Semi-structured complex numbers H was a number set developed to enable division by zero in ordinary algebraic equations. Its utility has been shown in mathematics and engineering. However, very little has been done to show its usefulness in computer science. Consequently, the aim of this paper was to show the utility of semi-structured complex numbers in computer science by developing a division by zero calculator. First two computer programs were written, one for a standard (STD) calculator and the other for a division by zero (DBZ) calculator. The programs were fed 20000 randomly generated arithmetic equations of varying lengths and the space and time complexity associated with processing these equations were measured and compared to determine the efficiency of each calculator. In the process, three major contributions were made: (1) A representation for semi-structured complex numbers that enables it to be easily used by a computer was developed; (2) It was demonstrated that the DBZ calculator outperforms the STD calculator in terms of efficiency; and, (3) It was shown that the number set H reduced the amount of error handling required to run a computer program. These results provide a firm foundation to advance the number set as a useful tool in computer science.
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.
Recently a new number p (called the unit unstructured number) was invented to solve “division by zero”. “p” was used to create a new number set called the semi-structured complex numbers and a new coordinate plane called the real-unstructured xz-plane. However, the definition of p led to a number set that had a poor theoretical foundation and lacked consistency in simple algebraic calculations producing contradictory results when dividing by zero. To overcome these issues, this research redefines the unit p and, in the process, shows that (1) the new definition of p produces a more theoretically grounded number set that forms a “field” and behaves consistently in algebraic calculations, (2) p can be used to find a viable solution to the logarithm of zero; (3) the new definition of p helps provide an unambiguous understanding of what 0/0=k means (where k is any real number). The research proves that 0/0=k simply represents 90^° clockwise rotation of the vector kp from the positive unstructured z-axis to k on the positive real x-axis along the real-unstructured xz-plane. These results serve to provide a firm foundation for Semi-structured Complex Numbers and support their practical use.
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