Reformulating and Strengthening the Theoretical Foundations of Semi-structured Complex Numbers

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.