Check Digit

• Check Digit

https://store.theartofservice.com/the-check-digit-toolkit.html

International Standard Book Number Check digits

1 A check digit is a form of redundancy check used for error detection, the

decimal equivalent of a binary check bit. It consists of a single digit

computed from the other digits in the message.


International Standard Book Number ISBN-10 check digits

1 The 2001 edition of the official manual of the International ISBN

Agency says that the ISBN-10 check digit – which is the last digit of the

ten-digit ISBN – must range from 0 to 10 (the symbol X is used for 10), and must be such that the sum of all the

ten digits, each multiplied by its (integer) weight, descending from 10

to 1, is a multiple of 11.https://store.theartofservice.com/the-check-digit-toolkit.html


1 Formally, using modular arithmetic,

we can say:



1 It is also true for ISBN-10's that the sum of all the ten digits, each

multiplied by its weight in ascending order from 1 to 10, is a multiple of

11. For this example:



1 The two most common errors in handling an ISBN (e.g., typing or writing it) are a single

altered digit or the transposition of adjacent digits. It can be proved that all possible valid ISBN-10's have at least two digits different from each other. It can also be proved that there are no pairs of valid ISBN-10's with eight identical digits and two transposed

digits. (These are true only because the ISBN is less than 11 digits long, and because 11 is

prime.)



1 The ISBN check digit method therefore ensures that it will always be possible to detect these two most common types of error, i.e. if either of these types of error has occurred, the result will never be a

valid ISBN - the sum of the digits multiplied by their weights will never be a multiple of

11. However, if the error occurs in the publishing house and goes undetected, the

book will be issued with an invalid ISBN.



1 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 number (although it is still unlikely).


International Standard Book Number ISBN-10 check digit calculation

1 Modular arithmetic is convenient for calculating the check digit using modulus 11. Each of the first nine digits of the ten-

digit ISBN – excluding the check digit, itself – is multiplied by a number in a sequence

from 10 to 2, and the remainder of the sum, with respect to 11, is computed. The

resulting remainder, plus the check digit, must equal 11; therefore, the check digit is 11 minus the remainder of the sum of the

products.https://store.theartofservice.com/the-check-digit-toolkit.html


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



1 Thus the check digit is 2, and the complete sequence is ISBN 0-306-

40615-2. The value required to satisfy this condition might be 10; if

so, an 'X' should be used.



1 The 2005 edition of the International ISBN Agency's official manual

describes how the 13-digit ISBN check digit is calculated.



1 The calculation of an ISBN-13 check digit begins with the first 12 digits of the

thirteen-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 (0) replaces a ten (10), so, in all cases, a single check digit results.



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



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



1 Formally, the ISBN-13 check digit calculation is:



1 The ISBN-10 formula uses the prime modulus 11 which avoids this blind

spot, but requires more than the digits 0-9 to express the check digit.



1 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).



1 public static boolean isISBN13Valid(String

isbn) {



1 check += Integer.valueOf(isbn.substring(i, i + 1));



1 function isValidISBN13(ISBNumbe

r) {



1 sum = transformed_digits.r

educe(:+)



1 check = (10 - (sum(int(digit) * (3 if idx % 2 else 1) for idx, digit in

enumerate(isbn[:12])) % 10)) % 10



1 FUNCTION validate_isbn_13(isb

n VARCHAR2) RETURN INTEGER IS



1 modular INTEGER;



1 IF modular = 0 THEN



1 reminder := 10 - modular;



1 IF TO_CHAR( reminder ) = SUBSTR(isbn, 13, 1 )

THEN



1 # Bourne-Again Shell


International Standard Music Number - Check digit

1 To calculate the check digit, each digit of the ISMN is multiplied by a weight, alternating 1 and 3 left to right. These weighted digits are

added together. The check digit is the integer between 0 and 9 that makes the sum a multiple of 10.


ISO/IEC 7812 - Check digit

1 The final digit is a check digit which is calculated using the Luhn

algorithm, defined in Annex B of ISO/IEC 7812-1.


ISO 6346 - Check Digit

1 The check digit consists of one numeric digit providing a means of

validating the recording and transmission accuracies of the owner

code and serial number.


