Bicomplex number

In abstract algebra, a bicomplex number is a pair (w, z) of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate ${\displaystyle (w,z)^{*}=(w,-z)}$, and the product of two bicomplex numbers as

${\displaystyle (u,v)(w,z)=(uw-vz,uz+vw).}$

"Tessarine" redirects here. For real tessarines, see Split-complex number.

Then the bicomplex norm is given by

${\displaystyle (w,z)^{*}(w,z)=(w,-z)(w,z)=(w^{2}+z^{2},0),}$ a quadratic form in the first component.

The bicomplex numbers form a commutative algebra over C of dimension two, which is isomorphic to the direct sum of algebras C ⊕ 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 the Brahmagupta–Fibonacci identity. This property of the quadratic form of a bicomplex number indicates that these numbers form a composition algebra. In fact, bicomplex numbers arise at the binarion level of the Cayley–Dickson construction based on ℂ with form z².

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

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 over C of dimension two, and since C is of dimension two over R, the bicomplex numbers are an algebra over R of dimension four. In fact the real algebra is older than the complex one; it was labelled tessarines in 1848 while the complex algebra was not introduced until 1892.

A basis for the tessarine 4-algebra over R specifies z = 1 and z = −i, giving the matrices ${\displaystyle k={\begin{pmatrix}0&i\\i&0\end{pmatrix}},\quad \ j={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$, which multiply according to the table given. When the identity matrix is identified with 1, then a tessarine t = w + z j .

As commutative hypercomplex numbers, the tessarine algebra has been advocated by Clyde M. Davenport (1978,[1] 1991,[2] 2008[3]) (exchange j and −k in his multiplication table). In particular Davenport notes the utility of the isomorphic correspondence between the bicomplex numbers and the direct sum of a pair of complex planes. Tessarines have also been applied in digital signal processing.[4][5][6]

In 2009 mathematicians proved a fundamental theorem of tessarine algebra: a polynomial of degree n with tessarine coefficients has n² roots, counting multiplicity.[7]

History

The subject of multiple imaginary units was examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 in Philosophical Magazine, William Rowan Hamilton communicated a system multiplying according to the quaternion group. In 1848 Thomas Kirkman reported[8] on his correspondence with Arthur Cayley regarding equations on the units determining a system of hypercomplex numbers.

Tessarines

In 1848 James Cockle introduced the tessarines in a series of articles in Philosophical Magazine.[9]

A tessarine is a hypercomplex number of the form

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

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

Bicomplex numbers

In 1892 Corrado Segre introduced[10] bicomplex numbers in Mathematische Annalen, which form an algebra isomorphic to the tessarines.

Corrado Segre read W. R. Hamilton's Lectures on Quaternions (1853) and the works of W. K. Clifford. Segre used some of Hamilton's notation to develop his system of bicomplex numbers: Let h and i be elements that square to −1 and that commute. Then, presuming associativity of multiplication, the product hi must 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

${\displaystyle g=(1-hi)/2,\quad g'=(1+hi)/2}$   are idempotents.

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

The University of Kansas has contributed to the development of bicomplex analysis. In 1953, Ph.D. student James D. Riley's thesis "Contributions to the theory of functions of a bicomplex variable" was published in the Tohoku Mathematical Journal (2nd Ser., 5:132–165). In 1991 G. Baley Price published a book[11] on bicomplex numbers, multicomplex numbers, and their function theory. Professor Price also gives some history of the subject in the preface to his book. Another book developing bicomplex numbers and their applications is by Catoni, Bocaletti, Cannata, Nichelatti & Zampetti (2008).[12]

Quotient rings of polynomials

One comparison of bicomplex numbers and tessarines uses the polynomial ring R[X,Y], where XY = YX. The ideal ${\displaystyle A=(X^{2}+1,\ Y^{2}-1)}$ then provides a quotient ring representing tessarines. In this quotient ring approach, elements of the tessarines correspond to cosets with respect to the ideal A. Similarly, the ideal ${\displaystyle B=(X^{2}+1,\ Y^{2}+1)}$ produces a quotient representing bicomplex numbers.

A generalization of this approach uses the free algebra R⟨X,Y⟩ in two non-commuting indeterminates X and Y. Consider these three second degree polynomials ${\displaystyle X^{2}+1,\ Y^{2}-1,\ XY-YX}$. Let A be the ideal generated by them. Then the quotient ring R⟨X,Y⟩/A is isomorphic to the ring of tessarines.

