In this chapter we define the entropy and the conformal module of conjugacy classes of braids. We formulate our Main Theorem, stating that the entropy of each conjugacy class of braids is inversely proportional with factor \(\frac {\pi }{2}\) to its conformal module. Here we prove this theorem in the case of irreducible braids. The general case will be proved in the next two chapters. We also define the conformal module of conjugacy classes of elements of the fundamental group of complex manifolds, in particular, of the twice punctured complex plane. A version of the definition for elements of fundamental groups, not merely of their conjugacy classes, appears more effective for applications to quantitative versions of the Gromov-Oka Principle that will be given in later Chapters. In this chapter we compute these quantities explicitly in a number of examples.