Talk:Bicomplex number

Deletion review
If the review is to go ahead, some further steps need to be taken (see here). H.G. 09:59, 16 September 2008 (UTC)


 * I was advised at WP math not to go ahead with the review. Unless there is someone else interested in pursuing this.  Katzmik (talk) 10:03, 16 September 2008 (UTC)
 * In that case, I'll remove the tag for the time being. H.G. 10:26, 16 September 2008 (UTC)

Direct sum
The following line was inserted in the lead:
 * The term "tessarine" is archaic and no longer used much, as the algebra of tessarines is isomorphic to the sum of two copies of the complex numbers.

The direct sum $$C \oplus C$$ involves elements (w,z) which do not comingle their components in products. The tessarine product does mix components upon multiplication. The assertion is false. The contribution is being removed.Rgdboer (talk) 02:18, 13 February 2010 (UTC)
 * Correction. A proof has been supplied to show the isomorphism with $$C \oplus C$$. Dialogue welcomed here in Talk:Tessarine on topic.Rgdboer (talk) 23:02, 13 February 2010 (UTC)
 * I've re-inserted the sentence about them being archaic, as it's correct and more importantly it dissuades anyone coming here and spending too long reading the article thinking it is on a modern area of maths.-- JohnBlackburne wordsdeeds 23:38, 16 February 2010 (UTC)
 * Added references from 2004, 2006, 2007 on use of tessarines in DSP. Please remove the disparaging remark.Rgdboer (talk) 03:21, 18 February 2010 (UTC)
 * It is still correct (if it said they were never used it would be wrong). It's notable that the source for tessarines in e.g. the 2006 paper is one of Cockle's papers from 150 years ago, suggesting there was no mention of them in literature between then and now. And I don't see how a statement about a mathematical term can be disparaging.-- JohnBlackburne wordsdeeds 10:48, 18 February 2010 (UTC)

I've studied these algebras intensively and have never heard the term 'tessarine' until now. Even 'bicomplex number' is rather rare (though vastly more common!), since modern mathematicians simply say we're talking about the associative algebra $$ \mathbb{C} \otimes_\mathbb{R} \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C} $$, or if you prefer, the complex Clifford algebra $$ \mathbb{C}l_1 $$. So, I've rewritten the beginning of the article to reflect the perspective most mathematicians would take on this algebra.

John Baez (talk) 07:36, 25 July 2014 (UTC)

Aren't these split-quaternions?
Call me stupid but after reading this article I don't see why it's not incorporated in the split-quaternions article and is stand alone. Any actual explanation would be welcomed. AlucardNoir 22:18, 22 August 2010 (UTC) —Preceding unsigned comment added by AlucardNoir (talk • contribs)
 * The tessarine algebra is commutative, split-quaternions are not. Also view the multiplication tables displayed at the beginning of each article. They show only one minus one on the diagonal for split-quaternions while tessarines show two.Rgdboer (talk) 19:52, 23 August 2010 (UTC)


 * On the other hand, we have $$R \oplus R$$ isomorphic to split-complex numbers, and $$H \oplus H$$ isomorphic to split-biquaternions. For consistency then the algebra isomorphic to $$C \oplus C$$ might be called "split-quaternions". There is an argument for calling tessarines by the alternative name split-quaternion. The algebra currently described under that name was originally coquaternions. The move was made some time ago; my opinion is that the name given by the discoverer James Cockle be restored, and split-quaternion made into a redirect to this article. A consistent naming convention where split algebras are direct sums would then be in order.Rgdboer (talk) 23:08, 24 August 2010 (UTC)


 * For information on the source that has been for employing the term split-quaternion elsewhere see Talk:Split-quaternion.Rgdboer (talk) 21:31, 25 August 2010 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: page moved per discussion below. The histories at the two pages were swapped, just as Arthur Rubin described below. - GTBacchus(talk) 01:12, 29 November 2010 (UTC)

Tessarine → bicomplex number — Unlike zoological names, we are not bound to the first use. They are bicomplex numbers, and that is probably the most common modern usage. The historical references could be moved to history of quaternions, as already noted above. — Arthur Rubin (talk) 20:41, 29 October 2010 (UTC)
 * Agree. From the refs "bicomplex numbers" seems far more common, and it's also far more descriptive, i.e. indicative of what they are and consistent with other hypercomplex numbers. A "tessarine" could be anything, i.e. it's an odd name for a mathematical concept and it's not surprising it's fallen out of use.-- JohnBlackburne wordsdeeds 21:49, 29 October 2010 (UTC)
 * Unsure. From the cited articles it seems that the most common name is 'Commutative hypercomplex algebra'. Is there any proof that 'bicomplex number' is more common?--MathFacts (talk) 22:27, 29 October 2010 (UTC)


