Bicomplex Number

Bicomplex Number

In mathematics, a tessarine is a hypercomplex number of the form

where

The tessarines are best known for their subalgebra of real tessarines, also called split-complex numbers, which express the parametrization of the unit hyperbola. James Cockle introduced the tessarines in 1848 in a series of articles in Philosophical Magazine. 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."

In 1892 Corrado Segre introduced bicomplex numbers in Mathematische Annalen, which form an algebra equivalent to the tessarines (see section below). As commutative hypercomplex numbers, the tessarine algebra has been advocated by Clyde M. Davenport (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 (see Pei (2004) and Alfsmann (2006,7). In 2009 mathematicians proved a fundamental theorem of tessarine algebra: a polynomial of degree n with tessarine coefficients has n2 roots, counting multiplicity.