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10. This system thus detects all single-digit substitution and transposition errors (including jump transpositions), but at the cost of the check digit possibly being 10, represented by "X". (An alternative is simply to avoid using the serial numbers which result in an "X" check digit.) ISBN-13 instead uses the GS1 algorithm used in EAN numbers.
Each of the first nine digits of the 10-digit ISBN—excluding the check digit itself—is multiplied by its (integer) weight, descending from 10 to 2, and the sum of these nine products found. The value of the check digit is simply the one number between 0 and 10 which, when added to this sum, means the total is a multiple of 11.
ISBN-10 and ISBN-13 use different check digit algorithms to compute the last digit of the ISBN. The following tools will convert any ISBN-10 to an ISBN-13 along with the correct check digit value. Nihiltres' ISBN tool; ISBN converter of the U.S. Library of Congress; ISBN converter of ISBN.org; Stylistically, please:
The check digit is computed as follows: Drop the check digit from the number (if it's already present). This leaves the payload. Start with the payload digits. Moving from right to left, double every second digit, starting from the last digit. If doubling a digit results in a value > 9, subtract 9 from it (or sum its digits).
The checksum digit is the digit which must be added to this checksum to get a number divisible by 10 (i.e. the additive inverse of the checksum, modulo 10). [8] See ISBN-13 check digit calculation for a more extensive description and algorithm. The Global Location Number (GLN) also uses the same method.
Check digit. To detect transcription errors in which digits have been altered or transposed, the last digit of a pre-2007 ISBN must match the result of a mathematical formula incorporating the other 9 digits (ISBN-10 check digits).
Verhoeff had the goal of finding a decimal code—one where the check digit is a single decimal digit—which detected all single-digit errors and all transpositions of adjacent digits. At the time, supposed proofs of the nonexistence [6] of these codes made base-11 codes popular, for example in the ISBN check digit.
In 2007, the length of an ISBN changed from 10 to 13 digits, and a new 3-digit prefix (978 or 979) was added in front of 10-digit ISBNs. [2] The following registration groups are compatible with or without a 978- prefix: 0–5; 600–639; 64–69; 7; 80–94; 950–989; 9900–9989; 99900–99999; The following must have a 979- prefix: