This study has investigated prospective secondary mathematics teachers’ understanding of the concept of irrationality. Understanding of irrational numbers is essential for the extension and reconstruction of the concept of number from the system of rational numbers to the system of real numbers. The concept itself is inherently difficult. Historical development of the notion was a journey paved with epistemological obstacles of many kinds, from the discovery of incommensurable lengths to the acceptance of transcendental numbers into the system of real numbers by the decree of completeness axiom more than two thousand years later. Yet, these two types of numbers come as a lump sum to the learner sitting in a contemporary classroom, usually under the title “irrational numbers”. This study could be understood as an inquiry into the obstacles - epistemological, intuitive, and didactic - which replicate in individual learners as cognitive obstacles, affecting the understanding of irrationality in particular and the notion of real number in general.