It is well-known that positional representation systems often admit multiple representations forthe same real value, as expressed by identities such as 0.99999... = 1. In the following paper, movingfrom this identity, we introduce a reformulated positional numerical representation system, basedon an abstract 4-tuple definition of a positional representation. We then develop several comparisonmethods, defining various equivalence relations, and an evaluation map into the real numbers, provingits convergence. We then establish two propositions representing the system’s fundamental structure,delving into the consequences of the falsehood of their converses. These results provide the foundationfor further developments of the system and culminate in a final theorem, which proves the existence ofan alternative periodic representation for all non-zero real numbers with a finite positional expansion.Moreover, the final proposition gives a conceptual interpretation of the theorem.
Giovanni Minenna (Fri,) studied this question.