Bicomplex number

Inabstract algebra,abicomplex numberis a pair(w,z)ofcomplex numbersconstructed by theCayley–Dickson processthat defines the bicomplex conjugate $(w,z)^{*}=(w,-z)$ ,and the product of two bicomplex numbers as

(u,v)(w,z)=(uw-vz,uz+vw).

Then thebicomplex normis given by

(w,z)^{*}(w,z)=(w,-z)(w,z)=(w^{2}+z^{2},0),

aquadratic formin the first component.

The bicomplex numbers form a commutativealgebra overCof dimension two that isisomorphicto thedirect sum of algebrasC⊕C.

The product of two bicomplex numbers yields a quadratic form value that is the product of the individual quadratic forms of the numbers: a verification of this property of the quadratic form of a product refers to theBrahmagupta–Fibonacci identity.This property of the quadratic form of a bicomplex number indicates that these numbers form acomposition algebra.In fact, bicomplex numbers arise at the binarion level of the Cayley–Dickson construction based on $\mathbb {C}$ with norm z².

The general bicomplex number can be represented by the matrix ${\begin{pmatrix}w&iz\\iz&w\end{pmatrix}}$ ,which hasdeterminant $w^{2}+z^{2}$ .Thus, the composing property of the quadratic form concurs with the composing property of the determinant.

Bicomplex numbers feature two distinctimaginary units.Multiplication being associative and commutative, the product of these imaginary units must have positive one for its square. Such an element as this product has been called ahyperbolic unit.^[1]

As a real algebra

Tessarine multiplication
×	1	i	j	k
1	1	i	j	k
i	i	−1	k	−j
j	j	k	-1	-i
k	k	−j	-i	1

Bicomplex numbers form an algebra overCof dimension two, and sinceCis of dimension two overR,the bicomplex numbers are an algebra overRof dimension four. In fact the real algebra is older than the complex one; it was labelledtessarinesin 1848 while the complex algebra was not introduced until 1892.

Abasisfor the tessarine 4-algebra overRspecifiesz= 1 andz= −i,giving the matrices $k={\begin{pmatrix}0&i\\i&0\end{pmatrix}},\quad \ j={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$ ,which multiply according to the table given. When the identity matrix is identified with 1, then a tessarinet=w+z j.

History

The subject of multipleimaginary unitswas examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 inPhilosophical Magazine,William Rowan Hamiltoncommunicated a system multiplying according to thequaternion group.In 1848Thomas Kirkmanreported on his correspondence withArthur Cayleyregarding equations on the units determining a system of hypercomplex numbers.^[2]

Tessarines

In 1848James Cockleintroduced the tessarines in a series of articles inPhilosophical Magazine.^[3]

Atessarineis a hypercomplex number of the form

t=w+xi+yj+zk,\quad w,x,y,z\in \mathbb {R}

where $ij=ji=k,\quad i^{2}=-1,\quad j^{2}=+1.$ Cockle used tessarines to isolate the hyperbolic cosine series and the hyperbolic sine series in the exponential series. He also showed howzero divisorsarise in tessarines, inspiring him to use the term "impossibles". The tessarines are now best known for their subalgebra ofreal tessarines $t=w+yj\$ , also calledsplit-complex numbers,which express the parametrization of theunit hyperbola.

Bicomplex numbers

In a 1892Mathematische Annalenpaper,Corrado Segreintroducedbicomplex numbers,^[4]which form an algebra isomorphic to the tessarines.^[5]

Segre readW. R. Hamilton'sLectures on Quaternions(1853) and the works ofW. K. Clifford.Segre used some of Hamilton's notation to develop his system ofbicomplex numbers:Lethandibe elements that square to −1 and that commute. Then, presumingassociativityof multiplication, the producthimust square to +1. The algebra constructed on the basis{ 1,h,i,hi}is then the same as James Cockle's tessarines, represented using a different basis. Segre noted that elements

g=(1-hi)/2,\quad g'=(1+hi)/2

areidempotents.

When bicomplex numbers are expressed in terms of the basis{ 1,h,i,−hi},their equivalence with tessarines is apparent, particularly if the vectors in this basis are reordered as{ 1,i,−hi,h}.Looking at the linear representation of theseisomorphicalgebras shows agreement in the fourth dimension when the negative sign is used; consider the sample product given above under linear representation.

Bibinarions

The modern theory ofcomposition algebraspositions the algebra as a binarion construction based on another binarion construction, hence thebibinarions.^[6]The unarion level in the Cayley-Dickson process must be a field, and starting with the real field, the usual complex numbers arises as division binarions, another field. Thus the process can begin again to form bibinarions.Kevin McCrimmonnoted the simplification of nomenclature provided by the termbinarionin his textA Taste of Jordan Algebras(2004).

