I summarize the essential steps of the demonstration leading to Shannon's definition. Assuming a random source with an event space, X, comprising N elements or sym- bols with probabilities pi (i = 1...N), the unknown function H should meet three conditions: (1) H = H(p1, P2, PN) is a continuous function of the probability set Pi; (2) If all probabilities were equal (namely, Pi = 1/N), the function H should be monotonously increasing with N;5 (3) If any occurrence breaks down into two successive possibilities, the original H should break down into a weighed sum of the corresponding individual values of Shannon's formal demonstration (see Appendix B) shows that the unique function sat- isfying the three requirements (1)-(3) is the following: (4.5) where K is an arbitrary positive constant, which we can set to K = 1, since the logarithm definition applies to any choice of base (K = logp x / log X with p # q being positive real numbers). It is clear that an equivalent notation of Eq. (4.5) is (4.6) where x is a symbol from the source X and I(x) is the associated information measure (as defined in Chapter 3). C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J., 27 (1948), 379-423, 623-56. This paper can be freely downloaded from http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html. Based on the fact that equally likely occurrences provide more choice, then higher uncertainty. Say there initially exist two equiprobable events (a, b) with p(a) = p(b) = 1/2, SO H is the function H I(1/2, 1/2); assume next that event (b) corresponds to two possibilities of probabilities 1/3 and 2/3, respectively. According to requirement (3), we should have H = H(1/2, 1/2) + (1/2)H(1/3, 2/3) with the coefficient 1/2 in the second right-hand-side term being justified by the fact that the event (b) occurs only half of the time, on average. 4.2 Entropy in dice The function H, which Shannon called entropy, is seen to be formally identical to the function of entropy defined by Boltzmann. However, we should note that S