ISBN

An International Standard Book Number (ISBN) is a code used to uniquely identify a book together. It also uniquely encodes the book's publisher and includes information about its language of authorship. The original 10-"digit" ISBN-10 (where a "digit" consists of a decimal digit 0-9 for the first 9 places and 0-9 or X for the tenth place, corresponding to a mixed base string), in use for more than 30 years, was officially replaced with a 13-digit ISBN-13 (where each place is truly a decimal digit) as of Jan. 1, 2007.

The digits of an ISBN are arranged in four groups (for an ISBN-10) or five groups (for an ISBN-13), which are sometimes (but not always) separated by hyphens. At present, an ISBN-13 is always prefixed by the digits 978 (US ISBN Agency). The first group in ISBN-10 or the second group for an ISBN-13 is a single digit which encodes country or language in which a publisher is incorporated: 0 for English, 2 for French, 3 for German, 4 for Japanese, 8 for Indian publishers, etc. The next group of digits specifies the publisher, and may range in length from two to seven digits, with fewer digits used for larger publishers. Some publishers with offices in more than one country (at least when different languages are spoken in those countries) have multiple publisher codes and initial digits.

The next group of digits specifies an individual book, and may be from one to six digits in length. The actual number is eight minus the number of digits in the publisher group, so that small publishers may have only 10 books while large ones can have up to a millions books. The last digit is a check digit which may be in the range 0-9 or X (where X is the Roman numeral for 10) for an ISBN-10, or 0-9 for an ISBN-13.

For ISBN-10, the check digit is computed from the equation

(1)

For example, the ISBN-10 for the first edition of the printed version of MathWorld is 0-8493-9640-9, and

ISBN10			(2)
			(3)
			(4)
			(5)

where denotes a dot product and is the vector composed of the first 9 digits of the ISBN-10.

The scheme used by 978- and (future) 979-prefixed ISBN-13, is instead given by

(6)

(Book Industry Study Group). Therefore, the ISBN-13 corresponding to the ISBN-10 above would have check digit

ISBN13			(7)
			(8)
			(9)
			(10)

and so would be 978-0-8493-9640-3.

The ISBN is error-detecting, but not error-correcting (unless it is known that only a single digit is erroneous). The ISBN detects any single-digit error, as well as most two-digit error resulting from transposing two digits.