The last digit of an ISBN-13 is a check digit calculated from the first 12 digits. Alternating weights of 1 and 3 are applied, the products are added, and the final digit is chosen so the total becomes a multiple of 10. If a digit is mistyped, the check usually fails.
The thirteenth digit is not another piece of publisher or title information. It is a compact arithmetic test attached to the identifier. The International ISBN Agency’s ISBN-13 method alternates weights of 1 and 3 across the first 12 digits, then chooses a final digit that makes the weighted total divisible by 10.
Start with the first 12 digits
Take the example used in the ISBN manual: 978-0-11-000222-?. Ignore the hyphens for the calculation. Reading from the left, multiply the first digit by 1, the second by 3, the third by 1, and continue alternating.
For 978011000222, the products are 9, 21, 8, 0, 1, 3, 0, 0, 0, 6, 2, and 6. Their sum is 56.
Find what the total needs to reach
Next, look at the remainder when 56 is divided by 10. The remainder is 6. The check digit must bring the total to the next multiple of 10, so 10 minus 6 gives 4. The complete ISBN is therefore 978-0-11-000222-4.
There is one small edge case: if the subtraction would produce 10, the check digit is written as 0. Another way to express the same rule is to choose the digit from 0 through 9 that makes the full weighted sum divisible by 10.
Verify an ISBN instead of recalculating its identity
To check an ISBN you already have, include the thirteenth digit with weight 1 after applying the alternating weights to the first 12 digits. If the final sum is divisible by 10, the number passes the check-digit test. That does not prove the ISBN belongs to the book in your hand; it only shows that the digit pattern is internally consistent.
What the check digit can—and cannot—catch
A checksum is designed for error detection, not error correction. Many ordinary single-digit typing mistakes will make the total fail the divisibility test, alerting a catalog, bookseller, or data-entry system that something is wrong. The calculation does not tell you which digit was wrong, and not every possible multi-digit error is guaranteed to be detected.
That modest job is exactly why the method is useful. A tiny amount of arithmetic gives a 13-digit identifier a built-in consistency check without requiring a database lookup.