Multicomponent hybrid numbers: On algebraic properties and matrix representations of hybrid-hyperbolic numbers

Bahar Doğan Yazıcı and Murat Tosun
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 1, Pages 26–40
DOI: 10.7546/nntdm.2022.28.1.26-40
Full paper (PDF, 178 Kb)

Details

Authors and affiliations

Bahar Doğan Yazıcı
Department of Mathematics, Bilecik Seyh Edebali University
11200, Bilecik, Turkey

Murat Tosun
Department of Mathematics, Sakarya University
54050, Sakarya, Turkey

Manuscript history

• Received: 29 January 2021
• Revised: 22 December 2022
• Accepted: 18 January 2022
• Online First: 7 February 2022

Abstract

In this study, the hybrid-hyperbolic numbers are introduced. This number system is a more general form of the hybrid number system, which is an interesting number system, as well as a number system that includes multicomponent number systems (i.e., complex-hyperbolic, dual-hyperbolic and bihyperbolic numbers). In this paper, we give algebraic properties of hybrid-hyperbolic numbers. In addition, 2 × 2 and 4 × 4 hyperbolic matrix representations of hybrid-hyperbolic numbers are given and some properties of them are examined.

Keywords

• Hybrid numbers
• Complex-hyperbolic numbers
• Dual-hyperbolic numbers
• Bihyperbolic numbers
• Hybrid-hyperbolic numbers

2020 Mathematics Subject Classification

• 13A18
• 53A17

References

1. Motter, A. E., & Rosa, M. A. F. (1998). Hyperbolic Calculus. Advances in Applied Clifford Algebras, 8, 109–128.
2. Sobczyk, G. (1995). The hyperbolic number plan. The College Mathematics Journal, 26(4), 268–280.
3. Harkin, A. A., & Harkin, J. B. (2004). Geometry of Generalized Complex Numbers. Mathematics Magazine, 77(2), 118–129.
4. Kantor, I. L., & Solodovnikov, A. S. (1989). Hypercomplex Numbers, Springer-Verlag, New York.
5. Özdemir, M. (2018). Introduction to Hybrid Numbers. Advances in Applied Clifford Algebras, 28, Art. No. 11.
6. Majernik, V. (1996). Multicomponent Number Systems. Acta Physica Polonica A, 90(3), 491–498.
7. Cihan, A., Azak, A. Z., Güngör, M. A., & Tosun, M. (2019). A Study of Dual Hyperbolic Fibonacci and Lucas numbers. Analele stiintifice ale Universitatii Ovidius Constanta, 27(1), 35–48.
8. Akar, M., Yüce, S., & Şahin, S. (2018). On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers. Journal of Computer Science and Computational Mathematics, 8(1), 1–6.
9. Bilgin, M., & Ersoy, S. (2020). Algebraic Properties of Bihyperbolic Numbers. Advances in Applied Clifford Algebras, 30, Art. No. 13.
10. Fjelstad, P., & Gal, S. G. (1998). n-dimensional Hyperbolic Complex Numbers. Advances in Applied Clifford Algebras, 8, 47–68.
11. Messelmi, F., (2013). Dual Complex Numbers and Their Holomorphic Functions. Preprint. DOI: 10.5281/zenodo.22961.
12. Rochon, D., & Shapiro, M., (2004). On Algebraic Properties of Bicomplex and Hyperbolic numbers. Analele Universităţii din Oradea. Fascicola Matematică, 11, 71–110.