On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication
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 disc...
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| Published in: | PloS one Vol. 19; no. 10; p. e0312502 |
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| Language: | English |
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| ISSN: | 1932-6203, 1932-6203 |
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| Abstract | 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 extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new
π
-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing. |
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| AbstractList | 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 extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing. 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 extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new [qi]-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing. 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 extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π -periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing. 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 extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing.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 extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing. |
| Audience | Academic |
| Author | Singh, Pushpendra Gupta, Anubha Joshi, Shiv Dutt |
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| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/39453926$$D View this record in MEDLINE/PubMed |
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| Cites_doi | 10.2307/1970147 10.3390/axioms10040250 10.1007/BF02940993 10.1090/S0002-9904-1958-10166-4 10.32323/ujma.423045 10.1088/0143-0807/5/1/007 10.1007/s12045-016-0358-9 10.1002/mma.5831 10.2307/1967865 10.1090/S0002-9904-1958-10225-6 10.1073/pnas.44.3.280 10.1007/s00006-013-0386-4 |
| ContentType | Journal Article |
| Copyright | Copyright: © 2024 Singh et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. COPYRIGHT 2024 Public Library of Science 2024 Singh et al. This is an open access article distributed under the terms of the Creative Commons Attribution License: http://creativecommons.org/licenses/by/4.0/ (the “License”), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. 2024 Singh et al. This is an open access article distributed under the terms of the Creative Commons Attribution License: http://creativecommons.org/licenses/by/4.0/ (the “License”), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| DOI | 10.1371/journal.pone.0312502 |
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| References_xml | – volume: 26 start-page: 315 issue: 173 year: 1845 ident: pone.0312502.ref002 article-title: On a connection between the general theory of normal couples and the theory of complete quadratic functions of two variables publication-title: Philosophical Magazine – volume: 72 start-page: 20 year: 1960 ident: pone.0312502.ref009 article-title: On the non-existence of elements of Hopf invariant one publication-title: Ann. of Math doi: 10.2307/1970147 – volume: 10 start-page: 250 issue: 4 year: 2021 ident: pone.0312502.ref016 article-title: Powers of elliptic scator numbers publication-title: Axioms doi: 10.3390/axioms10040250 – volume: 8 start-page: 123 year: 1930 ident: pone.0312502.ref007 article-title: Theorie der alternativen Ringe publication-title: Abh Math Sem Univ Hamburg doi: 10.1007/BF02940993 – ident: pone.0312502.ref006 – volume: 64 start-page: 87 year: 1958 ident: pone.0312502.ref011 article-title: On the parallelizability of the spheres publication-title: Bull Amer Math Soc doi: 10.1090/S0002-9904-1958-10166-4 – volume: 1 start-page: 80 issue: 2 year: 2018 ident: pone.0312502.ref017 article-title: Product associativity in scator algebras and the quantum wave function collapse publication-title: Universal Journal of Mathematics and Applications doi: 10.32323/ujma.423045 – volume: 5 start-page: 25 year: 1984 ident: pone.0312502.ref013 article-title: The quaternion group and modern physics publication-title: European Journal of Physics doi: 10.1088/0143-0807/5/1/007 – volume: 21 start-page: 529 year: 2016 ident: pone.0312502.ref014 article-title: Algebra and geometry of Hamilton’s quaternions: ‘Well, Papa, can you multiply triplets?’ publication-title: Resonance doi: 10.1007/s12045-016-0358-9 – volume: 26 start-page: 208 issue: 172 year: 1845 ident: pone.0312502.ref003 article-title: On Jacobi’s Elliptic functions, in reply to the Rev. Brice Bronwin; and on Quaternions publication-title: Philosophical Magazine – volume: 43 start-page: 1017 issue: 3 year: 2020 ident: pone.0312502.ref015 article-title: Components exponential scator holomorphic function publication-title: Mathematical Methods in the Applied Sciences doi: 10.1002/mma.5831 – volume: 2 start-page: 424 issue: 1843 ident: pone.0312502.ref001 article-title: On a new species of imaginary quantities connected with a theory of quaternions publication-title: Proc. Royal Irish Acad – volume: 20 start-page: 155 issue: 3 year: 1919 ident: pone.0312502.ref004 article-title: On quaternions and their generalization and the history of the eight square theorem publication-title: Ann. Math doi: 10.2307/1967865 – volume: 64 start-page: 279 year: 1958 ident: pone.0312502.ref008 article-title: On the nonexistence of elements of Hopf invariant one publication-title: Bull. Amer. Math. Soc doi: 10.1090/S0002-9904-1958-10225-6 – volume: 44 start-page: 280 year: 1958 ident: pone.0312502.ref010 article-title: Non-parallelizability of the n sphere for n > 7 publication-title: Proc. Nat. Acad. Sci. USA doi: 10.1073/pnas.44.3.280 – volume: 23 start-page: 639 issue: 3 year: 2013 ident: pone.0312502.ref012 article-title: A hyperbolic non distributive algebra in 1 + 2 dimensions publication-title: Advances in Applied Clifford Algebras doi: 10.1007/s00006-013-0386-4 – volume: 84 start-page: 1 year: 1878 ident: pone.0312502.ref005 article-title: Über lineare Substitutionen und bilineare publication-title: Formen J Reine Angew Math |
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| SubjectTerms | Algebra Algebra, Abstract Algorithms Complex numbers Dimensional analysis Dividing (mathematics) Group theory Image processing Mathematical analysis Mathematical research Mathematics Methods Models, Theoretical Multiplication Multiplication & division Number systems Numbers, Complex Periodic functions Quantum physics Quaternions State vectors Vector space |
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| Title | On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication |
| URI | https://www.ncbi.nlm.nih.gov/pubmed/39453926 https://www.proquest.com/docview/3120835605 https://www.proquest.com/docview/3120911230 http://dx.doi.org/10.1371/journal.pone.0312502 |
| Volume | 19 |
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