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Journal articles on the topic 'Hypercomplex numbers'

Author: Grafiati

Published: 29 July 2024

Last updated: 25 July 2025

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1

Quadling, Douglas, I. L. Kantor, A. S. Solodovnikov, and A. Shenitzer. "Hypercomplex Numbers." Mathematical Gazette 74, no. 470 (1990): 399. http://dx.doi.org/10.2307/3618163.

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2

Votiakova, Lesia, and Lіudmila Nakonechna. "The Normed Algebra of Binary Numbers." Mathematical and computer modelling. Series: Physical and mathematical sciences 26 (December 26, 2024): 5–19. https://doi.org/10.32626/2308-5878.2024-26.5-19.

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The richness of the theory of functions of a complex variable, the effectiveness of its methods have always served as a stimulus and a source of ideas when constructing a theory of the function of a hypercomplex variable. It should be noted that hypercomplex number systems are an extension of the field of complex numbers. Modern hypercomplex studies can be divided into algebraic and analytical; the latter are often called hypercomplex analysis in the broad sense. Hypercomplex systems are an effective tool in mathematical modeling that allows representing complex multidimensional data and opera

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3

Ibrayev, Alpamys T. "Method for Constructing a Commutative Algebra of Hypercomplex Numbers." Symmetry 15, no. 9 (2023): 1652. http://dx.doi.org/10.3390/sym15091652.

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Until now, it was believed that, unlike real and complex numbers, the construction of a commutative algebra of quaternions or octonions with division over the field of real numbers is impossible in principle. No one questioned the existing theoretical assertion that quaternions, octonions, and other hypercomplex numbers cannot have the commutativity property. This article demonstrates the following for the first time: (1) the possibility of constructing a normed commutative algebra of quaternions and octonions with division over the field of real numbers; (2) the possibility of constructing a

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4

Gu, Ying-Qiu. "Clifford Algebras, Hypercomplex Numbers and Nonlinear Equations in Physics." Geometry, Integrability and Quantization 25 (2023): 47–72. http://dx.doi.org/10.7546/giq-25-2023-47-72.

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Hypercomplex number systems are vector algebras with the definition of multiplication and division of vectors, satisfying the associativity and distributive law. In this paper, some new types of hypercomplex numbers and their fundamental properties are introduced, the Clifford algebra formalisms of hydrodynamics and gauge field equations are established, and some novel consistent conditions helpful to understand the properties of solutions to nonlinear physical equations are derived. The coordinate transformation and covariant derivatives of hypercomplex numbers are also discussed. The basis e

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5

Chaitin-Chatelin, F., and T. Meškauskas. "Computation with hypercomplex numbers." Nonlinear Analysis: Theory, Methods & Applications 47, no. 5 (2001): 3391–400. http://dx.doi.org/10.1016/s0362-546x(01)00454-0.

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6

Gu, Ying-Qiu. "Hypercomplex Numbers and Roots of Algebraic Equation." Journal of Geometry and Symmetry in Physics 64 (2022): 9–22. http://dx.doi.org/10.7546/jgsp-64-2022-9-22.

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By means of hypercomplex numbers, in this paper we discuss algebraic equations and obtain some interesting relations. A structure equation $A^2=nA$ of a group is derived. The matrix representation of a group constitutes the basis elements of a hypercomplex number system. By a canonical real matrix representation of a cyclic group, we define the cyclic number system, which is exactly the solution space of the higher order algebraic equations, and thus can be used to solve the roots of algebraic equations. Hypercomplex numbers are linear algebras with definition of multiplication and division, s

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7

Labunets, V. G., E. V. Kokh, and E. Ostheimer. "ALGEBRAIC MODELS AND METHODS OF COMPUTER IMAGE PROCESSING. PART 1. MULTIPLET MODELS OF MULTICHANNEL IMAGES." Computer Optics 42, no. 1 (2018): 84–95. http://dx.doi.org/10.18287/2412-6179-2018-42-1-84-95.

