Positive rational numbers (operator interpretation)

As is known, in pedagogical literature three different interpretations of the concept of "natural number" are commonly used. Namely: quantitative, ordinal, and "number as a measure of magnitude." This note develops the fourth approach proposed earlier by one of the authors. Namely, a natural number is considered as an operator acting on a onedimensional vector space. With this approach, it is possible to significantly simplify the introduction of rational numbers, also interpreting them as operators on a one-dimensional vector space and at the same time getting rid of cumbersome calculations.