In the talk I will give a brief introduction to the theory of rational and irrational numbers and their representation by integers. As the representation of irrational numbers must be infinite, we are led to the problem of approximation of real numbers by a sequence of rational numbers. The measure of irrationality of a real number gives us information how well the real number is approximable by rational numbers. In the talk I will give several examples, applications and results concerning the irrationality measure.