In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate
(w,z)*=(w,-z)
(u,v)(w,z)=(uw-vz,uz+vw).
Then the bicomplex norm is given by
(w,z)*(w,z)=(w,-z)(w,z)=(w2+z2,0),
The bicomplex numbers form a commutative algebra over C of dimension two that is isomorphic to the direct sum of algebras .
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
The general bicomplex number can be represented by the matrix
\begin{pmatrix}w&iz\ iz&w\end{pmatrix}
w2+z2
Bicomplex numbers feature two distinct imaginary 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 a hyperbolic unit.
| width=15 | × | width=15 | 1 | width=15 | i | width=15 | j | width=15 | k |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | i | j | k | |||||
| i | i | −1 | k | −j | |||||
| j | j | k | -1 | -i | |||||
| k | k | −j | -i | 1 |
A basis for the tessarine 4-algebra over R specifies z = 1 and z = −i, giving the matrices
k=\begin{pmatrix}0&i\ i&0\end{pmatrix}, j=\begin{pmatrix}0&1\ 1&0\end{pmatrix}
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.[1]
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+xi+yj+zk, w,x,y,z\inR
ij=ji=k, i2=-1, j2=+1.
t=w+yj
In a 1892 Mathematische Annalen paper, Corrado Segre introduced bicomplex numbers,[3] which form an algebra isomorphic to the tessarines.
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 is then the same as James Cockle's tessarines, represented using a different basis. Segre noted that elements
g=(1-hi)/2, g'=(1+hi)/2
The modern theory of composition algebras positions the algebra as a binarion construction based on another binarion construction, hence the bibinarions. 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 McCrimmon noted the simplification of nomenclature provided by the term binarion in his text A Taste of Jordan Algebras (2004).
Write 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:
| n | |
| \sum | |
| k=1 |
(ak,bk)(u,v)k =
| n | |
| \left({\sum | |
| k=1 |
aiuk},
| n | |
| \sum | |
| k=1 |
bkvk\right).
In consequence, when a polynomial equation
f(u,v)=(0,0)
u1,u2,...,un, v1,v2,...,vn.
(ui,vj)
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.
Bicomplex number appears as the center of CAPS (complexified algebra of physical space), which is Clifford algebra
Cl(3,C)
Tessarines have been applied in digital signal processing.[6] [7] [8]
Bicomplex numbers are employed in fluid mechanics. The use of bicomplex algebra reconciles two distinct applications of complex numbers: the representation of two-dimensional potential flows in the complex plane and the complex exponential function.[9]