Talk:Multicomplex number

Hello, 8OYscLu9. Your earlier remarks are fully justified in my eyes. The statement that the bicomplex numbers are a special case holds true, however, both articles use different reference material I believe. Well, and then I may have made a mistake altogether, who knows. While I was glad to create the multicomplex number stub from the reference given, the bicomplex number article uses a web reference with different definitions. In short, if instead of real number coefficients you use another multicomplex number as coefficients, you'll again end up with a multicomplex number. So, I find your "clean-up please" remark appropriate and would leave it out here ... until someone has the time to pick it up ... Thanks, Jens Koeplinger 14:57, 20 January 2007 (UTC)

I have taken the liberty of making a few changes to address this issue. I think they work well, but please let me know if I am mistaken, or whether I have made any errors in the math expressions. --8OYscLu9 (talk) 12:47, 6 July 2008 (UTC)

It's a bit better now I think
I've taken Jheald's latest additions and grouped it a bit. I think the article is getting better now. Maybe some general properties or so, and it should be a nice article. Thanks, Jens Koeplinger 12:01, 1 April 2007 (UTC)


 * PS: But when direct sum and when outer product should be used to indicate isomorphism. Koeplinger 12:16, 1 April 2007 (UTC)


 * The two products $$\oplus$$ and $$\otimes$$ are very different things. If we take $$\mathbb{C}\otimes\mathbb{C}\otimes\mathbb{C}$$ that means that each base of the algebra can be factored into a power of $$i_1$$ times $$i_2$$ times $$i_3$$, so each number in $$\mathbb{C}\otimes\mathbb{C}\otimes\mathbb{C}$$ is a linear combination over eight basis elements.


 * On the other hand, when we write $$\mathbb{MC}_8 = \mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}$$ that means each of its 8 basis elements can be written as a particular vector of 4 complex numbers, with the product of two vectors defined by the product (?or should that be a product?) of only the corresponding components, taken pair by pair.


 * Are the two isomorphic? Can one write $$\mathbb{C}\otimes\mathbb{C}\otimes\mathbb{C} \sim \mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}\oplus\mathbb{C}$$ ? I'm not sure.  Can one find an element of the first space such that $$\alpha^8 = -1$$, and none of the numbers from $$\alpha ... \alpha^4$$ can be expressed as linear combinations of their predecessors?  I don't know; $$exp(i_1 \pi/4) exp(i_2 \pi/4) exp(i_3 \pi/4)$$ might be a candidate.  Can one find eight different members u of the second space such that u2 = -1 ?  ... Left as an excercise for the reader.  :-)   Jheald 14:06, 1 April 2007 (UTC)


 * Ok. I've added the word "also" to the isomorphism that uses an outer product (diff) to indicate that this is shown with purpose (and not a mistake). For n=4 we have $$\mathbb{C}\otimes\mathbb{C} = \mathbb{C}\oplus\mathbb{C}$$, but this is a special case. Thanks for the details, Jens Koeplinger 14:31, 1 April 2007 (UTC)

Content
At present the content of this article does not correspond to the content of the book by GB Price cited. The other source is not useful. There are then no references for the content, some of which may be questionable. If no one objects, the content should be changed to correspond to that used by Price after introduction by Corrado Segre. The multicomplex number of these and many other authors extends or continues the series complex number, bicomplex number, ... The content should correspond to the most common usage.Rgdboer (talk) 23:09, 8 July 2010 (UTC)

According to GB Price, the multicomplex number systems Cn are defined inductively as follows: Let C0 be the real number system. For every n > 0 let in be a square root of minus one. Then $$C_{n+1} = \lbrace z = x + y i_n : x,y \in C_n \rbrace .$$ In the multicomplex number systems one also requires when n &ne; m that $$i_n i_m = i_m i_n$$ (commutative property). Then C1 is the complex number system, C2 is the bicomplex number system, C3 is the tricomplex number system of Corrado Segre, and Cn is the multicomplex number system of order n.

The multicomplex number systems are not to be confused with Clifford numbers (elements of a Clifford algebra), since Clifford's square roots of minus one anti-commute ($$i_n i_m + i_m i_n = 0$$ for Clifford).Rgdboer (talk) 22:01, 16 July 2010 (UTC)


 * Today the content was corrected.Rgdboer (talk) 22:02, 17 July 2010 (UTC)

Which root?
80YscLu9 asks:
 * Excuse me if I misunderstand, but don't you mean "the nth root of minus one" rather than "a square root of minus one"? 8OYscLu9 (talk) 18:16, 30 March 2011 (UTC)

Note has been made at Talk:Tricomplex number of the conflicting meanings. See Tricomplex number for the case n = 3 of such a development of multicomplex numbers. This article is on the topic of Segre's counting of the orders. Note that roots of minus one never get off the complex plane, so such a method doesn't lead to higher dimensions as does Segre's.Rgdboer (talk) 00:54, 31 March 2011 (UTC)


 * Thank you for your reply. Two questions: 1. What exactly do you mean when you say they don't get off the complex plane? Could you elaborate? If the numbers have three independent units or axes, then they are off the plane in the ordinary sense, it would seem. Not to mention, say, 27 axes. 2. Also, aren't you restricting this article arbitrarily--since you admit that n = 3 is a multicomplex number, and the title of this article is "Multicomplex Number"? 8OYscLu9 (talk) 02:06, 1 April 2011 (UTC)

Answers: 1)The circle group lies in the complex plane. Whatever N you pick, the Nth roots of unity all lie on this circle, as does the finite group they generate. (2)Restricting only to the sense of the term "Multicomplex number" as used by Professors Segre and Price.Rgdboer (talk) 20:27, 1 April 2011 (UTC)


 * I understand what you are saying. I assure you I am not confusing the circle group with hypercomplex numbers. However, I come at hypercomplex numbers from another angle, and I see now that the numbers you describe are possibly the same ones that I am interested in, except that I concentrate on real coefficients. I therefore have another question for you: Do you consider the following to be a form of "multicomplex number"?


