Bicomplex number

As a real algebra

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 k = ( 0 i i 0 ) ,   j = ( 0 1 1 0 ) , 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, 1991, 2008) (exchange j and −k in his multiplication table). 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.

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

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 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.

A tessarine is a hypercomplex number of the form

where i j = j i = k , i 2 = − 1 , 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.

Bicomplex numbers

In 1892 Corrado Segre introduced 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

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, 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 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).

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 ,   X Y − Y X . 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 ( X Y ) 2 + 1 ∈ A  note that

Now consider the alternative ideal B generated by X 2 + 1 ,   Y 2 + 1 ,   X Y − Y X . In this case one can prove ( X Y ) 2 − 1 ∈ B . The ring isomorphism R⟨X,Y⟩/A ≅ R⟨X,Y⟩/B involves a change of basis exchanging Y ↔ X Y .

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:

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 , … , u n ,   v 1 , v 2 , … , 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.