$q$-deformed rationals and irrationals

The concept of $q$-deformation, or $q$-analogue'' 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, analysis, and many other parts of mathematics. 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. The goal of this lecture is to explain what is a $q$-rational and a $q$-irrational, demonstrate beautiful properties of these objects, and describe their relations to many different areas. We also tried to describe some applications of $q$-numbers.