The concept of irrational numbers in high-school students and prospective teachers

It has been assumed, on historical and psychological grounds, that the concept of irrational numbers faces two major intuitive obstacles: a) the difficulty to accept that two magnitudes (two line segments) may be incommensurable (no common unit may be found); and b) the difficulty to accept that the set of rational numbers, though everywhere dense, does not cover all the points in an interval: one has to consider also the more rich infinity of irrational points. In order to assess the presence and the effects of these obstacles, three groups of subjects were investigated: students in grades 9 and 10 and prospective teachers.The results did not confirm these hypotheses. Many students are ignorant when asked to classify various numbers (rational, irrational, real) but only a small part of the subjects manifest genuine intuitive biases. It has been concluded that such erroneous intuitions (a common unit can always be found by indefinitely decreasing it and in an interval it is impossible to have twodifferent infinite sets of points or numbers]) have not a primitive nature. They imply a certain intellectual development.