Resolving the Centuries-Old Mathematical Dispute: Clarifying Conceptual Ambiguities via Analytic Philosophy on the Equivalence Proposition of 0.999... and 1

The proposition 0.999... = 1, a classic statement in mathematics, is universally recognized within mainstream theoretical systems, yet it has long run counter to learners’ intuitive perception. This discrepancy does not arise from misunderstandings among learners, but from a host of semantic confusions embedded in existing basic mathematical concepts. Adopting the conceptual analysis method of analytic philosophy, this paper conducts a systematic investigation into various classic proofs of the proposition, and sorts out six typical conceptual problems: the equality sign bears diverse semantic implications while no differentiated symbols are adopted; fractions and infinite recurring decimals confuse operational processes with static numerical values; the inverse operation of infinite division is essentially the cancellation of a process rather than the reconstruction of numerical values; multiplication of infinite decimals involving carry operations differs from conventional arithmetic; the transitivity of equality becomes invalid due to restrictions imposed by semantic contexts; and proofs based on infinite series apply limit theory to address infinitesimals. The study demonstrates that in accordance with the concepts and operational rules of elementary mathematics, 0.999... is not equivalent to 1. The equivalence of the two numbers is established only when advanced mathematics reconstructs relevant semantics based on limit theory. The equation 0.999... = 1 is a conclusion derived under the exclusive semantic rules of limit theory, and it does not apply to the logical framework of elementary arithmetic. Learners’ intuitive judgment that 0.999... < 1 is reasonable. This paper defines the conceptual boundaries between the two major mathematical systems, thereby effectively resolving the century-old academic and cognitive disputes over this proposition.