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Unformatted text preview: 1 Z 11 and ISBN numbers As explained earlier, Z 11 is a set consisting of 11 members, namely, the integers 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 and 10. The members of Z 10 are called digits . The members of Z 2 are called bits . By analogy with these two widely used names, we are going to call the mem bers of Z 11 elvits . So an elvit is one of the numbers 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10. Notice that there are eleven elvits . 1.1 ISBN numbers ISBN numbers are used to catalog and identify books. Each book has an ISBN number. An ISBN number is a sequence ( m 1 ,m 2 ,m 3 ,m 4 ,m 5 ,m 6 ,m 7 ,m 8 ,m 9 ,m 10 ) of ten elvits. When we work with ISBN numbers, the name of the elvit “10” is “ X ”, so Z 11 = n , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ,X o . The first nine of these ten elvits are required to belong to the set n , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 o , so m 10 is the only elvit in an ISBN number that is permitted to have the value X . Furthermore, m 10 is a redundant symbol , included for errorcorrecting purposes. It is always given by the formula m 10 = 9 X j =1 jm j , (1.1.1) that is, m 10 = m 1 + 2 m 2 + 3 m 3 + 4 m 4 + 5 m 5 + 6 m 6 + 7 m 7 + 8 m 8 + 9 m 9 . Why do we use Z 11 rather than Z 10 ? And why do we use Equation (1.1.1)? I will explain this eventually. But before we do that, let us verify the formula for some books. EXAMPLE 1. Our textbook has the ISBN number 0534399002. Let us verify (1.1.1), making sure we remember that we are working in Z 11 , so all 1 the operations are performed modulo 11. m 1 + 2 m 2 + 3 m 3 + 4 m 4 + 5 m 5 + 6 m 6 + 7 m 7 + 8 m 8 + 9 m 9 = 0 + 2 × 5 + 3 × 3 + 4 × 4 + 5 × 3 + 6 × 9 + 7 × 9 + 8 × 0 + 9 × = 0 + X + 9 + 5 + 4 + X + 8 + 0 + 0 = 0 + 10 + 9 + 5 + 4 + 10 + 8 + 0 + 0 reduced modulo 11 = 46 reduced modulo 11 = 2 . So it works! EXAMPLE 2. The book “An Introduction to Analysis,” by William R. Wade, has the ISBN number 013093089X. Let us verify (1.1.1). m 1 + 2 m 2 + 3 m 3 + 4 m 4 + 5 m 5 + 6 m 6 + 7 m 7 + 8 m 8 + 9 m 9 = 0 + 2 × 1 + 3 × 3 + 4 × 0 + 5 × 9 + 6 × 3 + 7 × 0 + 8 × 8 + 9 × 9 = 0 + 2 + 9 + 0 + 45 + 18 + 0 + 64 + 81 = 11 + 1 + 7 + 9 + 4 = 1 + 7 + 9 + 4 = 21 = 10 = X . Once again, it works! I suggest you look at a few books that you can find at home or in the library, look at heir ISBN numbers, and check that in all cases Formula (1.1.1) is true. Now that you are convinced that ISBN numbers do indeed obey Formula (1.1.1), the three questions that you ought to be asking yourself are: 1. Why do we use Z 11 ?, 2. Why do we include m 10 , which contains absolutely no new information, since it is completely determined by the first 9 elvits?, 3. Why do we use Formula (1.1.1)?...
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 Spring '08
 ctw
 Math, Integers, Prime number, Universal quantification, International Standard Book Number, check digit, ISBN numbers

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