Abstract
This thesis deals with the roles - effective and possible - of the notions of square root and decimal number in the construction of the real number concept in secondary education. We highlight the creation, by certain transposition choices, of "didactic gaps" in the transition from decimal numbers to real numbers. First, we analyze, with the tools of Chevallard's anthropological approach, the relationships to numbers that have developed in scholarly institutions at different times, as well as the evolution of the didactic transposition of the real number concept in secondary education since 1850. Secondly, we studied the personal relationships of current pupils and teachers with this mathematical concept. These different analyzes allowed us to formulate hypotheses on the learning conditions of real numbers and the role of the square root and decimal numbers in the reduction of didactic gaps. Finally, we propose a set of didactic situations, in the sense of Brousseau, which satisfies some of the previous conditions.