 * Comment has been in existence since 2004 as a separate article, it was only redirected here in January. If there has been content swapped between the two articles in the 6 years of their mutual existences, there will be WP:GFDL issues to resolve. 76.66.203.138 (talk) 04:34, 30 October 2010 (UTC)
 * That needs to be dealt with whichever article remains primary. If absolutely necessary, we could move:
 * Bicomplex number to Bicomplex number (temp)
 * Tessarine to Bicomplex number (over dummy redirect)
 * Bicomplex number (temp) to Tessarine (suppressing redirect)
 * and edit Tessarine to point to Bicomplex number
 * — Arthur Rubin (talk) 07:37, 18 November 2010 (UTC)
 * Note I intended to refer to the Principle of Priority in my initial reasoning, but I wasn't aware of our article on the subject. — Arthur Rubin  (talk) 04:35, 30 October 2010 (UTC)
 * Don't Move: The tessarine of 1848 arose in the wake of Hamilton's lectures on biquaternions, and thus represents the English-algebrists movement of that era; Philosophical Magazine was the standard venue for scientific innovation. The bicomplex number label dates from Corrado Segre's a later Italian article in a German publication. Considering priority and the English publication, there is insufficient reason to make the Move.Rgdboer (talk) 00:58, 5 November 2010 (UTC)
 * As I said, we are not restricted by the Principle of Priority. — Arthur Rubin  (talk) 02:18, 5 November 2010 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Tessarine problem
In the section Quotient rings of polynomials, this passage appears:

"A modern approach to tessarines uses the polynomial ring R[X,Y] in two indeterminates X and Y. Consider these three second degree polynomials X^2 + 1, Y^2 - 1, XY - YX. Let A be the ideal generated by them. Then the quotient ring R[X,Y]/A is isomorphic to the ring of tessarines."

But in the definition of polynomial ring, polynomial multiplication is commutative. So the polynomial XY - YX is just the 0 polynomial, and can be omitted.

The inclusion of XY - YX in this version of tessarines tends to suggest that the author of this passage believes that polynomial rings are non-commutative.

I hope someone knowledgeable about both polynomial rings and tessarines can rewrite this passage so that it makes sense.Daqu (talk) 23:13, 23 March 2015 (UTC)


 * Here the polynomial ring has two indeterminates X and Y which may or may not commute according to specification. The appropriate section in the polynomial ring article is "Polynomial ring in several variables". Nothing is said there about commutation, but there is no presumption that the indeterminants do commute. In abstract algebra, given experience with matrix multiplication, there is no presumption of commutation. The statement you are referring to has commutation in a polynomial ring with a single indeterminant.Rgdboer (talk) 03:01, 24 March 2015 (UTC)


 * I agree with Daqu's concern. Phrasing in terms of quotients of free algebras (aka noncommutative polynomial rings) when the relation XY-YX is going to be included anyway is both convoluted and distracting, so I replaced it with regular polynomial rings. Rschwieb (talk) 16:20, 26 March 2015 (UTC)


 * For abstract algebra generally, consider the quaternion ring where XY + YX goes into the ideal for a quotient. Any study of rings ought to catch this important instance, and presumption of "regular polynomial rings" is inappropriate.Rgdboer (talk) 02:44, 27 March 2015 (UTC)


 * In the case of representing the quaternions, using the free algebra is justifiable. In our commutative case here, it's unnecessary generality. If I suggested that we express all rings as quotients of free nonassociative rings and include relations that make them associative, I would hope you would protest.
 * The cost of accessibility seems to outweigh the (slight) benefit of having an analogous picture to the representation of quaternions. This representation of the quaternions does not even appear in the quaternion article (when I looked just now) so it's even unclear that the "benfit" is even user facing. Polynomial rings are more widely known and understood compared to free algebras. Rschwieb (talk) 13:16, 27 March 2015 (UTC)


 * Surely all concerns will be addressed if we simply add one word: "A modern approach to tessarines uses the commutative polynomial ring R[X,Y] in two indeterminates X and Y."? —Quondum 14:49, 27 March 2015 (UTC)


 * Some assertions without reference:"In the definition of polynomial ring, polynomial multiplication is commutative." Also, "Polynomial rings are more widely known and understood compared to free algebras." Where is there a reference that says indeterminates commute? The use of polynomial quotients in this article is not specially developed here, it is a standard method in modern algebra, and serves as one of the unifying theories in mathematics.  Mention of associativity as a property that could be specified by quotients takes us well outside convention. The issue of commutativity is extremely common, even designated as "Abelian", though in this context the original writeup considered the property in terms of a generator of the ideal making a quotient. Quondom's suggestion to explain it away might use the preamble, "Presuming that indeterminates X and Y commute", but that weakens the illustration of the quotient ring concept, which includes quotient ring. Considering the power of this method to explain many hypercomplex number systems, appeal to the method should be in its natural, broad context of non-commuting indeterminates.Rgdboer (talk) 21:20, 27 March 2015 (UTC)


 * The deletion of XY&minus;YX was reverted. There still is much to discuss here.Rgdboer (talk) 20:38, 30 March 2015 (UTC)


 * Agreed. We should determine whether the method of non-commutative polynomials is frequently used to generate hypercomplex numbers in general. —Quondum 23:12, 30 March 2015 (UTC)


 * "In the definition of polynomial ring, polynomial multiplication is commutative." Yes, after more than a decade of reading ring theory I can confidently say that the term "polynomial ring R[X,Y]" without other qualification connotes that the indeterminates commute. When the indeterminates are not assumed to commute, it's possible to call it the free algebra on X and Y, or at the very least say "polynomials in noncommuting indeterminates X,Y."


