the sum of the positional values: a6*2^6 + a5*2^5 +...+ a0*2^0 and the a's are either 0 or 1.
Often it is necessary to encode other than numerical information. If one wants a digital system to play checkers, then it must have a representation of the checker position. This must also be encoded into sequences of bits. For instance, one encoding is based on the fact that each of the 32 squares of the checkerboard can be in any of five situations:
The checker position in Figure 2 would be coded as shown. This requires a total of 3 * 32 = 96 bits for the representation: In the figure we wrote it as 32 3-bit sequences; we could also have written it as two 48-bit sequences:
As long as the digital system that processed these bits assumed the correct interpretation of each of the bits, it would make no difference.
There is more than one way to construct a representation of some situation -- more than one way to encode it into bit sequences. For instance, we could have coded the checker position as a 7 bit sequence for each checker man.
Five bits are required for the board position, since there are 32 possible squares (2^5). Since there are 24 men (12 for each side), the total number of bits for this representation is 24 * 7 = 168. This is much larger than the 96 for the original encoding, though it represents the same set of situations and thus has the same amount of information.(3)
Any informational situation may be represented in terms of sequences of bits. As long as one can be definite about the things to be represented and the variety of these different things that can exist, one can invent some sort of correspondence to state these same things in terms of bit sequences. The limits are the total variety of things that have to be represented (e.g., 7 bits only distinguishes 128 things) and whether you can be definite about your description of them.
Standard ways exist for representing information in sequences of bits. These
3. Can you find out why they differ? Actually, both representations take more bits than are necessary. Can you find better representations? What is the minimum number of bits required? Such questions are just intellectual games, of course, but they help to understand the issue of encoding. Actually, don't work too hard on the minimum; no one knows it exactly.