We state and justify a conjecture concerning the comparative complexity of the representation of real numbers on the line versus other representations. The Introduction provides the setting of the subject and states the conjecture. Part[1] deals with the controversial line’s «nature» and depicts several interpretations of the straight line. Part 2 analyses the particular phenomenology related to the assignment of real numbers to the line’s points, in order to extract some limitations of such an assignment. Part 3 compares several representations of real numbers with the representation on the line, and presents a set of distinctive features for the latter.This paper, theoretically oriented, comes from a broader research work addressed to uncover epistemological obstacles related to the representation of real numbers on the straight line. We are doing a work field, involving students of several levels, on this subject.