ISBN

The original 9-digit standard book number (SBN) had no registration group identifier, but prefixing a zero to a 9-digit SBN creates a valid 10-digit ISBN.

Publishers receive blocks of ISBNs, with larger blocks allotted to publishers expecting to need them; a small publisher may receive ISBNs of one or more digits for the registration group identifier, several digits for the registrant, and a single digit for the publication element. Once that block of ISBNs is used, the publisher may receive another block of ISBNs, with a different registrant element. Consequently, a publisher may have different allotted registrant elements. There also may be more than one registration group identifier used in a country. This might occur once all the registrant elements from a particular registration group have been allocated to publishers.

It is also true for ISBN-10s that the sum of all ten digits, each multiplied by its weight in ascending order from 1 to 10, is a multiple of 11. For this example:

In contrast, it is possible for other types of error, such as two altered non-transposed digits, or three altered digits, to result in a valid ISBN (although it is still unlikely).

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.

For example, the check digit for an ISBN-10 of 0-306-40615-? is calculated as follows:

For example, the check digit for the ISBN-10 of 0-306-40615-? is calculated as follows:

// Returns ISBN error syndrome, zero for a valid ISBN, non-zero for an invalid one.

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. A zero replaces a ten, so, in all cases, a single check digit results.

For example, the ISBN-13 check digit of 978-0-306-40615-? is calculated as follows:

s = 9×1 + 7×3 + 8×1 + 0×3 + 3×1 + 0×3 + 6×1 + 4×3 + 0×1 + 6×3 + 1×1 + 5×3 = 9 + 21 + 8 + 0 + 3 + 0 + 6 + 12 + 0 + 18 + 1 + 15 = 93
93 / 10 = 9 remainder 3
10 – 3 = 7

Thus, the check digit is 7, and the complete sequence is ISBN 978-0-306-40615-7.

Additionally, if the sum of the 2nd, 4th, 6th, 8th, 10th, and 12th digits is tripled then added to the remaining digits (1st, 3rd, 5th, 7th, 9th, 11th, and 13th), the total will always be divisible by 10 (i.e., end in 0).

An ISBN-10 is converted to ISBN-13 by prepending "978" to the ISBN-10 and recalculating the final checksum digit using the ISBN-13 algorithm. The reverse process can also be performed, but not for numbers commencing with a prefix other than 978, which have no 10-digit equivalent.