Irrational numbers: The gap between formal and intuitive knowledge

This report focuses on prospective secondary mathematics teachers' understanding of irrational numbers. Various dimensions of participants' knowledge regarding the relation between the two sets, rational and irrational, are examined. Three issues are addressed: richness and density of numbers, the fitting of rational and irrational numbers on the real number line, and operations amongst the elements of the two sets. The results indicate that there are inconsistencies between participants' intuitions and their formal and algorithmic knowledge. Explanations used by the vast majority of participants relied primarily on considering the infinite non-repeating decimal representations of irrationals, which provided a limited access to issues mentioned above. © Springer Science+Business Media, Inc. 2007.