To see that ${\displaystyle (XY)^{2}+1\in A}$  note that

${\displaystyle XY^{2}X=X(Y^{2}-1)X+(X^{2}+1)-1,}$ so that

${\displaystyle XY^{2}X+1=X(Y^{2}-1)X+(X^{2}+1)\in A.}$ But then

${\displaystyle XY(XY-YX)+XY^{2}X+1\in A.}$ as required.

Now consider the alternative ideal B generated by ${\displaystyle X^{2}+1,\ Y^{2}+1,\ XY-YX}$. In this case one can prove ${\displaystyle (XY)^{2}-1\in B}$. The ring isomorphism R⟨X,Y⟩/A ≅ R⟨X,Y⟩/B involves a change of basis exchanging ${\displaystyle Y\leftrightarrow XY}$.

Alternatively, suppose the field C of ordinary complex numbers is presumed given, and C[X] is the ring of polynomials in X with complex coefficients. Then the quotient C[X]/(X² + 1) is another presentation of bicomplex numbers.

Polynomial roots

Write ²C = C ⊕ C and represent elements of it by ordered pairs (u,v) of complex numbers. Since the algebra of tessarines T is isomorphic to ²C, the rings of polynomials T[X] and ²C[X] are also isomorphic, however polynomials in the latter algebra split:

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

In consequence, when a polynomial equation ${\displaystyle f(u,v)=(0,0)}$ in this algebra is set, it reduces to two polynomial equations on C. If the degree is n, then there are n roots for each equation: ${\displaystyle u_{1},u_{2},\dots ,u_{n},\ v_{1},v_{2},\dots ,v_{n}.}$ Any ordered pair ${\displaystyle (u_{i},v_{j})\!}$ from this set of roots will satisfy the original equation in ²C[X], so it has n² roots.

Due to the isomorphism with T[X], there is a correspondence of polynomials and a correspondence of their roots. Hence the tessarine polynomials of degree n also have n² roots, counting multiplicity of roots.

References

1. Davenport, Clyde M. (1978). An Extension of the Complex Calculus to Four Real Dimensions, with an Application to Special Relativity (M.S. thesis). Knoxville, Tennessee: University of Tennessee, Knoxville.
2. Davenport, Clyde M. (1991). A Hypercomplex Calculus with Applications to Special Relativity. Knoxville, Tennessee: University of Tennessee, Knoxville. ISBN 0-9623837-0-8.
3. Davenport, Clyde M. (2008). "Commutative Hypercomplex Mathematics". Archived from the original on 2 October 2015.
4. 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. IEEE. 52 (7): 2012–2031. doi:10.1109/TSP.2004.828901. ISSN 1941-0476.
5. Alfsmann, Daniel (4–8 September 2006). On families of 2^N dimensional hypercomplex algebras suitable for digital signal processing (PDF). 14th European Signal Processing Conference, Florence, Italy: EURASIP.CS1 maint: location (link)
6. Alfsmann, Daniel; Göckler, Heinz G. (2007). On Hyperbolic Complex LTI Digital Systems (PDF). EURASIP.
7. Poodiack, Robert D.; LeClair, Kevin J. (November 2009). "Fundamental theorems of algebra for the perplexes". The College Mathematics Journal. MAA. 40 (5): 322–335. doi:10.4169/074683409X475643. JSTOR 25653773.
8. Thomas Kirkman (1848) "On Pluquaternions and Homoid Products of n Squares", London and Edinburgh Philosophical Magazine 1848, p 447 Google books link
9. James Cockle in London-Dublin-Edinburgh Philosophical Magazine, series 3
   • 1848 On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra, 33:435–9.
   • 1849 On a New Imaginary in Algebra 34:37–47.
   • 1849 On the Symbols of Algebra and on the Theory of Tessarines 34:406–10.
   • 1850 On the True Amplitude of a Tessarine 36:290-2.
   • 1850 On Impossible Equations, on Impossible Quantities and on Tessarines 37:281–3.
   Links from Biodiversity Heritage Library.
10. 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: 413–467, doi:10.1007/bf01443559. (see especially pages 455–67)
11. G. Baley Price (1991) An Introduction to Multicomplex Spaces and Functions, Marcel Dekker ISBN 0-8247-8345-X
12. F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti. (2008) The Mathematics of Minkowski Space-Time with an Introduction to Commutative Hypercomplex Numbers, Birkhäuser Verlag, Basel ISBN 978-3-7643-8613-9