nLab tessarines

Redirected from "bicomplex number".
Contents

Context

Arithmetic

Algebra

Contents

Definition

The tessarines are an algebra over the real numbers. Every tessarine tt may be represented as

t=a 0+a 1i+a 2j+a 3k t = a_0 + a_1 i + a_2 j + a_3k

where the basis elements satisfy the following products:

×\timesijk
i-1k-j
jk+1i
k-ji-1

and conjugation t *=a 0a 1ia 2ja 3kt^* = a_0 - a_1 i - a_2 j - a_3k.

These are closely related to the quaternions, as the generators satisfy similarly-looking relations.

The tessarines are also referred to as bicomplex numbers since they may be obtained using the generalization of the Cayley-Dickson construction applied on the complex numbers.

References

  • Elena Luna-Elizarrarás, Michael Shapiro, Daniele Struppa (2013) Bicomplex Holomorphic Functions: the algebra, geometry and analysis of bicomplex numbers. Birkhauser ISBN 978-3-319-24868-4

  • Fidelis Zanetti de Castro, Marcos Eduardo Valle. A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks. Neural Networks 122, February 2020, Pages 54-67. (doi)

Last revised on November 3, 2023 at 05:26:17. See the history of this page for a list of all contributions to it.