As we assume a total order on the alphabet , we can speak of the first letter, the second, etc. Let n = || be the size of the alphabet, and write value(x)=i when x is the ith letter. Now it is possible to calculate the number associated with a given string ak-1ak-2...a0, namely by means of the formula i=0k-1 value(ai) ni. This was the subject of homework 3.5. One can think of the first letter as the digit 1, the second letter as the digit 2, etc. This way they correspond with their values. Now the word corresponding to a certain natural number is called the n-adic representation of that number.
The n-adic number representation is a representation in base n, just like the n-ary representation. In both representation schemes the number associated to a string ak-1ak-2...a0 is given by i=0k-1 value(ai) ni. The difference is that in the n-ary representation the values of the digits range from 0 to n-1, and in the n-adic representation from 1 to n. The n-ary representation yields a unique number for every string, but not a unique string for every number. This is due to the number 0. In the decimal representation for instance, the strings 00587 and 587 denote the same number, because initial 0's do not count. This representational inefficiency is eliminated by trading the digit 0 for the digit n, while still every natural number has a representation.
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