The International Standard Book Numbering system is an interesting example of a code that detects two sorts of errors: a single mistaken digit or a transposition of two digits. The ISBN is a 10-digit code representing a book. Say the code for a particular book is ABCDEFGHIJ. Then:

• the first digit indicates the language in which the book is published (not necessarily the language in which the book is written: for example, Spanish textbooks published in the United States have an ISBN starting with 0, meaning English, even if the book is an immersion textbook, written entirely in Spanish);

• the next three or four digits are a code for the publisher;

• the next few digits—all but the last—indicate the particular book;

• the last digit, J, is chosen so that

                                               10A+9B+8C+7D+6E +5F +4G+3H +2I +J

is divisible by 11.

There is a possible problem here. Suppose 10A + 9B + 8C + 7D + 6E + 5F + 4G+3H +2I leaves a remainder 1 on division by 11. It is necessary to have J = 10,  and there is no single digit available. The standard remedy is to write X for the last digit in this case.

Suppose a digit is written incorrectly in an ISBN. For example, suppose that instead of ABCDEFGHIJ, one writes ABCdEFGHIJ, where D and d are different.

Then when you test for divisibility by 11, the check sum 10A+9B+8C+7d +6E +5F +4G+3H +2I +J is calculated. Now

                     10A+9B+8C+7d +6E +5F +4G+3H +2I +J

                          = 10A+9B+8C+7D+6E +5F +4G+3H +2I +J +7(d −D),

and as d = D this difference 7(d −D) is a product of two numbers both less than 11 in size. So it is not divisible by 11. So the supposed check sum is not divisible by 11. The same argument applies if any one digit is written incorrectly.

transposed. The argument does not depend on which pair; as an example, suppose the C and H are interchanged. Then the check sum is

                              10A+9B+8H+7D+6E +5F +4G+3C+2I +J

                               = 10A+9B+8C+7D+6E +5F +4G+3H +2I +J +5(H −C);

and again the check sum is wrong, because H −C is less than 11.

The standard way of writing an ISBN is with dashes after the first, fifth and ninth digit: for example, 0-8176-8319-X.

Sample Problem 1.1 Find the check digit for an ISBN number that starts 0- 6693-3907.

Solution. Multiplying, 10×0+9×6+8×6+7×9+6×3+5×3+4×9+3× 0+2×7 = 0+54+48+63+18+15+36+0+14= 248. Now 248 = 242+6= 11×22+6, So we must add 5 to obtain a multiple of 11, and the check digit is 5.

In 2007, a new version of the ISBN was introduced. It has 13 digits, so it is referred to as the ISBN-13, and the original is now called the ISBN-10. The ISBN starts with either the three digits 978 or 979. The next nine digits are the same as the first nine digits of the ISBN-10. Finally, the check digit is appended. It is defined as follows: the sum of the even-position digits plus three times the sum of the odd-position digits must be divisible by 10. The use of 10, rather than 11,  eliminates the need for the symbol X as a possible check digit, but more errors may pass undetected.

The ISBN-13s are a class of International Article Numbers, also called EANs,  13-digit codes that are allocated to a large number of different retail articles. (They were originally called European Article Numbers, whence the abbreviation.) The introductory code numbers 978 and 979 are allocated to book manufacturers, and other introductory codes are assigned to other products. These numbers are used in making the barcodes that are scanned when items are purchased.

Sample Problem 1.2 What is the ISBN-13 corresponding to the ISBN 0-8176- 8319-X, assuming that the first three digits are 978?

Solution. The first 12 digits are 978081768319. Suppose the check digit is x.