hypercomplex number is given by as an element of a unital, but not necessarily associative or commutative, finite-dimensional algebra over the real numbers. Elements are generated with real number coefficients $$(a_0, \dots, a_n)$$ for a basis $$\{ 1, i_1, \dots, i_n \}$$. Where possible, it is conventional to choose the basis so that $$i_k^2 \in \{ -1, 0, +1 \}$$. A technical approach to hypercomplex numbers directs attention first to those of dimension two.

Two-dimensional real algebras
Theorem:  Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, and the dual numbers. In particular, every 2-dimensional unital algebra over the reals is associative and commutative.

Proof: Since the algebra is 2-dimensional, we can pick a basis $\{1, u\}$. Since the algebra is closed under squaring, the non-real basis element u squares to a linear combination of 1 and u:
 * $$u^2 = a_0 + a_1 u$$

for some real numbers a0 and a1.

Using the common method of completing the square by subtracting a1u and adding the quadratic complement a$2 1$/4 to both sides yields
 * $$u^2 - a_1 u + \frac{1}{4}a_1^2 = a_0 + \frac{1}{4}a_1^2.$$

Thus $\left(u - \frac{1}{2}a_1\right)^2 = \tilde{u}^2$ where $\tilde{u}^2~ = a_0 + \frac{1}{4}a_1^2.$ The three cases depend on this real value:
 * If 4a0 = −a12, the above formula yields ũ2 = 0. Hence, ũ can directly be identified with the nilpotent element $$\varepsilon$$ of the basis $$\{ 1, ~\varepsilon \}$$ of the dual numbers.
 * If 4a0 > −a12, the above formula yields ũ2 > 0. This leads to the split-complex numbers which have normalized basis $$\{ 1, ~j \}$$ with $$j^2 = +1$$. To obtain j from ũ, the latter must be divided by the positive real number $a \mathrel{:=} \sqrt{a_0 + \frac{1}{4}a_1^2}$ which has the same square as ũ has.
 * If 4a0 < −a12, the above formula yields ũ2 < 0. This leads to the complex numbers which have normalized basis $$\{ 1, ~i \}$$ with $$i^2 = -1$$. To yield i from ũ, the latter has to be divided by a positive real number $a \mathrel{:=} \sqrt{\frac{1}{4}a_1^2 - a_0}$ which squares to the negative of ũ2.

The complex numbers are the only 2-dimensional hypercomplex algebra that is a field. Algebras such as the split-complex numbers that include non-real roots of 1 also contain idempotents $\frac{1}{2}(1 \pm j)$ and zero divisors $$(1 + j)(1 - j) = 0$$, so such algebras cannot be division algebras. However, these properties can turn out to be very meaningful, for instance in describing the Lorentz transformations of special relativity.

In a 2004 edition of Mathematics Magazine the 2-dimensional real algebras have been styled the "generalized complex numbers". The idea of cross-ratio of four complex numbers can be extended to the 2-dimensional real algebras.

Clifford algebras
A Clifford algebra is the unital associative algebra generated over an underlying vector space equipped with a quadratic form. Over the real numbers this is equivalent to being able to define a symmetric scalar product, u ⋅ v = $1⁄2$(uv + vu) that can be used to orthogonalise the quadratic form, to give a basis $\{e_{1}, ..., e_{k}\}$ such that: $$\frac{1}{2} \left(e_i e_j + e_j e_i\right) = \begin{cases} -1, 0, +1 & i = j, \\ 0 & i \not = j. \end{cases}$$

Imposing closure under multiplication generates a multivector space spanned by a basis of 2k elements, $\{1, e_{1}, e_{2}, e_{3}, ..., e_{1}e_{2}, ..., e_{1}e_{2}e_{3}, ...\}$. These can be interpreted as the basis of a hypercomplex number system. Unlike the basis $\{e_{1}, ..., e_{k}\}$, the remaining basis elements need not anti-commute, depending on how many simple exchanges must be carried out to swap the two factors. So e1e2 = −e2e1, but e1(e2e3) = +(e2e3)e1.

Putting aside the bases which contain an element ei such that ei2 = 0 (i.e. directions in the original space over which the quadratic form was degenerate), the remaining Clifford algebras can be identified by the label Clp,q($$\mathbb{R}$$), indicating that the algebra is constructed from p simple basis elements with ei2 = +1, q with ei2 = −1, and where $$\mathbb{R}$$ indicates that this is to be a Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers.

These algebras, called geometric algebras, form a systematic set, which turn out to be very useful in physics problems which involve rotations, phases, or spins, notably in classical and quantum mechanics, electromagnetic theory and relativity.

Examples include: the complex numbers Cl0,1($$\mathbb{R}$$), split-complex numbers Cl1,0($$\mathbb{R}$$), quaternions Cl0,2($$\mathbb{R}$$), split-biquaternions Cl0,3($$\mathbb{R}$$), split-quaternions Cl1,1($\mathbb{R}$) ≈ Cl2,0($\mathbb{R}$) (the natural algebra of two-dimensional space); Cl3,0($$\mathbb{R}$$) (the natural algebra of three-dimensional space, and the algebra of the Pauli matrices); and the spacetime algebra Cl1,3($$\mathbb{R}$$).

