The Natural Number System in 3 Dimensions: Redundancy, Symmetry and Decimal Emergence,

This work introduces a three-dimensional extension of the BBF number system, in which every natural number is represented by coordinates (b,a,z). The classical two-dimensional BBF representation (b,a) places each number uniquely in the BBF matrix: the exponent a determines the dyadic height, while the coordinate b encodes the odd core through no=1+2b. The three-dimensional system BBFz augments this structure by incorporating decimal scaling. Using the decomposition 10z=2z⋅5z, the dyadic component 2z is absorbed into the exponent a, while the remaining odd factor 5z becomes the new odd core. This yields b=(5z−1)/2, which explains the systematic relationship between the 2D and 3D coordinates. The article compares both coordinate systems, illustrates their interaction through explicit examples, and shows how the BBF matrix extends naturally into a three-dimensional structure. The resulting framework provides a unified representation for integers and rational numbers within a consistent geometric encoding scheme.