Parabolic Numbers: A New Perspective

Abstract

Thus far, many studies have been conducted on $p$-complex numbers. Depending on the sign of $p$, there are three cases: hyperbolic, dual, and elliptic. In the literature, dual numbers are called parabolic numbers, but they do not parameterize parabolas. Therefore, a number system that parameterizes parabolas is worth studying. This paper defines $p$ as a function of the coordinate $y$ and obtains a number system named parabolic numbers whose circles are parabolas. These parabolic numbers complete the set of number systems where circles are conic sections. Finally, this paper discusses the prospect of further research.

Keywords

• Parabolic numbers
• $p$-complex numbers
• coordinate dependence

References

1. M. E. Peskin, D. V. Schroeder, An introduction to quantum field theory, CRC Press, Boca Raton, 2018.
2. D. Dürr, S. Teufel, Bohmian mechanics: The physics and mathematics of quantum theory, Springer Berlin, Heidelberg, 2009.
3. D. J. Griffiths, D. F. Schroeter, Introduction to quantum mechanics, 3rd Edition, Cambridge University Press, Cambridge, 2018.
4. I. M. Yaglom, Complex numbers in geometry, Academic Press, New York, 1968.
5. F. S. Dündar, A use of elliptic complex numbers in Newtonian gravity, Advances in Applied Clifford Algebras 32 (2022) Article Number 20 7 pages.
6. W. D. Richter, On lp-complex numbers, Symmetry 12 (6) (2020) 877 9 pages.
7. W. D. Richter, Short remark on (p1, p2, p3)-complex numbers, Symmetry 16 (1) (2024) 9 15 pages.
8. Y. Kulaç, M. Tosun, Some equations on p-complex Fibonacci numbers, AIP Conference Proceedings 1926 (1) (2018) 020024 6 pages.

9. J. A. Shuster, J. Köplinger, Elliptic complex numbers with dual multiplication, Applied Mathematics and Computation 216 (12) (2010) 3497–3514.
10. M. A. Güngör, O. Tetik, De-Moivre and Euler formulae for dual-complex numbers, Universal Journal of Mathematics and Applications 2 (3) (2019) 126–129.
11. N. Gürses, G. Y. Şentürk, S. Yüce, A study on dual-generalized complex and hyperbolic-generalized complex numbers, Gazi University Journal of Science 34 (1) (2021) 180–194.
12. K. E. Özen, On the trigonometric and p-trigonometric functions of elliptical complex variables, Communications in Advanced Mathematical Sciences 3 (3) (2020) 143–154.
13. N. Gürses, S. Y¨uce, One-parameter planar motions in generalized complex number plane CJ , Advances in Applied Clifford Algebras 25 (4) (2015) 889–903.
14. N. Gürses, M. Akbiyik, S. Yüce, One-parameter homothetic motions and Euler-Savary formula in generalized complex number plane CJ , Advances in Applied Clifford Algebras 26 (2016) 115–136.
15. F. Catoni, R. Cannata, V. Catoni, P. Zampetti, Two-dimensional hypercomplex numbers and related trigonometries and geometries, Advances in Applied Clifford Algebras 14 (1) (2004) 47– 68.
16. A. A. Harkin, J. B. Harkin, Geometry of generalized complex numbers, Mathematics Magazine 77 (2) (2004) 118–129. Journal of New Theory 49 (2024) 7-15 / Parabolic Numbers: A New Perspective 15
17. F. Catoni, R. Cannata, V. Catoni, P. Zampetti, N-dimensional geometries generated by hypercomplex numbers, Advances in Applied Clifford Algebras 15 (1) (2005) 1–25.
18. I. Kantor, A. Solodovnikov, Hypercomplex numbers: An elementary introduction to algebras, Springer, New York, 1989.
19. I. M. Yaglom, A simple non-Euclidean geometry and its physical basis: An elementary account of Galilean geometry and the Galilean principle of relativity, Springer, New York, 1979.
20. F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti, The mathematics of Minkowski space-time: With an introduction to commutative hypercomplex numbers, Birkhäuser Verlag, Basel, 2008.
21. M. Jafari, Y. Yayli, Generalized quaternions and their algebraic properties, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics 64 (1) (2015) 15–27.