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We present a new theoretical framework for multichannel image processing using commutative hypercomplex algebras. Hypercomplex algebras generalize the algebras of complex numbers. The main goal of the work is to show that hypercomplex algebras can be used to solve problems of multichannel (color, multicolor, and hyperspectral) image processing in a natural and effective manner. In this work, we suppose that the animal brain operates with hypercomplex numbers when processing multichannel retinal images. In our approach, each multichannel pixel is considered not as an K–D vector, but as an K–D h

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8

Da¸sdemir, A. "On Hadamard Product of Hypercomplex Numbers." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 104, no. 4 (2021): 68–73. http://dx.doi.org/10.31489/2021m4/68-73.

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Certain product rules take various forms in the set of hypercomplex numbers. In this paper, we introduce a new multiplication form of the hypercomplex numbers that will be called «the Hadamard product», inspired by the analogous product in the real matrix space, and investigate some algebraic properties of that, including the norm of inequality. In particular, we extend our new definition and its applications to the complex matrix theory.

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9

Alpay, Daniel, and Ilwoo Cho. "Operators induced by certain hypercomplex systems." Opuscula Mathematica 43, no. 3 (2023): 275–333. http://dx.doi.org/10.7494/opmath.2023.43.3.275.

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In this paper, we consider a family \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\) of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations \(\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}\) of the hypercomplex system \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\), and study the realizations \(\pi_{t}(h)\) of hypercomplex numbers \(h \in \mathbb{H}_{t}\), as \((2\times 2)\)-matrices acting on \(\mathbb{C}^{2}\), for an arbitrarily fixed scale \(t\in\mathbb{R}\). Algebraic, operator-theoreti

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10

Kisil, Vladimir V. "Induced Representations and Hypercomplex Numbers." Advances in Applied Clifford Algebras 23, no. 2 (2012): 417–40. http://dx.doi.org/10.1007/s00006-012-0373-1.

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11

Cheng, Daizhan, Zhengping Ji, Jun-e. Feng, Shihua Fu, and Jianli Zhao. "Perfect hypercomplex algebras: Semi-tensor product approach." Mathematical Modelling and Control 1, no. 4 (2021): 177–87. http://dx.doi.org/10.3934/mmc.2021017.

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<abstract><p>The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebras (PHAs) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product (STP) of matrices are reviewed. The zero sets are defined for non-invertible hypercomplex numbers in a given PHA, and characteristic functions are proposed for calculating zero sets. Then PHA of various dimensions are considered. First, classification of $ 2 $-dimensional PHAs are investigated. Second, all the $ 3 $-dimensional PHAs are obtained and the corres

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12

SUNDHEIM, PAUL. "A MULTIPLICATIVE DETERMINANT FOR 2m-DIMENSIONAL MATRICES." Journal of Algebra and Its Applications 13, no. 01 (2013): 1350067. http://dx.doi.org/10.1142/s0219498813500679.

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A multiplication for a specific nested collection of multidimensional matrices is defined by association with a system of n = 2m-dimensional hypercomplex numbers. A totally symmetric and multiplicative determinant is then derived from the system which extends the Cayley hyperdeterminant to these higher dimensions. The determinant is related to the zero divisors of the system of hypercomplex numbers. Properties of the determinant are then discussed.

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13

Catoni, Francesco, Roberto Cannata, Enrico Nichelatti, and Paolo Zampetti. "Commutative hypercomplex numbers and functions of hypercomplex variable: a matrix study." Advances in Applied Clifford Algebras 15, no. 2 (2005): 183–212. http://dx.doi.org/10.1007/s00006-005-0011-2.

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14

SUNDHEIM, PAUL. "AN INFINITE SYSTEM OF HYPERCOMPLEX NUMBERS." Journal of Algebra and Its Applications 11, no. 06 (2012): 1250117. http://dx.doi.org/10.1142/s0219498812501174.

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A system of hypercomplex numbers is defined for each dimension that is a power of 2. In each of these dimensions, the numbers are associative and commutative but their advantages lie in the ease of multiplication, how closely their properties mirror the properties of the complex numbers in 2 dimensions as well as the location and properties of the zero divisors. In addition, the basis forms a group under multiplication providing a multiplication table that relates all of the roots of –1 and real and non-real roots of 1.

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15

Brewer, Sky. "Projective Cross-ratio on Hypercomplex Numbers." Advances in Applied Clifford Algebras 23, no. 1 (2012): 1–14. http://dx.doi.org/10.1007/s00006-012-0335-7.

