![]() ![]() Each additional symbol received adds more information, but the additional information varies in size, depending on symbol probabilities in the chosen alphabet. Shannon information is, by definition, additive. The information transmitted by each symbol involves a degree of receiver surprise the letter Z conveys more information than the letter E and so forth. For example, the letter E is more likely to occur than other letters in English text. State size can also be expressed by T, the number of bits corresponding to W.Ī message’s Shannon information is defined in terms of the probability of occurrence of each symbol produced by the sending system (whether human or machine). To be more precise, we may call W the size of the die's macro-state to distinguish it from the micro-states, which are determined by the locations and velocities of the elementary particles forming the die. The size of state information is defined simply as W, the number of distinct states that the system can be in. We only require that the system be found in one of the 128 discrete states, independent of the cause or means of observation. This state description is independent of the causes or probabilities of occurrence of each state, which might be the result of a simple die toss, deliberate placement by an intelligent being, or some hidden process. If, for example, W = 128, each system state can be unequivocally labeled by a single positive integer in the range 1 through 128. ![]() A polyhedron die is a three-dimensional object with W flat surfaces. If a single cubical die is placed or tossed on a surface, the six possible states of the system are 1 through 6. Let’s begin our discussion of states with simple die systems. Thus, for example, ceiling function (6.2) = 7 bits or buckets no partly filled buckets or hanging chads are allowed. The binary transformation rule adopts the ceiling function of computer science, the provision that non-integer values of T are to be rounded up to the nearest integer. This equation has no fixed scientific meaning it is simply the rule for transforming any base-10 number W into a binary number T, analogous to a simple mathematical rule like A x B = B x A. The right side is read, “the base 2 logarithm of W”. Solving this equation for T yields the minimum number of bits (buckets or binary digits) needed to store the number W: T = Log 2W. As shown in my first figure, a T-bit binary number can represent base-10 integers up to the maximum number W = 2 T – 1. I adopt the symbol T (after computer scientist Alan Turing) to indicate the number of buckets in the line. Empty buckets are designated 0 and sand-filled buckets are designated 1 fractional fills are not allowed. Imagine sand buckets placed in a straight line on a beach.
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