Abstract—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.

Index Terms—Division by zero, semi-structured complex numbers, singular matrices.

Peter Jean-Paul is with Department of Economic Development, Ministry of Finance, Economic Development and the Youth Economy, Castries, Saint Lucia. Shanaz Wahid is with Mathematics & Statistics, Faculty of Science and Technology, University of the West Indies, St. Augustine, Trinidad and Tobago.

Cite:Peter Jean-Paul, Shanaz Wahid, "Reformulating and Strengthening the Theoretical Foundations of Semi-structured Complex Numbers," International Journal of Applied Physics and Mathematics vol. 12, no. 4, pp. 34-58, 2022.

Copyright © 2022 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).