Rational approaches to arithmetic and real numbers in the Italian textbooks and programs for the classical and the scientific curricula

Rational arithmetic was traditional in the Italian Classical Lycée, considered in someway parallel to rational geometry, and was maintained also in the Modern Lycée. The comments to the programmes before 1923 suggest anyway a not too rational approach to arithmetic and real numbers; on the other hand, textbooks seem to shift towards a rational treatment of these topics, also for the Technical Institutes. With the birth of the Scientific Lycée in 1923 rational arithmetic disappeared from the programmes, while the introduction of real numbers proved to be still a problem, which will became more evident looking at the frequent rearrangements of this part of the programmes in the successive years. On the one hand real numbers had to be introduced geometrically, and this required a detailed introduction of ratios of magnitudes and proportions; on the other hand there was the need to introduce quite early geometry of coordinates, which required real numbers