The International Standard Book Number Check Digit Scheme
Starting in the years 1968-72, all books published were assigned a ten digit number or an International Standard Book Number (ISBN). This system was developed as book publishers and wholesalers began to computerize their inventories. Identifying a book by its author, title, edition, etc., was replaced by an internationally recognized number. This made book ordering easier and more efficient, overcame language barriers (ordering books from foreign countries), and cleared up many other problems.
The presentation of an ISBN varies slightly from book to book. While each ISBN is always ten digits long, digits can be arranged slightly differently. The first number in an ISBN is the group or country number, and it identifies the language area, national, or geographic grouping of publishers. For example, a 0 indicates that the book was published in the English speaking world (United Kingdom, United States, Australia, etc.), a 2 indicates the book was published in the French speaking world (Belgium, France, etc.), a 3 indicates it was published in the German speaking world (Austria, Germany, etc.), the number 87 indicates Denmark, and the number 90 indicates Holland. The second number in an ISBN identifies the publisher. This number usually is between two to five digits in length, but can be longer.Examples
0-19-853287-3 is the ISBN for the book Codes and Cryptography, by Dominic Welsh. The publisher code is 19 and denotes Oxford University Press.
0-471-51001-7 is the ISBN for the book Modern Algebra, An Introduction, by John Durbin. The publisher code is 471 and denotes John Wiley & Sons, Inc.
0-8176-3805-9 is the ISBN for the book Math Into Latex, An Introduction To Latex and AMS-Latex, by George Gratzer. The publisher code is 8176 and denotes Birkhauser.
0-86720-498-2 is the ISBN for the book Introduction To Linear Algebra, by Geza Schay. The publisher code is 86720 and denotes Jones and Bartlett Publishers.
The third number in the ISBN is the code the publisher has chosen for the book. Note from the examples above, that a longer publisher code leaves fewer digits available for book titles. Thus a publisher who has a large number of book titles will want a shorter publisher code. The last number is always a single digit and is the check digit.
For example, the following two ISBNs are used to identify Neal
Koblitz's book A Course in Number Theory and Cryptography.
ISBN #1: 0-387-96576-9 ISBN #2:3-540-96576-9
In the first number, the leading 0 indicates the version of the book published in English. The second number 387 is the code for the Spinger-Verlag publishing company in New York City. The third number 96576 identifies the book and is the same for both versions. The last digit 9 is the check digit. In the second number, the leading 3 indicates the version of the book published in German. The second number 540 is the code for the Spinger-Verlag publishing company in Berlin. The third number 96576 identifies the book and is the same for both versions. The last digit 9 is the check digit.
The ISBN Check Digit Scheme
For a1a2a3a4a5a6a7a8a9a10, the ten digit ISBN, the check digit a10 is appended to the nine digit identification number a1a2a3a4a5a6a7a8a9 such that a10 satisfies the equation
10a1 + 9a2 + 8a3 + 7a4 + 6a5 + 5a6 + 4a7 + 3a8 + 2a9 + a10 = 0 (mod 11)
If the check digit a10 is 10, the letter X is used instead.
The remainder when a number is divided by 11 could be any digit from 0 to 9 or the number 10. Since the ISBN scheme uses modulo 11 arithmetic and wants the check digit a10 to be a single character, it assigns a10 the value of X when 10 is to the check digit. The ISBN for the book Linear Algebra and its Applications, by David Lay, is 0-201-52032-X. The X indicates that the check digit is the number 10.
Examples:
Valid ISBN: 0-201-52032-X
Calculation: 10a1 + 9a2 + 8a3 + 7a4 + 6a5 + 5a6 + 4a7 + 3a8 + 2a9 + a10 (mod 11)
= 10· 0+9· 2 + 8· 0 + 7· 1 + 6· 5 + 5· 2 + 4· 0 + 3· 3 + 2· 2 + 10(mod 11)
= 0 + 18 + 0 + 7 + 30 + 10 + 0 + 9 + 4 + 10 (mod 11)
= 88 (mod 11)
= 0
Since the remainder is 0 when 88 is divided by 11, this is a valid ISBN.
Invalid ISBN: 3-357-02001-4
Calculation: 10a1 + 9a2 + 8a3 + 7a4 + 6a5 + 5a6 + 4a7 + 3a8 + 2a9 + a10 (mod 11)
= 10· 3+9· 3 + 8· 5 + 7· 7 + 6· 0 + 5· 2 + 4· 0 + 3· 0 + 2· 1 + 4 (mod 11)
= 30 + 27 + 40 + 49 + 0 + 10 + 0 + 0 + 2 + 4 (mod 11)
= 162 (mod 11)
= 8
Since the remainder is 8 (not 0) when 162 is divided by 11, this is not a valid ISBN.
The ISBN scheme does catch all single digit and transposition of adjacent digit errors. However, this does come at a cost. The ISBN check digit schemes introduces a new character into the ISBN, an X, and only works for identification numbers that have 10 digits in them.
Paper Assignment: You are employed by an organization that wants to use a check digit scheme. Assume that you are only familiar with the UPC and the ISBN schemes. Your job is to recommend to the company, in a report called a feasibility study, either the UPC or ISBN scheme. To do this, compare these two schemes, paying attention to the following factors:
ease of use (the length of, time involved with, and complexity of the calculations),
strength (in terms of the errors it can detect).
In making your evaluation, consider also the type of organization that will employ the check digit scheme.
More information on ISBN's can be found at the following sites:
ISBN Info
List of all Group Number
