checkdigit

A check digit is a form of redundancy used in identification numbers to help detect errors in data entry, transcription, or transmission. It is a single digit added to the end of a numeric sequence and computed from the other digits in the sequence. The goal is to allow quick validation of the entire number and to catch common mistakes such as mistyped digits or swapped adjacent digits.

Common algorithms rely on modular arithmetic. The most widely used family is modulus-10 schemes, including the Luhn algorithm, which is employed for credit card numbers and many other identifiers. In Luhn, every second digit from the right is doubled; if the result is greater than 9, 9 is subtracted. The sum of all digits, including the check digit, should be a multiple of 10. ISBN-13 and EAN-13 use a related mod-10 check with alternating weights of 1 and 3. Modulus-11 schemes also exist, where the check digit can be 0–10 (often represented as X). Some systems use larger moduli or more complex rules to detect a broader range of errors.

Applications and limitations: Check digits are used in ISBNs and ISSNs, UPCs and EANs, bank account numbers, and various product and library identifiers. They provide a simple, fast error-detection mechanism requiring little computation. However, no check-digit scheme guarantees error-free data; certain transpositions or multiple-digit errors can go undetected, and check digits do not provide data security or authentication.