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16

Pulver, Sandra. "Quaternions: The hypercomplex number system." Mathematical Gazette 92, no. 525 (2008): 431–36. http://dx.doi.org/10.1017/s0025557200183639.

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Are there solutions of the equation x2 + 1 = 0 ? Carl Fredrich Gauss (1777–1855) conjectured that there was a solution and that it was the square root of - 1 . But since the squares of all real numbers, positive or negative, are positive, Gauss introduced a fanciful idea. His solution to this equation was , which he named i. He integrated i with the real numbers to form a set known as , the complex numbers, where each element in that set was of the form a + bi, where a, . Gauss illustrated this on a graph, the horizontal axis became the real axis and represented the real coefficient, while the

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17

Valkova-Jarvis, Zlatka, Maria Nenova, and Dimitriya Mihaylova. "Hypercomplex Numbers—A Tool for Enhanced Efficiency and Intelligence in Digital Signal Processing." Mathematics 13, no. 3 (2025): 504. https://doi.org/10.3390/math13030504.

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Mathematics is the wide-ranging solid foundation of the engineering sciences which ensures their progress by providing them with its unique toolkit of rules, methods, algorithms and numerical systems. In this paper, an overview of the numerical systems that have currently found an application in engineering science and practice is offered, while also mentioning those systems that still await full and comprehensive applicability, recognition, and acknowledgment. Two possible approaches for representing hypercomplex numbers are proposed—based on real numbers and based on complex numbers. This ma

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18

Kalinovsky, Ya A., Yu E. Boyarinova, Ya V. Khitsko, and A. S. Sukalo. "Use of Methods for Generating Isomorphic Hypercomplex Number Systems to Increase the Efficiency of Multiplying Hypercomplex Numbers." Èlektronnoe modelirovanie 40, no. 5 (2018): 27–40. http://dx.doi.org/10.15407/emodel.40.05.027.

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19

Kisil, Vladimir V. "Symmetry, Geometry and Quantization with Hypercomplex Numbers." Geometry, Integrability and Quantization 18 (2017): 11–76. http://dx.doi.org/10.7546/giq-18-2017-11-76.

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20

Hase-Liu, Matthew, and Adam Sheffer. "Sum–product phenomena for planar hypercomplex numbers." European Journal of Combinatorics 89 (October 2020): 103162. http://dx.doi.org/10.1016/j.ejc.2020.103162.

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21

Catoni, Francesco, Roberto Cannata, Vincenzo Catoni, and Paolo Zampetti. "N-dimensional geometries generated by hypercomplex numbers." Advances in Applied Clifford Algebras 15, no. 1 (2005): 1–25. http://dx.doi.org/10.1007/s00006-005-0001-4.

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22

BERKOVICH, Y., and A. SHENKMAN. "HYPERNION NUMBERS AND THEIR USE IN THE ANALYSIS OF NETWORKS DRIVEN BY NONSINUSOIDAL SOURCES." Journal of Circuits, Systems and Computers 13, no. 01 (2004): 65–76. http://dx.doi.org/10.1142/s0218126604001192.

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A method of using hypercomplex numbers for the analysis of linear electric circuits with nonsinusoidal voltages and currents has been proposed. Similar to the complex number method for circuits with sinusoidal voltages and currents, the proposed method reduces the analysis of nonsinusoidal circuits to the analysis of direct-current circuits. A special system of hypercomplex numbers, called hypernions, has been created in order to obtain a new efficient method for analyzing nonsinusoidal networks. This system is interesting in that it expands the concept of numbers and attaches meaning to equat

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23

А.В., Коганов. "Лучевые числа и алгебры". Труды НИИСИ РАН 9, № 1 (2019): 99–107. http://dx.doi.org/10.25682/niisi.2019.1.0013.

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В статье рассмотрен новый подход к теории гиперкомплексных чисел, при котором отри-цательные числа рассматриваются как комплексное расширение неотрицательных чисел. Такие алгебры, названные лучевыми, обладают неожиданными свойствами, и классические комплексные и гиперкомплекс-ные алгебры получаются из них путем факторизации. В частности, будет показано, что в лучевых алгебрах имеется несколько различных операций сопряжения. Некоторые из этих операций могут быть использованы в математической и теоретической физике. Подробно рассмотрен случай модификации алгебры комплексных чисел The paper consi

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24

Cação, Isabel, Maria Irene Falcão, and Helmuth Malonek. "Hypercomplex Polynomials, Vietoris’ Rational Numbers and a Related Integer Numbers Sequence." Complex Analysis and Operator Theory 11, no. 5 (2017): 1059–76. http://dx.doi.org/10.1007/s11785-017-0649-5.