The elements of the algebra Clp,q($$\mathbb{R}$$) form an even subalgebra Cl$[0] q+1,p$($$\mathbb{R}$$) of the algebra Clq+1,p($$\mathbb{R}$$), which can be used to parametrise rotations in the larger algebra. There is thus a close connection between complex numbers and rotations in two-dimensional space; between quaternions and rotations in three-dimensional space; between split-complex numbers and (hyperbolic) rotations (Lorentz transformations) in 1+1-dimensional space, and so on.

Whereas Cayley–Dickson and split-complex constructs with eight or more dimensions are not associative with respect to multiplication, Clifford algebras retain associativity at any number of dimensions.

In 1995 Ian R. Porteous wrote on "The recognition of subalgebras" in his book on Clifford algebras. His Proposition 11.4 summarizes the hypercomplex cases:
 * Let A be a real associative algebra with unit element 1. Then
 * 1 generates $$\mathbb{R}$$ (algebra of real numbers),
 * any two-dimensional subalgebra generated by an element e0 of A such that e02 = −1 is isomorphic to $$\mathbb{C}$$ (algebra of complex numbers),
 * any two-dimensional subalgebra generated by an element e0 of A such that e02 = 1 is isomorphic to $$\mathbb{R}$$2 (pairs of real numbers with component-wise product, isomorphic to the algebra of split-complex numbers),
 * any four-dimensional subalgebra generated by a set $\{e_{0}, e_{1}\}$ of mutually anti-commuting elements of A such that $$e_0 ^2 = e_1 ^2 = -1$$ is isomorphic to $$\mathbb{H}$$ (algebra of quaternions),
 * any four-dimensional subalgebra generated by a set $\{e_{0}, e_{1}\}$ of mutually anti-commuting elements of A such that $$e_0 ^2 = e_1 ^2 = 1$$ is isomorphic to M2($$\mathbb{R}$$) (2 × 2 real matrices, coquaternions),
 * any eight-dimensional subalgebra generated by a set $\{e_{0}, e_{1}, e_{2}\}$ of mutually anti-commuting elements of A such that $$e_0 ^2 = e_1 ^2 = e_2 ^2 = -1$$ is isomorphic to 2$$\mathbb{H}$$ (split-biquaternions),
 * any eight-dimensional subalgebra generated by a set $\{e_{0}, e_{1}, e_{2}\}$ of mutually anti-commuting elements of A such that $$e_0 ^2 = e_1 ^2 = e_2 ^2 = 1$$ is isomorphic to M2($$\mathbb{C}$$) (2 × 2 complex matrices, biquaternions, Pauli algebra).

Cayley–Dickson construction
All of the Clifford algebras Clp,q($$\mathbb{R}$$) apart from the real numbers, complex numbers and the quaternions contain non-real elements that square to +1; and so cannot be division algebras. A different approach to extending the complex numbers is taken by the Cayley–Dickson construction. This generates number systems of dimension 2n, n = 2, 3, 4, ..., with bases $$\left\{1, i_1, \dots, i_{2^n-1}\right\}$$, where all the non-real basis elements anti-commute and satisfy $$i_m^2 = -1$$. In 8 or more dimensions (n ≥ 3) these algebras are non-associative. In 16 or more dimensions (n ≥ 4) these algebras also have zero-divisors.

The first algebras in this sequence are the four-dimensional quaternions, eight-dimensional octonions, and 16-dimensional sedenions. An algebraic symmetry is lost with each increase in dimensionality: quaternion multiplication is not commutative, octonion multiplication is non-associative, and the norm of sedenions is not multiplicative.

The Cayley–Dickson construction can be modified by inserting an extra sign at some stages. It then generates the "split algebras" in the collection of composition algebras instead of the division algebras:
 * split-complex numbers with basis $$\{ 1,\, i_1 \}$$ satisfying $$\ i_1^2 = +1$$,
 * split-quaternions with basis $$\{ 1,\, i_1,\, i_2,\, i_3 \}$$ satisfying $$\ i_1^2 = -1,\, i_2^2 = i_3^2 = +1$$, and
 * split-octonions with basis $$\{ 1,\, i_1,\, \dots,\, i_7 \}$$ satisfying $$\ i_1^2 = i_2^2 = i_3^2 = -1$$, $$\ i_4^2 = i_5^2 = i_6^2 = i_7^2 = +1 .$$

Unlike the complex numbers, the split-complex numbers are not algebraically closed, and further contain nontrivial zero divisors and nontrivial idempotents. As with the quaternions, split-quaternions are not commutative, but further contain nilpotents; they are isomorphic to the square matrices of dimension two. Split-octonions are non-associative and contain nilpotents.

Tensor products
The tensor product of any two algebras is another algebra, which can be used to produce many more examples of hypercomplex number systems.

In particular taking tensor products with the complex numbers (considered as algebras over the reals) leads to four-dimensional bicomplex numbers $$\mathbb{C} \otimes_\mathbb{R} \mathbb{C}$$ (isomorphic to tessarines $$\mathbb{C} \otimes_\mathbb{R} D$$), eight-dimensional biquaternions $$\mathbb{C} \otimes_\mathbb{R} \mathbb{H}$$, and 16-dimensional complex octonions $$\mathbb{C} \otimes_\mathbb{R} \mathbb{O}$$.

Further examples

 * bicomplex numbers: a 4-dimensional vector space over the reals, 2-dimensional over the complex numbers, isomorphic to tessarines.
 * multicomplex numbers: 2n-dimensional vector spaces over the reals, 2n−1-dimensional over the complex numbers
 * composition algebra: algebra with a quadratic form that composes with the product

title: Hypercomplex number