Bicomplex number

In abstract algebra, a tessarine or bicomplex number is a hypercomplex number in a commutative, associative algebra over real numbers with two imaginary units.

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[1] on his correspondence with Arthur Cayley regarding equations on the units determining a system of hypercomplex numbers.

In 1848 James Cockle introduced the tessarines in a series of articles in Philosophical Magazine.[2] A tessarine is a hypercomplex number of the form

\( t = w + x i + y j + z k, \quad w, x, y, z \in \mathbb{R} \)

where \( i j = j i = k, \quad i^2 = -1, \quad j^2 = +1 \) . 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 \( t = w + y j \ \) , also called split-complex numbers, which express the parametrization of the unit hyperbola.

In 1892 Corrado Segre introduced[3] bicomplex numbers in Mathematische Annalen, which form an algebra isomorphic to the tessarines (see section below). As commutative hypercomplex numbers, the tessarine algebra has been advocated by Clyde M. Davenport (1991, 2008) (exchange j and −k in his multiplication table).[4][5] Davenport has noted the isomorphism with the direct sum of the complex number plane with itself. Tessarines have also been applied in digital signal processing.[6][7][8] In 2009 mathematicians proved a fundamental theorem of tessarine algebra: a polynomial of degree n with tessarine coefficients has n2 roots, counting multiplicity.[9]

Linear representation

For the tessarine \(t = w + xi + yj + zk, \ \) note that \(t = (w + xi) + (y + zi) j \ \)since ij = k . The mapping

\( t \mapsto \begin{pmatrix} p & q \\ q & p \end{pmatrix}, \quad p = w + xi, \quad q = y + zi \)

is a linear representation of the algebra of tessarines as a subalgebra of 2 × 2 complex matrices. For instance, ik = i(ij) = (ii)j = −j in the linear representation is

\( \begin{pmatrix} i & 0 \\ 0 & i \end{pmatrix} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} . \)

Note that unlike most matrix algebras, this is a commutative algebra. This algebra has dimension two over the complex numbers with basis { 1, j }.

Bicomplex number

Corrado Segre read W. R. Hamilton's Lectures on Quaternions (1853) and the works of William Kingdon 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

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

When bicomplex numbers are expressed in terms of the basis { 1, h, i, −hi }, the equivalence between them and 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, a 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, emeritus professor G. Baley Price published a book[10] 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).[11]
Direct sum C ⊕ C

The direct sum of the complex field with itself is denoted C ⊕ C. The product of two elements (a \oplus b) and (\( c \oplus d) \) is \( a c \oplus b d \) in this direct sum algebra.

Proposition: The algebra of tessarines is isomorphic to C ⊕ C.

Proof: Every tessarine has an expression t = u + v j \ where u and v are complex numbers. Now if \(s = w + z j \ \) is another tessarine, their product is

\( t s = (u w + v z) + (u z + v w) j \).

The isomorphism mapping from tessarines to C ⊕ C is given by

\( t \mapsto (u+v) \oplus (u - v) , \quad s \mapsto (w + z) \oplus (w - z). \)

In C ⊕ C, the product of these images, according to the algebra-product of C ⊕ C indicated above, is

\( (u + v)(w + z) \oplus (u - v)(w - z). \)

This element is also the image of ts under the mapping into C ⊕ C. Thus the products agree, the mapping is a homomorphism; and since it is bijective, it is an isomorphism.
Conic quaternion / octonion / sedenion, bicomplex number

When w and z are both complex numbers

w :=~a + ib

z :=~c + id

(with a, b, c, d real) then t algebra is isomorphic to conic quaternions \( a + bi + c \varepsilon + d i_0 \), to bases \(\{ 1,~i,~\varepsilon ,~i_0 \} \), in the following identification:

\( 1 \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \qquad i \equiv \begin{pmatrix} i & 0 \\ 0 & i\end{pmatrix} \qquad \varepsilon \equiv \begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix} \qquad i_0 \equiv \begin{pmatrix} 0 & i \\ i & 0\end{pmatrix}. \)

They are also isomorphic to "bicomplex numbers" (from multicomplex numbers) to bases \{ 1,~i_1, i_2, j \} \) if one identifies:

\( 1 \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \qquad i_1 \equiv \begin{pmatrix} i & 0 \\ 0 & i\end{pmatrix} \qquad i_2 \equiv \begin{pmatrix} 0 & i \\ i & 0\end{pmatrix} \qquad j \equiv \begin{pmatrix} 0 & -1 \\ -1 & 0\end{pmatrix}. \) \)

Note that j in bicomplex numbers is identified with the opposite sign as j from above.

