Quaternions and Octonions Review, Relationship with Cayley-Dickson Construction, and Contribution to Quantum Mechanics and 3D Rotation

الملخص

Mathematics has always been the mother of sciences. The main reasons behind this are the broadness of mathematics and its compelling ability to translate theory into laws and algorithms to help us understand the universe better. The discovery of imaginary numbers was a critical moment in the history of mathematics, extending its horizon by solving undefinable polynomials with such a revolutionary idea. This paper aims to clear the common misconception about the existence of a finite number of numerical systems, explain their applications, and extend basic algebraic properties to conclude their origin. The focus of this paper is on the abstract mathematical approach to higher-dimensional complex systems, or hyper- complex number systems, of Quaternions and Octonions, discussing the historical background of these systems, the related fundamental algebraic concepts, their construction, properties, operations, and finally their real-life applications. Hyper- complex number systems are not only beneficial in computer science and theoretical physics but also groundbreaking within the fields of mathematics. Accordingly, this paper summarizes the findings throughout the history of hyper- complex numbers and demonstrates their ability to be applied in physics, quantum mechanics, computer graphics, and more.