The Number of Numbers: Zero, Infinity, and Mathematical Boundaries

Abstract

Background: Traditional mathematics treats zero and infinity as well-defined concepts, yet their practical attainment remains elusive. Zero represents the convergence point of number lines, while infinity symbolizes divergence. In this paper, we explore the theory that zero and infinity are bounded but unattainable in practical life, forming the core of an n-dimensional sphere comprising infinite number lines. Objectives: Our study aims to redefine the understanding of zero and infinity within number theory, demonstrating their roles in the bounded, yet infinite, structure of mathematical constructs. Methods: We employ mathematical modeling, logical analysis, and theoretical exploration to investigate the properties and implications of zero and infinity. Results: Through rigorous derivation and analysis, we present the concept of the n-dimensional sphere of numbers, illustrating how infinite number lines converge at zero and diverge at infinity. Conclusions: Zero and infinity, while theoretically bounded, remain practically unattainable due to inherent constraints such as the Heisenberg Uncertainty Principle. Our research advances the theoretical framework of number theory, providing a new perspective on these fundamental concepts. Furthermore, Miller's Law posits that the human brain, and by extension computational systems, can effectively handle only around 7 ± 2 distinct chunks of information at a time, limiting practical applications involving complex numerical constructs.

Keywords: Zero, Infinity, Number Theory, N-Dimensional Sphere, Mathematical Boundaries, Convergence, Divergence Subject Descriptors: G.1.0 [Numerical Analysis]: General-Numerical Algorithms, G.2.0 [Discrete Mathematics]: General-Mathematical Logic Categories: F.1.1 [Computation by Abstract Devices]: Models of Computation, F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic