De Moivre and Euler Formulas for Hyper-Dual Numbers

Year 2025, Issue: 53, 47 - 53, 31.12.2025

İskender Öztürk , Hasan Çakır , Mustafa Özdemir

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

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

Dual numbers , hyper-dual numbers , De Moivre's formula , roots of hyper-dual numbers

References

• I. Niven, The roots of a quaternion, American Mathematical Monthly 49 (6) (1942) 386--388.
• M. Özdemir, The roots of a split quaternion, Applied Mathematics Letters 22 (2) (2009) 258--263.
• M. Özdemir, Finding n-th roots of a $2 \times 2$ real matrix using De Moivre’s formula, Advances in Applied Clifford Algebras 29 (1) (2019) 2.
• Ö. Bektaş, S. Yüce, De Moivre’s and Euler’s formulas for the matrices of octonions, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences 89 (1) (2019) 113–127.
• M. Özdemir, A. A. Ergin, Rotations with unit timelike quaternions in Minkowski 3-space, Journal of Geometry and Physics 56 (2) (2006) 322--336.
• İ. Öztürk, M. Özdemir, On geometric interpretations of split quaternions, Mathematical Methods in the Applied Sciences 46 (1) (2022) 408--422.
• E. Cho, Euler's formula and De Moivre's formula for quaternions, Missouri Journal of Mathematical Sciences 11 (2) (1999) 80--83.
• J. Fike, J. Alonso, The development of hyper-dual numbers for exact second-derivative calculations, 49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, 2011.
• J. Fike, S. Jongsma, J. Alonso, E. van der Weide, Optimization with gradient and hessian information calculated using hyper-dual numbers, 29th AIAA Applied Aerodynamics Conference, Honolulu, 2011.
• J. A. Fike, Multi-objective optimization using hyper-dual numbers, Doctoral Dissertation Stanford University (2013) Stanford.
• A. Cohen, M. Shoham, Application of hyper-dual numbers to rigid bodies equations of motion, Mechanism and Machine Theory 111 (2017) 76--84.
• A. Cohen, M. Shoham, Application of hyper-dual numbers to multibody kinematics, Journal of Mechanisms and Robotics 8 (1) (2016) 011015.
• A. Cohen, M. Shoham, Principle of transference -- An extension to hyper-dual numbers, Mechanism and Machine Theory 125 (2018) 101--110.
• S. Aslan, Kinematic applications of hyper-dual numbers, International Electronic Journal of Geometry 14 (2) (2021) 292--304.
• Ö. Bektaş, H. Çakır, Hyper dual number matrices, Maejo International Journal of Science and Technology 17 (3) (2023) 187--200.
• P. Rehner, G. Bauer, Application of generalized (hyper-) dual numbers in equation of state modeling, Frontiers in Chemical Engineering 3 (2021) 758090.