ARTICLE | doi:10.20944/preprints202303.0368.v1
Subject: Engineering, Automotive Engineering Keywords: semi-structured complex numbers; trigonometry; Euler formulas; quaternions; singularities; engineering; science
Online: 21 March 2023 (04:30:05 CET)
Recently, a paper was written to reformulate and strengthen the theory of semi-structured complex numbers H (a new number set invented to enable division by zero). Whilst the paper had profound results, the application of this number set in many areas of engineering and science has not been fully explored. Consequently, the aim of this paper was to establish the number set H as a useful mathematical tool by providing an amalgam of results arising from its application in several areas of engineering and science. In the process, this paper makes four major contributions: these are (1) determining the product of the gradients of a horizontal and vertical line; (2) developing new Euler formulas relating exponential, trigonometric and hyperbolic functions in the 3D semi-structured complex number Euclidean space; (3) establishing semi-structured complex numbers as a better extension of the complex number set than quaternions; (4) resolving singularities that may arise in engineering and science equations (because of division by zero) to develop reasonable conclusions in the absence of experimental data. These results and their applications provide a firm foundation to advance the number set H as a useful mathematical tool.