Multiplicity of Scator Roots and the Square Roots in $\mathbb{S}^{1+2}$

Year 2023, Issue: 42, 29 - 42, 31.03.2023

Manuel Fernandez-guasti

https://doi.org/10.53570/jnt.1188215

Abstract

This paper presents the roots of elliptic scator numbers in $\mathbb{S}^{1+n}$, which includes both the fundamental $2\pi$ symmetry and the $\pi$-pair symmetry for $n\geq2$. Here, the scator set $\mathbb{S}^{1+n}$ is a subset of $\mathbb{R}^{1+n}$ with the scator product and the multiplicative representation. These roots are expressed in terms of both additive (rectangular) and multiplicative (polar) variables. Additionally, the paper provides a comprehensive description of square roots in $\mathbb{S}^{1+2}$, which includes a geometrical representation in three-dimensional space that provides a clear visualization of the concept and makes it easier to understand and interpret. Finally, the paper handles whether the aspects should be further investigated.

Keywords

Roots, non-distributive algebras, hypercomplex numbers

Supporting Institution

Universidad Autonoma Metropolitana - Iztapalapa

References

• I. Sabadini, M. Shapiro, F. Sommen, Hypercomplex Analysis, Springer, London, 2009.
• M. Özdemir, The Roots of a Split Quaternion, Applied Mathematics Letters 22 (2) (2009) 258–263.
• A. Kobus, J. L. Cieslinski, On the Geometry of the Hyperbolic Scator Space in 1+2 Dimensions, Advances in Applied Clifford Algebras 27 (2) (2017) 1369–1386.
• S. J. Sangwine, N. L. Bihan, Quaternion Polar Representations with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form, Advances in Applied Clifford Algebras 20 (2010) 111-120.
• M. Fernandez-Guasti, Roots of Elliptic Scator Numbers, Axioms 10 (4) (2021) 321 20 pages.
• I. Niven, The Roots of a Quaternion, The American Mathematical Monthly 49 (6) (1942) 386–388.
• M. Fernandez-Guasti, Powers of Elliptic Scator Numbers, Axioms 10 (4) (2021) 250 23 pages.
• M. Fernandez-Guasti, Associativity in Scator Algebra and the Quantum Wavefunction Collapse, Universal Journal of Mathematics and Applications 1 (2) (2018) 80–88.