De Moivre and Euler Formulas for Hyper-Dual Numbers

İskender Öztürk; Hasan Çakır; Mustafa Özdemir

doi:10.53570/jnt.1791828

De Moivre and Euler Formulas for Hyper-Dual Numbers

Abstract

The purpose of this study is to extend classical De Moivre and Euler formulas to the algebra of hyper-dual numbers and to investigate their implications for powers and roots. Hyper-dual numbers form a commutative ring with nilpotent elements that enable exact propagation of first- and second-order differentials. In addition to the fundamental operations, including conjugation, inversion, and Taylor expansion, it presents that every nonzero hyper-dual number admits a multiplicative normal form of the type $a(1+\theta_{1}\varepsilon)(1+\widehat{\theta}\,\varepsilon^{\ast})$. Based on this representation, Euler- and logarithm-type identities are derived, together with a general power formula valid for all integers. Using this framework, the existence and structure of $n$th roots are characterized: when the scalar part is positive and $n$ is even, two distinct roots occur; when $n$ is odd, a unique root exists; and when the scalar part vanishes, nilpotent root families appear in the quadratic case. Illustrative examples are provided to demonstrate the computation of roots and to verify consistency with the hyper-dual Taylor calculus. The findings extend known quaternionic and split-quaternionic results to the hyper-dual setting, contributing tools that combine symbolic manipulation of powers and roots with exact first- and second-order derivative propagation. These tools have potential applications in geometry, kinematics, and automatic differentiation.

Keywords

References

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Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

İskender Öztürk
0000-0001-5674-8219
Türkiye

Hasan Çakır ^*
0000-0003-4317-7968
Türkiye

Mustafa Özdemir
0000-0002-1359-4181
Türkiye

Publication Date

December 31, 2025

Submission Date

September 26, 2025

Acceptance Date

November 20, 2025

Published in Issue

Year 2025 Number: 53

DOI

https://doi.org/10.53570/jnt.1791828

IZ

https://izlik.org/JA83HN44PN

Cite

RIS / Bibtex

APA

Öztürk, İ., Çakır, H., & Özdemir, M. (2025). De Moivre and Euler Formulas for Hyper-Dual Numbers. Journal of New Theory, 53, 47-53. https://doi.org/10.53570/jnt.1791828

AMA

1.Öztürk İ, Çakır H, Özdemir M. De Moivre and Euler Formulas for Hyper-Dual Numbers. JNT. 2025;(53):47-53. doi:10.53570/jnt.1791828

Chicago

Öztürk, İskender, Hasan Çakır, and Mustafa Özdemir. 2025. “De Moivre and Euler Formulas for Hyper-Dual Numbers”. Journal of New Theory, nos. 53: 47-53. https://doi.org/10.53570/jnt.1791828.

EndNote

Öztürk İ, Çakır H, Özdemir M (December 1, 2025) De Moivre and Euler Formulas for Hyper-Dual Numbers. Journal of New Theory 53 47–53.

IEEE

[1]İ. Öztürk, H. Çakır, and M. Özdemir, “De Moivre and Euler Formulas for Hyper-Dual Numbers”, JNT, no. 53, pp. 47–53, Dec. 2025, doi: 10.53570/jnt.1791828.

ISNAD

Öztürk, İskender - Çakır, Hasan - Özdemir, Mustafa. “De Moivre and Euler Formulas for Hyper-Dual Numbers”. Journal of New Theory. 53 (December 1, 2025): 47-53. https://doi.org/10.53570/jnt.1791828.

JAMA

1.Öztürk İ, Çakır H, Özdemir M. De Moivre and Euler Formulas for Hyper-Dual Numbers. JNT. 2025;:47–53.

MLA

Öztürk, İskender, et al. “De Moivre and Euler Formulas for Hyper-Dual Numbers”. Journal of New Theory, no. 53, Dec. 2025, pp. 47-53, doi:10.53570/jnt.1791828.

Vancouver

1.İskender Öztürk, Hasan Çakır, Mustafa Özdemir. De Moivre and Euler Formulas for Hyper-Dual Numbers. JNT. 2025 Dec. 1;(53):47-53. doi:10.53570/jnt.1791828