The concept of "q-deformation" is applied in many parts of mathematics: in algebra and representation theory this is the origin of quantum groups, they are important for knot invariants, combinatorics, discrete geometry... In mathematical physics q-deformations are often understood as "quantization". The notion of q-deformed real number was introduced very recently. It is based on the geometric idea of invariance by a group action. In this introductory and accessible to everyone 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.