Reversible Arithmetic System: A Mathematical Framework Based on Computational History Tracing

This paper proposes a novel reversible arithmetic framework that addresses the irreversible information loss problem in traditional arith metic systems caused by specific operations such as multiplication by zero and division by zero. The system extends each numerical value into a mathematical object containing both the current value and com plete computational history, redefining the four fundamental arith metic operations to ensure traceability of computational processes. We establish the axiomatic foundation of this system, prove its key properties, and demonstrate its practical application value through er ror source tracing in differential equation solving, financial modeling, and machine learning. Theoretical analysis shows that while maintain ing computational reversibility, the system controls space complexity within acceptable limits through optimization techniques.