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25

Ibrayev, A., A. Alkhan, and A. Toktar. "PROBLEMS OF BUILDING AND APPLICATION OF HYPERCOMPLEX NUMBERS." EurasianUnionScientists 4, no. 2(83) (2021): 33–37. http://dx.doi.org/10.31618/esu.2413-9335.2021.4.83.1261.

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The article is dedicated to the problems of using multidimensional numbers for mathematical and computer modeling of complex physical processes and the design of knowledge-intensive devices, including digital image processing. The emphasis is on the issues of building the methods for processing three-dimensional signals. It is proposed to use three-dimensional variables presented in the form of hypercomplex numbers to formulate the three-dimensional Fourier transformation forms, which allows to analyze and process three-dimensional signals.

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26

L. Negulescu, Vlad. "The Vector Hypercomplex Numbers and the Matter Waves." International Journal of Applied Physics 9, no. 2 (2022): 12–16. http://dx.doi.org/10.14445/23500301/ijap-v9i2p103.

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27

Mikhailichenko, G. G., and R. M. Muradov. "Hypercomplex numbers in the theory of physical structures." Russian Mathematics 52, no. 10 (2008): 20–24. http://dx.doi.org/10.3103/s1066369x08100034.

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28

Kolodezhnov, V. N. "Three-component hypercomplex numbers on two-dimensional plane." Automation and Remote Control 71, no. 6 (2010): 1275–82. http://dx.doi.org/10.1134/s0005117910060287.

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29

Hamilton, J. J. "Hypercomplex numbers and the description of spin states." Journal of Mathematical Physics 38, no. 10 (1997): 4914–28. http://dx.doi.org/10.1063/1.531926.

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30

Vanwormhoudt, M. C. "Rings of hypercomplex numbers for NT Fourier transforms." Signal Processing 67, no. 2 (1998): 189–98. http://dx.doi.org/10.1016/s0165-1684(98)00036-x.

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31

Votiakova, Lesia, та Viktoriia Bodenchuk. "Matrix Algebra В as Euclidean Space". Mathematical and computer modelling. Series: Physical and mathematical sciences 24 (5 грудня 2023): 5–13. http://dx.doi.org/10.32626/2308-5878.2023-24.5-13.

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Representation of information by means of hypercomplex numerical systems is used in various problems of science and technology: in classical mechanics, solid body mechanics, electrodynamics, radio electronics, computer animation, and others [1]. Often a hypercomplex system (that is, a system whose elements are considered to be hypercomplex numbers) is understood as any finite-dimensional algebra over a field. An important place among such algebraic structures is occupied by matrix algebras. The impossibility of constructing algebras with division does not at all mean the impossibility of const

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32

Yefremov, Alexander P. "Physical theories in hypercomplex geometric description." International Journal of Geometric Methods in Modern Physics 11, no. 06 (2014): 1450062. http://dx.doi.org/10.1142/s0219887814500625.

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Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics.

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33

Borio, Daniele. "A Vector Representation of Multicomplex Numbers and Its Application to Radio Frequency Signals." Axioms 13, no. 5 (2024): 324. http://dx.doi.org/10.3390/axioms13050324.

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Hypercomplex numbers, which are multi-dimensional extensions of complex numbers, have been proven beneficial in the development of advanced signal processing algorithms, including multi-dimensional filter design, linear regression and classification. We focus on multicomplex numbers, sets of hypercomplex numbers with commutative products, and introduce a vector representation allowing one to isolate the hyperbolic real and imaginary parts of a multicomplex number. The orthogonal decomposition of a multicomplex number is also discussed, and its connection with Hadamard matrices is highlighted.

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34

Zaripov, R. G. "Representations of a group of difference information functions in the extended parastatistics of non-extensive systems." Izvestiya vysshikh uchebnykh zavedenii. Fizika, no. 7 (2022): 9–15. http://dx.doi.org/10.17223/00213411/65/7/9.

