Talk:Sedenion

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Mathematics grading:	Start Class	Low Importance	Field: Algebra
Lacks context for concept, needs cleanup shotwell 06:32, 6 October 2006 (UTC)

Could you describe what are they used for ? --Taw

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Also, what is the etymology of the name? Incidentally, I'm [RobertAtFM|http://wiki.fastmail.fm/wiki/index.php/RobertAtFm] on the [FastMail.FM Wiki|http://wiki.fastmail.fm/].

Yes, I agree: this page needs more information. I'd love to learn more about these things, but this page is barely more than a stub. --AlexChurchill 10:46, Jul 27, 2004 (UTC)

[edit] 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)

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)

[edit] 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)

[edit] Duo-tricenians??

This article says:

"Like (Cayley-Dickson) octonions, multiplication of Cayley-Dickson 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 being power-associative." In turn, is the property of being power-associative an operation that the duo-tricenians no longer have?? Georgia guy 19:31, 11 September 2006 (UTC)

Hi. All Cayley-Dickson construction products remain power associative, including the 32-ions. This does - of course - not apply to modified constructs, like e.g. the split-octonions, which contain nilpotents. Thanks, Jens Koeplinger 12:37, 15 September 2006 (UTC)