A Socratic methodological proposal for the study of the equality $0.999\ldots = 1$

Maria Angeles Navarro, Pedro PĂ©rez Carreras

Our objective is to produce a learning experience in the form of an interview which is structured in an introduction and three phases with the purpose of making the equality $0.999\ldots = 1$ acceptable providing the tail of dots ``$\dots$'' with a precise meaning. Essentially the interviewee will have to understand that a symbol is not what its aspect may suggest, but what we want it to be in a precise way. What we want will be the result of an evolutionary process of change of meaning, dictated by the context in which we move in each conceptual phase in which the experience is structured. In more detail, the Introduction will serve him/her to reflect on what a symbol is and to appreciate the usefulness of the positional system of numerical symbols being aware of the hidden character of the involved algebraic operations. The experience will run assigning different meanings to the symbol ``$\dots$'', each meaning reconciled with the previous one. Thus Phase 1 will extend the positional system of symbols to the rational numbers with the appearance of a new algebraic operation, the division. Phase 2 will state that the habitual algebraic operations are not sufficient to equip the symbol $0.999\dots$ with a numerical meaning, which will force us to the introduction of a new algebraic operation in Phase 3: interpreting the tail of dots either as a dynamic process (movement) or as its stabilization in an end product (rest), we will choose the last one and deal with the non trivial problem of how to formulate algebraically what means that a dynamic process becomes stabilized. To attain this objective, logical quantifiers need to appear in scene and the use of a suitable mathematical assistant will encourage their understanding through visualization.