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Algebraic and matrix representations of the group of information functions of the difference of nonextensive systems for three types of conformally generalized hypercomplex numbers are given. The corresponding geometries with metric functions are global Finsler geometries.

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35

Pavlov, Stanislav, Dmitry Kozlov, Mikhail Bakulin, Aleksandr Zuev, Andrey Latyshev, and Alexander Beliaev. "Generalization of Neural Networks on Second-Order Hypercomplex Numbers." Mathematics 11, no. 18 (2023): 3973. http://dx.doi.org/10.3390/math11183973.

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The vast majority of existing neural networks operate by rules set within the algebra of real numbers. However, as theoretical understanding of the fundamentals of neural networks and their practical applications grow stronger, new problems arise, which require going beyond such algebra. Various tasks come to light when the original data naturally have complex-valued formats. This situation is encouraging researchers to explore whether neural networks based on complex numbers can provide benefits over the ones limited to real numbers. Multiple recent works have been dedicated to developing the

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36

Kyrov, Vladimir Aleksandrovich. "Hypercomplex numbers in some geometries of two sets. II." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 5 (2020): 39–54. http://dx.doi.org/10.26907/0021-3446-2020-5-39-54.

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37

Mikhailichenko, G. G., and V. A. Kyrov. "Hypercomplex numbers in some geometries of two sets. I." Russian Mathematics 61, no. 7 (2017): 15–24. http://dx.doi.org/10.3103/s1066369x17070039.

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38

Kyrov, V. A. "Hypercomplex Numbers in Some Geometries of Two Sets. II." Russian Mathematics 64, no. 5 (2020): 31–48. http://dx.doi.org/10.3103/s1066369x20050047.

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39

Kamal, Aliaa, Moumen El-Melegy, Hassan El-Hawary, and Khaled Hussein. "Face Recognition by Principal Component Regression using Hypercomplex Numbers." Assiut University Journal of Multidisciplinary Scientific Research 51, no. 3 (2022): 268–78. http://dx.doi.org/10.21608/aunj.2022.131391.1006.

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40

Delsuc, Marl A. "Spectral representation of 2D NMR spectra by hypercomplex numbers." Journal of Magnetic Resonance (1969) 77, no. 1 (1988): 119–24. http://dx.doi.org/10.1016/0022-2364(88)90036-4.

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41

Catoni, Francesco, Roberto Cannata, Vincenzo Catoni, and Paolo Zampetti. "Two-dimensional hypercomplex numbers and related trigonometries and geometries." Advances in Applied Clifford Algebras 14, no. 1 (2004): 47–68. http://dx.doi.org/10.1007/s00006-004-0008-2.

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42

GÜRSES, NURTEN, and GÜLSÜM YELİZ ŞENTÜRK. "MATRIX THEORY OVER DGC NUMBERS." Journal of Science and Arts 23, no. 1 (2023): 209–28. http://dx.doi.org/10.46939/j.sci.arts-23.1-a17.

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Classical matrix theory for real, complex and hypercomplex numbers is a well-known concept. Is it possible to construct matrix theory over dual-generalized complex (DGC) matrices? The answer to this question is given in this paper. The paper is constructed as follows. Firstly, the fundamental concepts for DGC matrices are introduced and DGC special matrices are defined. Then, theoretical results related to eigenvalues/eigenvectors are obtained and universal similarity factorization equality (USFE) regarding to the dual fundamental matrix are presented. Also, spectral theorems for Hermitian and

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43

GARANT–PELLETIER, V., and D. ROCHON. "ON A GENERALIZED FATOU–JULIA THEOREM IN MULTICOMPLEX SPACES." Fractals 17, no. 03 (2009): 241–55. http://dx.doi.org/10.1142/s0218348x09004326.

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In this article we introduce the hypercomplex 3D fractals generated from Multicomplex Dynamics. We generalize the well known Mandelbrot and filled-in Julia sets for the multicomplex numbers (i.e. bicomplex, tricomplex, etc.). In particular, we give a multicomplex version of the so-called Fatou-Julia theorem. More precisely, we present a complete topological characterization in ℝ2n of the multicomplex filled-in Julia set for a quadratic polynomial in multicomplex numbers of the form w2 + c. We also point out the symmetries between the principal 3D slices of the generalized Mandelbrot set for tr

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44

Sangwine, S. J. "Fourier transforms of colour images using quaternion or hypercomplex, numbers." Electronics Letters 32, no. 21 (1996): 1979. http://dx.doi.org/10.1049/el:19961331.

