Zero-Centric Arithmetic in Communication Theory: A Unified Interpretation of Division-by-Zero Breakdown

Abstract

Communication systems rely heavily on ratio-based mathematical models such as signal-to-noise ratio (SNR), power gain, impedance relations, and capacity formulas. These expressions assume that denominators remain strictly non-zero so that the computed quantities preserve physical meaning. However, when noise power, input power, current, or bandwidth approach zero, the corresponding expressions become undefined. Traditional interpretations label these cases as producing “infinite SNR,” “infinite gain,” or “unbounded channel capacity,” even though no communication channel or RF system exhibits infinite physical behavior. This paper demonstrates that such contradictions arise not from extreme telecommunication phenomena, but from the collapse of the mathematical operation itself. Using the Zero-Centric Arithmetic framework, expressions of the form found in SNR, gain, and impedance formulations are reinterpreted as non-operations, not as infinite values. This reinterpretation restores coherence between theoretical models and measurable communication behavior, showing that limits such as noise → 0, input → 0, or bandwidth → 0 represent boundaries where classical formulas cease to apply rather than points where physical quantities diverge. The resulting perspective offers a consistent and physically meaningful foundation for analyzing singular behavior across communication systems. The proposed interpretation yields a unified and physically coherent understanding of undefined expressions in communication theory.