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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

Then the bicomplex norm is given by

The bicomplex numbers form a commutative algebra over ℂ of dimension two.

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 with form z² at the unarion level.

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.

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