A álgebra dos complexos/quatérnios/octônios e a construção de Cayley-Dickson

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Universidade Federal de Goiás


This research with theoretical approach seeks to investigate inmathematics, octonions,which is a non-associative extension of the quaternions. Its algebra division 8-dimensional formed on the real numbers is more extensive than can be obtained by constructing Cayley-Dickson. In this perspective we have as main goal to answer the following question: "What number systems allow arithmetic operations addition, subtraction, multiplication and division? " In the genesis of octonions is the Irish mathematician William Rowan Hamilton, motivated by a deep belief that quaternions could revolutionize mathematics and physics, was the pioneer of a new theory that transformed the modern world. Today, it is confirmed that the complexs/quaternions/octonions and its applications are manifested in different branches of science such as mechanics, geometry, mathematical physics, with great relevance in 3D animation and robotics. In order to investigate the importance of this issue and make a small contribution, we make an introduction to the theme from the numbers complex and present the rationale and motivations of Hamilton in the discovery of quaternions/octonions. Wemake a presentation of the algebraic structure and its fundamental properties. Then discoremos about constructing Cayley-Dickson algebras that produces a sequence over the field of real numbers, each with twice the previous size. Algebras produced by this process are known as Cayley-Dickson algebras; since they are an extension of complex numbers, that is, hypercomplex numbers. All these concepts have norm, algebra and conjugate. The general idea is that the multiplication of an element and its conjugate should be the square of its norm. The surprise is that, in addition to larger, the following algebra loses some specific algebraic property. Finally, we describe and analyze certain symmetry groups with multiple representations through matrixes and applications to show that This content has a value in the evolution of technology.



Números complexos, Quatérnios, Octônios, William Rowan Hamilton, Construção de Cayley-Dickson, Complex numbers, Quaternions, Octonions, William Rowan Hamilton, Cayley-Dickson Constructing


SANTOS, D. J. A álgebra dos complexos/quatérnios/octônios e a construção de Cayley-Dickson. 2016. 101 f. Dissertação (Mestrado Profissional em Matemática em Rede Nacional) - Universidade Federal de Goiás, Catalão, 2016.