Abstract 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.