Polynomial roots

Write²C=C⊕Cand represent elements of it by ordered pairs (u,v) of complex numbers. Since the algebra of tessarinesTis isomorphic to²C,therings of polynomialsT[X] and²C[X] are also isomorphic, however polynomials in the latter algebra split:

\sum _{k=1}^{n}(a_{k},b_{k})(u,v)^{k}\quad =\quad \left({\sum _{k=1}^{n}a_{i}u^{k}},\quad \sum _{k=1}^{n}b_{k}v^{k}\right).

In consequence, when a polynomial equation $f(u,v)=(0,0)$ in this algebra is set, it reduces to two polynomial equations onC.If the degree isn,then there arenrootsfor each equation: $u_{1},u_{2},\dots,u_{n},\ v_{1},v_{2},\dots,v_{n}.$ Any ordered pair $(u_{i},v_{j})\!$ from this set of roots will satisfy the original equation in²C[X], so it hasn²roots.^[7]

Due to the isomorphism withT[X], there is a correspondence of polynomials and a correspondence of their roots. Hence the tessarine polynomials of degreenalso haven²roots, countingmultiplicity of roots.

Applications

Bicomplex number appears as the center of CAPS (complexifiedalgebra of physical space), which is Clifford algebra $Cl(3,\mathbb {C} )$ .^[8]Since the linear space of CAPS can be viewed as the four dimensional space span { $1,e_{1},e_{2},e_{3}$ } over { $1,i,k,j$ }.

Tessarines have been applied indigital signal processing.^[9]^[10]^[11]

Bicomplex numbers are employed in fluid mechanics. The use of bicomplex algebra reconciles two distinct applications of complex numbers: the representation oftwo-dimensional potential flowsin the complex plane and thecomplex exponential function.^[12]

References

^M.E. Luna-Elizarrarás, M. Shapiro, D.C. Struppa (2013)Bicomplex Holomorphic Functions: the algebra, geometry and analysis of bicomplex numbers,page 6,Birkhauser ISBN 978-3-319-24868-4
^Thomas Kirkman(1848) "On Pluquaternions and Homoid Products ofnSquares ",London and Edinburgh Philosophical Magazine1848, p 447Google books link
^
James Cocklein London-Dublin-EdinburghPhilosophical Magazine,series 3
- 1848On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra,33:435–9.
- 1849On a New Imaginary in Algebra34:37–47.
- 1849On the Symbols of Algebra and on the Theory of Tessarines34:406–10.
- 1850On the True Amplitude of a Tessarine36:290-2.
- 1850On Impossible Equations, on Impossible Quantities and on Tessarines37:281–3.
Links fromBiodiversity Heritage Library.
^ Segre, Corrado(1892),"Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici"[The real representation of complex elements and hyperalgebraic entities],Mathematische Annalen,40(3): 413–467,doi:10.1007/bf01443559,S2CID 121807474,archived fromthe originalon 2013-09-12,retrieved2013-09-12(see especially pages 455–67)
^Abstract Algebra/Polynomial Ringsat Wikibooks
^Associative Composition Algebra/Bibinarionsat Wikibooks
^Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes",The College Mathematics Journal40(5):322–35.
^Baylis, W.E.; Kiselica, J.D. (2012).The Complex Algebra of Physical Space: A Framework for Relativity.Adv. Appl. Clifford Algebras.Vol. 22. SpringerLink. pp. 537–561.
^Pei, Soo-Chang; Chang, Ja-Han; Ding, Jian-Jiun (21 June 2004)."Commutative reduced biquaternions and their Fourier transform for signal and image processing"(PDF).IEEE Transactions on Signal Processing.52(7). IEEE: 2012–2031.doi:10.1109/TSP.2004.828901.ISSN 1941-0476.S2CID 13907861.
^Alfsmann, Daniel (4–8 September 2006).On families of 2^Ndimensional hypercomplex algebras suitable for digital signal processing(PDF).14th European Signal Processing Conference, Florence, Italy: EURASIP. Archived fromthe original(PDF)on 16 July 2011.Retrieved18 February2010.{{cite conference}}:CS1 maint: location (link)
^Alfsmann, Daniel; Göckler, Heinz G. (2007).On Hyperbolic Complex LTI Digital Systems(PDF).EURASIP.
^Kleine, Vitor G.; Hanifi, Ardeshir; Henningson, Dan S. (2022)."Stability of two-dimensional potential flows using bicomplex numbers".Proc. R. Soc. A.478(20220165).arXiv:2203.05857.Bibcode:2022RSPSA.47820165K.doi:10.1098/rspa.2022.0165.PMC9185835.PMID 35702595.

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