A check digit is a form of redundancy check used for error detection on identification numbers, such as bank account numbers, which are used in an application where they will at least sometimes be input manually. It is analogous to a binary number parity bit used to check for errors in computergenerated data. It consists of one or more Numerical digit computed by an algorithm from the other digits (or letters) in the sequence input.
With a check digit, one can detect simple errors in the input of a series of characters (usually digits) such as a single mistyped digit or some permutations of two successive digits.
Design
Check digit
are generally designed to capture
human transcription errors. In order of complexity, these include the following:

single digit errors, such as 1 → 2

transposition errors, such as 12 → 21

twin errors, such as 11 → 22

jump transpositions errors, such as 132 → 231

jump twin errors, such as 131 → 232

phonetic errors, such as 60 → 16 ("sixty" to "sixteen")
In choosing a system, a high probability of catching errors is traded off against implementation difficulty; simple check digit systems are easily understood and implemented by humans but do not catch as many errors as complex ones, which require sophisticated programs to implement.
A desirable feature is that leftpadding with zeros should not change the check digit. This allows variable length digits to be used and the length to be changed.
If there is a single check digit added to the original number, the system will not always capture multiple errors, such as two replacement errors (12 → 34) though, typically, double errors will be caught 90% of the time (both changes would need to change the output by offsetting amounts).
A very simple check digit method would be to take the sum of all digits (digital root) modulo operation 10. This would catch any singledigit error, as such an error would always change the sum, but does not catch any transposition errors (switching two digits) as reordering does not change the sum.
A slightly more complex method is to take the weighted sum of the digits, modulo 10, with different weights for each number position.
To illustrate this, for example if the weights for a four digit number were 5, 3, 2, 7 and the number to be coded was 4871, then one would take 5×4 + 3×8 + 2×7 + 7×1 = 65, i.e. 65 modulo 10, and the check digit would be 5, giving 48715.
Systems with weights of 1, 3, 7, or 9, with the weights on neighboring numbers being different, are widely used: for example, 31 31 weights in UPC codes, 13 13 weights in EAN numbers (GS1 algorithm), and the 371 371 371 weights used in United States bank routing transit numbers. This system detects all singledigit errors and around 90% of transposition errors. 1, 3, 7, and 9 are used because they are coprime to 10, so changing any digit changes the check digit; using a coefficient that is divisible by 2 or 5 would lose information (because 5×0 = 5×2 = 5×4 = 5×6 = 5×8 = 0 modulo 10) and thus not catch some singledigit errors. Using different weights on neighboring numbers means that most transpositions change the check digit; however, because all weights differ by an even number, this does not catch transpositions of two digits that differ by 5, (0 and 5, 1 and 6, 2 and 7, 3 and 8, 4 and 9), since the 2 and 5 multiply to yield 10.
The code instead uses modulo 11, which is prime, and all the number positions have different weights 1, 2, ... 10. This system thus detects all single digit substitution and transposition errors (including jump transpositions), but at the cost of the check digit possibly being 10, represented by "X". (An alternative is simply to avoid using the serial numbers which result in an "X" check digit.) instead uses the GS1 algorithm used in EAN numbers.
More complicated algorithms include the Luhn algorithm (1954), which captures 98% of single digit transposition errors (it does not detect 90 ↔ 09) and the still more sophisticated Verhoeff algorithm (1969), which catches all single digit substitution and transposition errors, and many (but not all) more complex errors. Similar is another abstract algebrabased method, the Damm algorithm (2004), that too detects all singledigit errors and all adjacent transposition errors. These three methods use a single check digit and will therefore fail to capture around 10% of more complex errors. To reduce this failure rate, it is necessary to use more than one check digit (for example, the modulo 97 check referred to below, which uses two check digits  for the algorithm, see International Bank Account Number) and/or to use a wider range of characters in the check digit, for example letters plus numbers.
Examples
UPC
The final digit of a Universal Product Code is a check digit computed as follows:

Add the digits in the oddnumbered positions (first, third, fifth, etc.) together and multiply by three.

Add the digits (up to but not including the check digit) in the evennumbered positions (second, fourth, sixth, etc.) to the result.

Take the remainder of the result divided by 10 (modulo operation). If the remainder is equal to 0 then use 0 as the check digit, and if not 0 subtract the remainder from 10 to derive the check digit.
For instance, the UPCA barcode for a box of tissues is "036000241457". The last digit is the check digit "7", and if the other numbers are correct then the check digit calculation must produce 7.

Add the odd number digits: 0+6+0+2+1+5 = 14.

Multiply the result by 3: 14 × 3 = 42.

Add the even number digits: 3+0+0+4+4 = 11.

Add the two results together: 42 + 11 = 53.

To calculate the check digit, take the remainder of (53 / 10), which is also known as (53 modulo 10), and if not 0, subtract from 10. Therefore, the check digit value is 7. i.e. (53 / 10) = 5 remainder 3; 10  3 = 7.
Another example: to calculate the check digit for the following food item "01010101010
x".

