The Independent Circular Geometric Number System (ICGNS): Redefining Numbers Through the Relation Between Radius and Angle

Abstract

Dr. Walid Nabil

This paper introduces a fully new numerical framework, the Independent Circular Geometric Number System (ICGNS), which redefines the concept of a number by transforming it from a one-dimensional linear value into a two-dimensional geometric entity within a circular space. In this system, every number is represented by three fundamental components: the logarithmic magnitude level (r), the angular phase (θ), and the sign (s), forming a unique radial point across a set of concentric circular layers. Unlike the classical geometric interpretation of complex numbers, the proposed system does not use geometry merely as a visualization tool. Instead, it provides a genuine circular number system capable of generating, classifying, and relating numbers through intrinsic radial–angular structures. The paper formulates the core transformation functions between the linear and circular domains, presents the rules for multiplication, addition, and geometric distance within the system, and provides practical examples demonstrating the circular distribution of natural numbers and multi-digit values such as calendar years. The study reveals that ICGNS offers strong explanatory power for periodic and wave-like patterns and has potential applications in mathematics, physics, astronomy, geography, data science, and signal analysis. The paper also discusses theoretical limitations of the system and shows how these limitations form natural pathways for future development, including algorithmic implementations and computational libraries based on circular numerical representation. Thus, ICGNS lays the foundation for a new numerical paradigm that re-conceptualizes the nature of numbers and opens novel research directions across mathematical and scientific disciplines.

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