The concept of q-deformation arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; q-deformations are important for knot invariants, combinatorial enumeration, discrete geometry... In mathematical physics, q-deformations are often understood as "quantizations".

The recently introduced notion of a q-deformed real number is based on the geometric idea of invariance by a modular group action. In this introductory and accessible talk, I will explain what is a q-rational and a q-irrational, demonstrate beautiful properties of these objects, and describe their relations to many different areas.