ISBN stands for International Standard Book Number and nearly all materials published anywhere in the world are now assigned such a number because of the many advantages in being able to track and identify books by their ISBN number. The ISBN number is a 10-digit number organized into 4 blocks which are separated by hyphens, which are an essential part of the system. The approach used in the ISBN number is that the 4 blocks can vary in size. This approach yields some advantages and some disadvantages. If codes have sections that can vary in length, one needs to have some way of being able to tell where one block ends and the next begins, which increases the length of the number of symbols used in total. Although the ISBN number has 10 information digits, its length is 13 because of the need for the 3 hyphens to separate the blocks. The hyphens could be eliminated if the blocks had specified lengths. (By way of example, the number of partitions of 10 into four nonzero ordered parts where the first part has length 1 or 2 and the last has length 1 is 13.) The first block is a language or country identifier and is partly aimed at giving the language spoken by the target audience for the book. English books for example are assigned a 0 in the first block, while the German language audience uses the code 3 and French the code 2. The code for Malta as a target audience is 99909. The second block is a publisher number, the third block an item number for that publisher, and the last digit is a check digit. Not surprisingly, large publishers have small identification numbers and small publishers have long ones. The pressure of so many publishers and books by publishers is already pushing the limits of this coding system.

The system for creating the check digit in the ISBN number system is simple but clever. The way that it works in general can easily be understood by looking at a specific example.

Example:

What check digit would be assigned to the book, *Mathematical Hierarchies and Biology* published by the American Mathematical Society?

This book is aimed at the English language audience so a 0 appears in the first block; the American Mathematical Society's publisher code is 8218 (e.g. all AMS books have this number in the second block), and the item number for this book, given to it by AMS, is 0762. Based on these 9 digits we seek the value C (for check digit) of the check digit, which will be placed in the tenth position of the code. We compute 10 times the first digit, 9 times the second, 8 times the third, etc. This gives the calculation:

10(0) + 9(8) + 8(2) + 7(1) + 6(8) + 5(0) + 4(7) + 3(6) + 2(2) + 1(C).

Simplifying we obtain:

0 + 72 + 16 + 7 + 48 + 0 + 28 + 18 + 4 + C.

We now choose C so that the sum is divisible by 11 (i.e. congruent to zero mod 11). Hence, C is 5. The ISBN number for this book is 0-8218-0762-5.

One problem with this system is that since C can be any number from 0 to 10, we are faced with the possibility that the check digit might have two digits, 1 and 0. To get around this problem when the check digit would be 10 the symbol X (for 10 in Roman numerals) is used. This nuisance in the system of having the ISBN consist of symbols that are not all numbers can be avoided by manipulating the item number so as to avoid having X as the check symbol. In practical terms this means that not all the item numbers would be available to a company for use. It is not difficult to prove that the ISBN check digit system catches all single digit substitution errors as well as errors caused by the switching of adjacent digits.

- Introduction
- Barcodes
- Zip codes
- The Universal Product Code
- ISBN Numbers
- The Ultimate System
- References