 * "Planar fourcomplex numbers and their operations can be represented by writing the
 * planar fourcomplex number (x, y, z, t) as u = x+ αy + βz + γt, where α, β and γ are bases
 * for which the multiplication rules are


 * α 2 = β, β 2 = − 1, γ 2 = − β, αβ = βα = γ, αγ = γα = − 1, βγ = γβ = − α. (3.273)


 * "Two planar fourcomplex numbers u = x + αy + βz + γt, u′ = x′ + αy′ + βz′ + γt′  are
 * equal, u = u′, if and only if x = x′, y = y′, z = z′, t = t′. If u = x + αy + βz + γt, u′  =

x′ + αy′ + βz′ + γt′ are planar fourcomplex numbers, the sum u + u′  and the product uu′ deﬁned above can be obtained by applying the usual algebraic rules to the sum (x+ αy + βz + γt) + (x′ + αy′ + βz′ + γt′) and to the product (x+ αy + βz + γt)(x′ + αy′ + βz′ + γt′), and grouping of the resulting terms," etc. ("3.3.1  Operations with planar fourcomplex numbers" from p. 90 of COMPLEX NUMBERS IN n DIMENSIONS by Silviu Olariu)


 * Does such a number, assuming real coefficients, reduce to some form of multicomplex number? Because I am especially interested in such numbers (whose bases are those roots of unity) and am wondering what to call them. 8OYscLu9 (talk) 21:34, 1 April 2011 (UTC)

Olariu. Thank you. His chapter 1 is about split-complex numbers. His tiny bibliography shows the material was developed in isolation. Need some time to read the book.Rgdboer (talk) 21:00, 2 April 2011 (UTC)CorrectionRgdboer (talk) 20:34, 5 April 2011 (UTC)


 * That would be great if you could do that. I am specifically interested in the four-dimensional number I cited above, and NOT any of his other examples. You will see that the bases α, β, etc., reduce to the fourth root of unity, and I am interested to know whether it reduces to a "multicomplex" number. If it does, could the same be said of other such numbers based on the third root, the fifth root, the 27th root, etc.? If it does not, what would you call the whole class of numbers based on the roots of unity? Do they have a name? 8OYscLu9 (talk) 21:00, 3 April 2011 (UTC)

The fourcomplex number you mention is isomorphic to tessarines, as the reviewer says (bicomplex number with &gamma; replaced by &minus;&gamma;). Corrected my earlier remark about chapter one. While the n-complex numbers of Olariu seem to be a "multicomplex" system, we cannot call them that without a reference. Probably an article is required with the title "N-complex number". This book is a major work on real algebras and goes beyond anything I have seen before. Since it dates from 2002 I regret not seeing it before. In particular, note the "trigonometric form" that Olariu comes to for each of his systems without any mention of Sophus Lie and the mathematical objects ascribed to him. Further, the cosexponential functions are a natural concept under-represented in the literature. The minimalist style of presentation becomes understandable once a reader knows the innovations involved.Rgdboer (talk) 20:34, 5 April 2011 (UTC)


 * It would seem to be different from the tessarines, assuming the Wikipedia article is correct. Notice that in Olariu's multiplication table, none of the bases squares to +1, whereas this is true for tessarines. 8OYscLu9 (talk) 21:04, 5 April 2011 (UTC)

Right you are. I regret the error. Let me try again. Note that &alpha; generates &beta;, together they generate &gamma;. So essentially &alpha; generates the other bases. Also note that &alpha; satisfies the polynomial $$X^4 +1 = 0$$. In abstract algebra this kind of an algebra is called a quotient ring R[X]/I where I is the ideal generated by the polynomial that &alpha; satisfies.Rgdboer (talk) 20:36, 6 April 2011 (UTC)


 * Yes. This description lays out the algebraic properties. What I am wondering is if there is a general term for the whole class of hypercomplex numbers generated by $$X^a + 1 = 0$$ where a is an integer and with real coefficients. Or if there is a term for the general class with any type of coefficient that includes this type of number. Perhaps there isn't one yet, but I am curious. 8OYscLu9 (talk) 21:13, 6 April 2011 (UTC)

Stepping back from the real field R to the field of rational numbers Q, there is a rich literature of algebra: the science of field extensions and algebraic number fields. The objects obtained are rupture fields or splitting fields of a polynomial in Q[X]. The lattice of field extensions corresponds to a lattice of Galois groups which describe the admissable permuations of the roots of the polynomial. Note the use of Q since it is a countable set and leaves room for lots of extensions without generating C. While these subjects are a canonical part of higher education in math (Galois theory), hypercomplex numbers are considered a traditional portal to study of algebra over a field, and treatment of particular instances is neglected. Olariu's book revitalizes the study and I don't know another source that gives names or comes close to Olariu's insight into the function theory of the structures he reveals.Rgdboer (talk) 19:35, 8 April 2011 (UTC)

Thank you very much for your insight. I will end this discussion now, since it threatens to drift into a discussion of the topic rather than the article qua article. You have been helpful to me, and I will keep checking the article for changes by you, the original posters, or anyone else. All the best, 8OYscLu9 (talk) 22:11, 8 April 2011 (UTC)