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
, and the product of two bicomplex numbers as Then the bicomplex norm is given by The bicomplex numbers form a commutative algebra over C of dimension two that 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.
In fact, bicomplex numbers arise at the binarion level of the Cayley–Dickson construction based on
The general bicomplex number can be represented by the matrix
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.
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
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.
[2] In 1848 James Cockle introduced the tessarines in a series of articles in Philosophical Magazine.
He also showed how zero divisors arise in tessarines, inspiring him to use the term "impossibles".
, also called split-complex numbers, which express the parametrization of the unit hyperbola.
In an 1892 Mathematische Annalen paper, Corrado Segre introduced bicomplex numbers,[4] which form an algebra isomorphic to the tessarines.
[5] 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.
Segre noted that elements 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 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 modern theory of composition algebras positions the algebra as a binarion construction based on another binarion construction, hence the bibinarions.
Kevin McCrimmon noted the simplification of nomenclature provided by the term binarion in his text A Taste of Jordan Algebras (2004).
Write 2C = C ⊕ C and represent elements of it by ordered pairs (u,v) of complex numbers.
Bicomplex number appears as the center of CAPS (complexified algebra of physical space), which is Clifford algebra
Tessarines have been applied in digital 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 of two-dimensional potential flows in the complex plane and the complex exponential function.