Check digit

A check digit is a form of redundancy check used for error detection, the decimal equivalent of a binary checksum. It consists of a single digit computed from the other digits in the message.

Examples

UPC

The final digit of a Universal Product Code is a check digit computed as follows:

1. Add the digits (up to but not including the check digit) in the odd-numbered positions (first, third, fifth, etc.) together and multiply by three.
2. Add the digits (up to but not including the check digit) in the even-numbered positions (second, fourth, sixth, etc.) to the result.
3. If the last digit of the result is 0, then the check digit is 0.
4. If the last digit of the result is not zero, then subtract the last digit from 10. The answer must equal the check digit.

For instance, the UPC-A barcode for a box of tissues is "036000291452". The last digit is the check digit "2", and if the other numbers are correct then the check digit calculation must produce 2.

1. We add the odd number digits 0+6+0+2+1+5 = 14, and multiply by 3 to get 14 × 3 = 42.
2. We add the even number digits to this result 42+3+0+0+9+4 = 58.
3. The last digit is 8, we subtract this from 10 to get 2.

ISBN 10

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 last digit (which is multiplied by 1) is the check digit, chosen to make the sum correct. It may need to have the value 10, which is represented as the letter X. For example, take the ISBN 0-201-53082-1. The sum of products is 0×10 + 2×9 + 0×8 + 1×7 + 5×6 + 3×5 + 0×4 + 8×3 + 2×2 = 98 modulo 11 (10) subtracted from 11 ≡ 1. So the ISBN is valid.

While this may seem more complicated than the first scheme, it can be validated very simply by adding all the products together then dividing by 11. If the result is an integer then the ISBN is valid.

ISBN 13 (in use January, 2007) is equal to the EAN-13 code found underneath a book's barcode. Its check digit is generated the same way as the UPC, except the even digits are multiplied by 3 instead of the odd digits (a UPC can be converted to EAN-13 by prefixing a 0).

Other examples of check digits

• The tenth digit of the National Provider Identifier for the healthcare industry more
• The Australian Tax File Number (based on modulo 11)
• The ninth digit of a Canadian Social Insurance Number (SIN)
• The North American CUSIP number
• The International SEDOL number
• The International Securities Identifying Number (ISIN)
• The International CAS registry number's final digit.
• Modulo 10 check digits in credit card account numbers, calculated with the Luhn algorithm.
  • Also used in the Norwegian KID (customer identification number) numbers used in bank giros (credit transfer).
• The final character encoded in a magnetic stripe card is a computed Longitudinal redundancy check
• final digit of a POSTNET code
• final digit of an ISSN code
• final digit of a DUNS number (though this is scheduled to change, such as that the final digit will be chosen freely in new allocations, rather than being a check digit)
• The Spanish fiscal identification number (número de identificación fiscal, NIF), (based on modulo 23).
• The ninth digit of a Vehicle Identification Number (VIN).
• The ninth digit of an Israeli Teudat Zehut (Identity Card) number.
• The 13th digit of Former Yugoslav Unique Master Citizen Number (JMBG)
• Last check digit in EAN/UPC serialisation of Global Trade Identification Number (GTINs). It applies to GTIN-8, GTIN-12, GTIN-13 and GTIN-14.
• The seventh character of a New Zealand NHI Number.
• The 18th digit of Aragon's IVH Number.

Compare to check bit.

External links

• Identification numbers and check digit schemes (a mathematical explanation of various check digit schemes)

References

Contemporary Abstract Algebra by Joseph Gallian, 5th edition, Houghton Mifflin, 2002, pp. 9-14.

"Modular Arithmetic in the Marketplace", by Joseph Gallian and Steven Winters, in The American Mathematical Monthly, Vol. 95, No. 6, 1988, pp. 548-551.de:Prüfziffer es:Dígito verificador eo:Kontrolkodo fr:Clé de contrôle he:ספרת ביקורת nl:Elfproef ja:チェックディジット pt:Dígito verificador ru:Контрольное число