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The final character of a ten-digit International Standard Book Number is a check digit computed so that multiplying each digit by its position in the number (counting from the right) and taking the sum of these products modulo 11 is 0. The digit the farthest to the right (which is multiplied by 1) is the check digit, chosen to make the sum correct.
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 second-to-last digit (not the last digit of the payload). If doubling a digit results in a value > 9, subtract 9 ...
The calculation of an ISBN-13 check digit begins with the first twelve digits of the 13-digit ISBN (thus excluding the check digit itself). Each digit, from left to right, is alternately multiplied by 1 or 3, then those products are summed modulo 10 to give a value ranging from 0 to 9. Subtracted from 10, that leaves a result from 1 to 10.
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:
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.
The Luhn mod N algorithm generates a check digit (more precisely, a check character) within the same range of valid characters as the input string. For example, if the algorithm is applied to a string of lower-case letters (a to z), the check character will also be a lower-case letter. Apart from this distinction, it resembles very closely the ...
The number is also part of the ISBN, for the later titles; the ISBN for a book (if it has one) can be formed by prefixing "0" for English language/US, and "441" (Ace's publisher number), to the serial number, and then calculating the last digit with an ISBN check digit calculator.