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

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

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 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. 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 ${\displaystyle \mathbb {C} }$ with norm 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.

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.^[1]