When w and z are both quaternions (to bases \( \{ 1,~i_1,~i_2,~i_3 \}) \), then t algebra is isomorphic to conic octonions; allowing octonions for w and z (to bases \( \{ 1,~i_1, \dots, ~i_7 \}) \) the resulting algebra is identical to conic sedenions.

Quotient rings of polynomials

One comparison of bicomplex numbers and tessarines uses the polynomial ring R[X,Y], where XY = YX. The ideal \( A = (X^2 + 1,\ Y^2 - 1) \) 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 \( B = (X^2 + 1,\ Y^2 + 1 ) \) 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 \( X^2 + 1,\ Y^2 - 1,\ XY - YX. \)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 \( (XY)^2 + 1 \in A \) note that

\( X Y^2 X = X(Y^2 - 1) X + (X^2 + 1) - 1 \), so that
\( X Y^2 X + 1 = X(Y^2 - 1)X + (X^2 + 1) \in A \). But then
\( XY(XY - YX) + X Y^2 X + 1 \in A \). as required.

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

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.

Algebraic properties

Tessarines with w and z complex numbers form a commutative and associative quaternionic ring (whereas quaternions are not commutative). They allow for powers, roots, and logarithms of \(j \equiv \varepsilon, which is a non-real root of 1 (see conic quaternions for examples and references). They do not form a field because the idempotents

\( \begin{pmatrix} z & \pm z \\ \pm z & z \end{pmatrix} \equiv z (1 \pm j) \equiv z (1 \pm \varepsilon) \)

have determinant / modulus 0 and therefore cannot be inverted multiplicatively. In addition, the arithmetic contains zero divisors

\(\begin{pmatrix} z & z \\ z & z \end{pmatrix} \begin{pmatrix} z & -z \\ -z & z \end{pmatrix} \equiv z^2 (1 + j )(1 - j) \equiv z^2 (1 + \varepsilon )(1 - \varepsilon) = 0. \)

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:

\( \sum_{k=1}^n (a_k , b_k ) (u , v)^k \quad = \quad \left({\sum_{k=1}^n a_i u^k} ,\quad \sum_{k=1}^n b_k v^k \right). \)

In consequence, when a polynomial equation f(u,v) = (0,0) in this algebra is set, it reduces to two polynomial equations on C. If the degree is n, then there are n roots for 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 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.

Notes and references

Thomas Kirkman (1848) "On Pluquaternions and Homoid Products of n Squares", London and Edinburgh Philosophical Magazine 1848, p 447 Google books link
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 Impossible Equations, on Impossible Quantities and on Tessarines 37:281–3.

Links from Biodiversity 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: 413–467, doi:10.1007/bf01443559. (see especially pages 455–67)
Clyde Davenport (1991) A Hypercomplex Calculus with Applications to Special Relativity ISBN 0-9623837-0-8
Clyde Davenport (2008) Commutative Hypercomplex Mathematics
Soo-Chang Pei, Ja-Han Chang & Jian-Jiun Ding (2004) "Commutative reduced biquaternions and their Fourier transform for signal and image processing", IEEE Transactions on Signal Processing 52:2012–31
Daniel Alfsmann (2006) On families of 2^N dimensional hypercomplex algebras suitable for digital signal processing, 14th European Signal Processing Conference, Florence, Italy
Daniel Alfsmann & Heinz G Göckler (2007) On Hyperbolic Complex LTI Digital Systems
Robert D. Poodiack & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", The College Mathematics Journal 40(5):322–35
G. Baley Price (1991) An introduction to multicomplex spaces and functions, Marcel Dekker ISBN 0-8247-8345-X
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

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