 * "Where is there a reference that says indeterminates commute?" Here is one, one which we should be consistent with, by the way. Where is a reference that says indeterminates do not commute in something you call just "a polynomial ring"?
 * "Considering the power of this method to explain many hypercomplex number systems" Power indeed. Quotients explain every ring in terms of generators, and it has virtually nothing to do with hypercomplex numbers.
 * At the very least I'll edit in the caveats about noncommutativity to mitigate confusion.
 * In the meantime, I look forward to seeing what references @Rgdboer has to support his claim. Rschwieb (talk) 18:50, 31 March 2015 (UTC)


 * Thank you for direction to free algebra which is the broader context. The edits today acknowledge your insight and corrections.Rgdboer (talk) 22:08, 1 April 2015 (UTC)


 * I'm a little confused. The ideals A and B given in Bicomplex number § Quotient rings of polynomials appear to be identical.  Some incomplete copy-and-edit?  The logic being followed is a little obscure to me. —Quondum 23:25, 1 April 2015 (UTC)


 * Well, they are at least not equal. If $$(X^2 + 1,\ Y^2 - 1)$$ and $$(X^2 + 1,\ Y^2 + 1)$$ were the same, then the ideal would contain $$Y^2+1-(Y^2-1)=2$$, so it would be the whole ring. Rschwieb (talk) 14:26, 3 April 2015 (UTC)


 * Oops, I need to read more carefully. Missed the sign difference.   —Quondum 20:08, 3 April 2015 (UTC)

Ideal details
The above discussion shows that it is necessary to demonstrate the importance of XY&minus;YX in the arithmetic of the ideal for tessarines. Let A = < X2 + 1, Y2 &minus;1, XY&minus;YX>, the ideal in question. Recall that sums and differences stay in the ideal, and that an ideal element multiplied by a ring element is in the ideal.

Note that XY2X = X(Y2 &minus;1)X + (X2 + 1) &minus; 1, so that
 * XY2X + 1 = X(Y2 &minus;1)X + (X2 + 1) ∈ A.

Therefore XY(XY &minus; YX) + XY2X + 1 ∈ A.

But this final element is (XY) 2 + 1, so that A contains the quadratic polynomial of a second classical imaginary unit, this one corresponding to XY, and the first corresponding to X.
 * Rgdboer (talk) 00:37, 28 March 2015 (UTC)

The quotient $$\mathbb R[X,Y]/(X^2+1, Y^2 - 1)$$ and the quotient $$\mathbb R\langle X,Y\rangle /(X^2+1, Y^2 - 1, XY-YX)$$ are identical. You might just as well argue that $$\mathbb R\langle X,Y\rangle/(X^2+1, Y-1, XY-YX)$$ is a "more important" representation of $$\mathbb C$$ than the normal one. Does this not illustrate the same objection that I have above? Rschwieb (talk) 19:17, 31 March 2015 (UTC)

Relevance of citation to Rochon's "Tetranumbers"
Since the IPs geolocated to Quebec are unwilling to start a section here, I will go ahead and do so.

I was not entirely which template was most suited for addressing the problem with this contribution, so I chose "relevance." What I really mean is that it is not encyclopedic content suitable for inclusion: this work does not have significance in its field. Rather, it comes off as promotion of someone's personal project (akin to No one cares about your garage band, except at a source level rather than an article level.)

Of course, the reasons for not including such things are obvious: who would want to read Wikipedia if it were cluttered with empty self-promotion? It also represents a verifiability problem. As an example, one of the cited articles appears to be in French and only has 13 citations in 19 years, many of which are also articles of Rochon. These are indicators that this content is not really fit for inclusion. But here others may present reasons why it is suitable. Rschwieb (talk) 20:23, 14 November 2017 (UTC)


 * The edit history shows four edits over a period of little more than 24 hours by to IPs 142.169.78.184 and 132.209.3.30. On the presumption that these are of home and university locations of the same individual, or of cooperating individuals, IMO this is a three-revert violation.  We do not need this kind of aggressive self-publishing on WP.
 * The addition is of some hobby application, and does not belong in the History section of a long-standing article such as this; if we allowed it, we would have our articles Real number, Complex number, Quaternion, Octonion, Clifford algebra, and hundreds of others littered with every student's pet hobby. Thus, I'll remove the additions as non-notable.  In particular, this material does not seem to be a contribution of value to the study of bicomplex numbers, and thus does not belong in this article.  —Quondum 02:52, 15 November 2017 (UTC)