Add the odd number digits: 0+0+0+0+0+0 = 0.

Multiply the result by 3: 0 x 3 = 0.

Add the even number digits: 1+1+1+1+1=5.

Add the two results together: 0 + 5 = 5.

To calculate the check digit, take the remainder of (5 / 10), which is also known as (5 modulo 10), and if not 0, subtract from 10: i.e. (5 / 10) = 0 remainder 5; (10  5) = 5. Therefore, the check digit x value is 5.
ISBN 10
The final character of a tendigit 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 operation 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. It may need to have the value 10, which is represented as the letter X. For example, take the : The sum of products is 0×10 + 2×9 + 0×8 + 1×7 + 5×6 + 3×5 + 0×4 + 8×3 + 2×2 + 1×1 = 99 ≡ 0 (mod 11). So the ISBN is valid. Note that positions can also be counted from left, in which case the check digit is multiplied by 10, to check validity: 0×1 + 2×2 + 0×3 + 1×4 + 5×5 + 3×6 + 0×7 + 8×8 + 2×9 + 1×10 = 143 ≡ 0 (mod 11).
ISBN 13
ISBN 13 (in use January 2007) is equal to the EAN13 code found underneath a book's barcode. Its check digit is generated the same way as the UPC except that the even digits are multiplied by 3 instead of the odd digits.
EAN (GLN, GTIN, EAN numbers administered by GS1)
EAN (European Article Number) check digits (administered by GS1) are calculated by summing each of the odd position numbers multiplied by 3 and then by adding the sum of the even position numbers. Numbers are examined going from right to left, so the first odd position is the last digit in the code. The final digit of the result is subtracted from 10 to calculate the check digit (or left asis if already zero).
A GS1 check digit calculator and detailed documentation is online at GS1's website.
Another official calculator page shows that the mechanism for GTIN13 is the same for Global Location Number/GLN.
Other examples of check digits
International

The International SEDOL number.

The final digit of an ISSN code or IMO Number.

The International Securities Identifying Number (ISIN).

Object Management Group FIGI standard final digit.
[http://openfigi.com]

The International CAS registry number's final digit.

Modulo 10 check digits in credit card account numbers, calculated by the Luhn algorithm.

Also used in the Norwegian KID (customer identification number) numbers used in bank giros (credit transfer),

Used in IMEI of mobile phones.

Last check digit in EAN/UPC serialisation of Global Trade Identification Number (GTIN). It applies to GTIN8, GTIN12, GTIN13 and GTIN14.

The 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 third and fourth digits in an International Bank Account Number (Modulo 97 check).

The final digit in an International Standard Text Code.

The final character encoded in a magnetic stripe card is a computed Longitudinal redundancy check.
In the USA

The tenth digit of the National Provider Identifier for the US healthcare industry.

The final digit of a POSTNET code.

The North American CUSIP number.

The final (ninth) digit of the ABA routing transit number, a bank code used in the United States.

The ninth digit of a Vehicle Identification Number (VIN).

Mayo Clinic patient identification numbers used in Arizona and Florida include a trailing check digit.

The eleventh digit of a Customs & Border Protection entry number.
In Central America

The Guatemalan Tax Number (NIT  Número de Identificación Tributaria) based on modulo operator 11.
In Eurasia

The Spanish fiscal identification number (número de identificación fiscal, NIF), (based on modulo 23).

The ninth digit of an Teudat Zehut (Identity Card) number.

The 13th digit of the and SFRY Unique Master Citizen Number (JMBG). (but not all of them, due to errors or nonresidency)

The last two digits of the 11digit Turkish Identification Number (TC Kimlik Numarası).

The ninth character in the 14character European Union cattle passport number (cycles from 1 to 7: see British Cattle Movement Service).

The ninth digit in an Kennitala (national ID number).

Modulo 97 check digits in a Belgium and bank account numbers. Serbia sometimes also uses modulo 11, for reference number.

The ninth digit in a Hungary TAJ number (social insurance number).

For the residents of India, the unique identity number named Aadhaar has a trailing 12th digit that is calculated with the Verhoeff algorithm.

The Intellectual Property Office of Singapore (IPOS) has confirmed a new format for application numbers of registrable Intellectual Property (IP, e.g., Trademark, , registered designs). It will include a check character calculated with the Damm algorithm.

The last digit of Chinese citizen ID number (second generation) is calculated by modulo 112 as specified in Chinese GuoBiao (aka. national standard) GB116431999 which adopts ISO 7064:1983. 'X' is used if the calculated checking digit is 10.
In Oceania

The Australian tax file number (based on modulo operator 11).

The seventh character of a New Zealand NHI Number.

The last digit in a New Zealand locomotive's Traffic Monitoring System (TMS) number.
Algorithms
Notable algorithms include:
See also

Casting out nines, similar modular sum check

Check bit, binary equivalent
External links