**ABSTRACT**

Recently a new number (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 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 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 means (where k is any real number). The research proves that simply represents clockwise rotation of the vector from the positive unstructured z-axis to the positive real x-axis along the real-unstructured -plane. These results serve to provide a firm foundation for Semi-structured Complex Numbers and support their practical use.

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