7/7 AN INTRODUCTION TO CYBERNETICS 



7/7. It will have been noticed that many of the exercises involved 

 the finding of products and high powers. Such computations are 

 often made easier by the use of logarithms. It is assumed that the 

 reader is familiar with their basic properties, but one formula will 

 be given for reference. If only logarithms to base a are available 

 and we want to find the logarithm to the base b of some number 

 TV, then 



In particular, log^A'' = 3-322 logioA''. 



The word variety, in relation to a set of distinguishable elements, 

 will be used to mean either (i) the number of distinct elements, or 

 (ii) the logarithm to the base 2 of the number, the context indicating 

 the sense used. When variety is measured in the logarithmic form 

 its unit is the "bit", a contraction of "Binary digiT". Thus the 

 variety of the sexes is 1 bit, and the variety of the 52 playing cards 

 is 5-7 bits, because logj 52 = 3-322 logjo 52 = 3-322 x 1-7160 = 5-7. 

 The chief advantage of this way of reckoning is that multiplicative 

 combinations now combine by simple addition. Thus in Ex. 7/6/2 

 the farmer can distinguish a variety of 3 bits, his wife 1 bit, and the 

 two together 3+1 bits, i.e. 4 bits. 



To say that a set has "no" variety, that the elements are all of one 

 type, is, of course, to measure the variety logarithmically; for the 

 logarithm of 1 is 0. 



Ex. 1 : In Ex. 7/6/4 how much variety, in bits, does each substance distinguish? 



Ex. 2: In Ex. 7/6/5: (i) how much variety in bits does each test distinguish? 

 (ii) What is the variety in bits of 2,000,000,000 distinguishable individuals? 

 From these two varieties check your previous answer. 



Ex. 3 : What is the variety in bits of the 26 letters of the alphabet ? 



Ex. 4: (Continued.) What is the variety, in bits, of a block of five letters (not 

 restricted to forming a word) ? Check the answer by finding the number of 

 such blocks, and then the variety. 



Ex. 5 : A question can be answered only by Yes or No ; (i) what variety is in the 

 answer? (ii) In twenty such answers made independently? 



Ex. 6: (Continued.) How many objects can be distinguished by twenty ques- 

 tions, each of which can be answered only by Yes or No? 



Ex. 7: A closed and single-valued transformation is to be considered on six 

 states : 



. a b c d e f 

 '''?????? 



in which each question mark has to be replaced by a letter. If the replace- 

 ments are otherwise unrestricted, what variety (logarithmic) is there in the 

 set of all possible such transformations? 



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