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45

Singh, Pushpendra, Anubha Gupta, and Shiv Dutt Joshi. "On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication." PLOS ONE 19, no. 10 (2024): e0312502. http://dx.doi.org/10.1371/journal.pone.0312502.

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Mathematics is the foundational discipline for all sciences, engineering, and technology, and the pursuit of normed division algebras in all finite dimensions represents a paramount mathematical objective. In the quest for a real three-dimensional, normed, associative division algebra, Hamilton discovered quaternions, constituting a non-commutative division algebra of quadruples. Subsequent investigations revealed the existence of only four division algebras over reals, each with dimensions 1, 2, 4, and 8. This study transcends such limitations by introducing generalized hypercomplex numbers e

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46

Hauser, Jochem, and Walter Dröscher. "Gravity beyond Einstein? Part III: numbers and coupling constants, contradictory experiments, hypercomplex gravity like-fields, propellantless space propulsion." Zeitschrift für Naturforschung A 77, no. 1 (2021): 13–86. http://dx.doi.org/10.1515/zna-2021-0147.

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Abstract This article, the last in a series of three articles, attempts to unravel the underlying physics of recent experiments regarding the contradictory properties of the neutron lifetime that has been a complete riddle for quite some time. So far, none of the advanced theories beyond the Standard Models (SMs) of particle physics and cosmology have shown sufficient potential to resolve this mystery. We also try to explain the blatant contradiction between the predictions of particle physics and experiments concerning the nature and properties of the (so far undetected) dark matter and dark

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Martinez, C. A. P., André L. M. Martinez, Manoel Ferreira Borges Neto, and Emerson Vitor Castelani. "SQUARE OF THE ERROR OCTONIONIC THEOREM AND HYPERCOMPLEX FOURIER SERIES." TEMA (São Carlos) 14, no. 3 (2013): 483. http://dx.doi.org/10.5540/tema.2013.014.03.0483.

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<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>The focus of this paper is to address some classical results for a class of hypercomplex numbers. More specifically we present an extension of the Square of the Error Theorem and a Bessel inequality for octonions. </span></p></div></div></div>

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48

Bill, Jeremiah, Lance Champagne, Bruce Cox, and Trevor Bihl. "Meta-Heuristic Optimization Methods for Quaternion-Valued Neural Networks." Mathematics 9, no. 9 (2021): 938. http://dx.doi.org/10.3390/math9090938.

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In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing t

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49

Et. al., Dr Indrajit Patra ,. "Shifts in the Foundation: The Continual Modification and Generalization of Axioms and the Search for the Mathematical Principles that Underlie our Reality." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 2 (2021): 1095–106. http://dx.doi.org/10.17762/turcomat.v12i2.1126.

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The study shall seek to explore the deep, underlying correspondence between the mathematical world of pure numbers and our physical reality. The study begins by pointing out that while the familiar, one-dimensional real numbers quantify many aspects of our day-to-day reality, complex numbers provide the mathematical foundations of quantum mechanics and also describe the behavior of more complicated quantum networks and multi-party correlations, and quaternions underlie Einsteinian special theory of relativity, and then poses the question whether the octonions could play a similar role in const

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Yefremov, Alexander P. "Associative Hypercomplex Algebras Arise over a Basic Set of Subgeometric One-Dimensional Elements." Mathematics 13, no. 13 (2025): 2105. https://doi.org/10.3390/math13132105.

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Abstract:

An abstract set of one-dimensional (spinor-type) elements randomly oriented on a plane is introduced as a basic subgeometric object. Endowing the set with the binary operations of multiplication and invertible addition sequentially yields a specific semi-group (for which an original Cayley table is given) and a generic algebraic system which is shown to generate, apart from algebras of real and complex numbers, the associative hypercomplex algebras of dual numbers, split-complex numbers, and quaternions. The units of all these algebras turn out to be composed of basic 1D elements, thus ensurin

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