<abstract xmlns="http://www.w3.org/1999/xhtml">For vectors in <bold>E</bold>3 we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications.Based on properties of hyperbolic (Clifford) complex numbers, we prove that the resulting algebra 𝕋 is an associative algebra over a field and contains a subring isomorphic to hyperbolic complex numbers. Moreover, the algebra 𝕋 is isomorphic to direct product ℂ<italic>×</italic>ℝ, and so it contains a subalgebra isomorphic to the Gaussian complex plane.</abstract>

For vectors in E3 we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and hint some applications.Based on properties of hyperbolic (Clifford) complex numbers, we prove that the resulting algebra 𝕋 is an associative algebra over a field and contains a subring isomorphic to hyperbolic complex numbers. Moreover, the algebra 𝕋 is isomorphic to direct product ℂ×ℝ, and so it contains a subalgebra isomorphic to the Gaussian complex plane.

A Three Dimensional Modification of the Gaussian Number Field

Siedlce University of Natural Sciences and Humanities

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"A Three Dimensional Modification of the Gaussian Number Field"}

For vectors in E3 we introduce an associative, commutative and distributive multiplication. We describe the related algebraic and geometrical properties, and...

A Three Dimensional Modification of the Gaussian Number Field

Published Online: Nov 15, 2019

Page range: 63 - 76

Received: Oct 03, 2019

DOI: https://doi.org/10.2478/tmmp-2019-0020

Keywords
Normed field, three dimensions, factor ring, generalized complex numbers

© 2019 Ján Haluška et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

A Three Dimensional Modification of the Gaussian Number Field

Published Online: Nov 15, 2019

Page range: 63 - 76

Received: Oct 03, 2019

DOI: https://doi.org/10.2478/tmmp-2019-0020

KeywordsNormed field, three dimensions, factor ring, generalized complex numbers

© 2019 Ján Haluška et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Normed field, three dimensions, factor ring, generalized complex numbers