This paper develops a novel mathematical framework that enables division by zero by introducing a measurement-based approach. We distinguish between an absolute zero and a measured zero, and introduce a transient state linked to the measurement process, thereby resolving the semantic ambiguity of the "zero" symbol that leads to classical paradoxes. We define a set of consistent algebraic axioms on an extended number system, S=R∪{0bm,0m,1t}, where 0bm is an absolute void, 0m is a contextual or measured zero, and 1t is a transient unit emerging from indeterminate operations. A rigorous analysis of the system's properties reveals that it forms a commutative semiring. This structure deliberately sacrifices the universal multiplicative inverse law to achieve operational closure for its target cases and high semantic fidelity. The framework is compared with existing approaches, highlighting its unique suitability for modeling physical and computational systems. We conclude by exploring specific future applications, supported by a proof-of-concept example, in symbolic knowledge representation, robust error modeling, and the design of fault-tolerant programming languages.