International Bank Account Number - Generating IBAN check digits

1 According to the ECBS "generation of the IBAN shall be the exclusive

responsibility of the bank/branch servicing the account". The ECBS document replicates part of the

ISO/IEC 7064:2003 standard as a method for generating check digits in

the range 02 to 98. Check digits in the ranges 00 to 96, 01 to 97, and 03

to 99 will also provide validation of an IBAN, but the standard is silent as to whether or not these ranges may

be used.



1 Check that the total IBAN length is correct as per the country. If not, the IBAN is invalid



1 Replace the two check digits by 00 (e.g. GB00 for the UK)



1 Move the four initial characters to the end of the string



1 Replace the letters in the string with digits, expanding the string as

necessary, such that A or a = 10, B or b = 11, and Z or z = 35. Each alphabetic character is therefore

replaced by 2 digits



1 Calculate mod-97 of the new number, which

results in the remainder



1 Subtract the remainder from 98, and use the result for the two check

digits. If the result is a single digit number, pad it with a leading 0 to

make a two-digit number


Codabar - Check digit

1 Because Codabar is self-checking, most standards do not define a check digit.http://mdn.morovia.com/manuals/bax3/shared.bartech.php#Symbolo

gy.Codabar



1 Some standards that use Codabar will define a check digit, but the

algorithm is not universal. For purely numerical data, such as the library barcode pictured above, the Luhn

algorithm is popular.http://www.makebarcode.co

m/specs/codabar.html



1 When all 16 symbols are possible, a simple modulo-16 checksum is

used.http://www.barcodesymbols.com/codabar.htm The values 10 through 19 are assigned to the

symbols –$:/+.ABCD, respectively.


Universal Product Code - Check digits

1 In the UPC-A system, the check

digit is calculated as follows:



1 # Add the digits in the Even and odd numbers|odd-numbered positions

(first, third, fifth, etc.) together and multiply by three.



1 # Add the digits in the even and odd numbers|even-numbered positions (second, fourth, sixth, etc.) to the

result.



1 # Find the result modulo operation|modulo 10 (i.e. the remainder when

divided by 10.. 10 goes into 58 5 times with 8 leftover).



1 # If the result is not zero, subtract the result

from ten.



1 For example, in a UPC-A barcode 03600029145'x' where 'x' is the unknown check digit, 'x' can be

calculated by



1 *adding the odd-numbered digits

(0+6+0+2+1+5 = 14),



1 *calculating modulo ten (58mod10 = 8),



1 *subtracting from ten (10minus;8 = 2).



1 This should not be confused with the numeral X which stands for a value of 10 in modulo 11, commonly seen in the International Standard Book

Number|ISBN check digit.


Code 128 - Check digit calculation

1 The remainder of the division is the check digit's 'value' which is then

converted into a character (following the instructions given Code

128#Conversion to char|below) and appended to the end of the barcode.



1 For example, in the following table, the code 128 variant A checksum

value is calculated for the alphanumeric string PJJ123C


Code 128 - Calculating check digit with multiple variants

1 As Code 128 allows multiple variants, as well as switching between

variants within a single barcode, it is important to remember that the

absolute Code 128 value of a character is completely independent of its value within a given variant. For instance the Variant C value 33 and

the Variant B value A are both considered to be a Code 128 value of

33, and the check digit would be computed based on the value of 33 times the character's position within

the barcode.


British Cattle Movement Service - Check Digit

1 The check digit for a cow’s ear tag is calculated by dividing the number obtained from the herd mark and

animal number by 7 and adding one to the remainder. Take UK herd

number 303565 cow number 01234. We work out the check digit as

follows:


MSI Barcode - Check digit calculation

1 The MSI barcode uses one of five possible schemes for calculating a check digit:


MSI Barcode - Check digit calculation

1 * Modulo operation|Mod 10 (most

common)


MSI Barcode - Mod 10 Check Digit

1 When using the Mod 10 check digit algorithm, a string to be encoded

1234567 will be printed with a check digit of 4:



1 [http://publib.boulder.ibm.com/infocenter/printer/v1r1/index.jsp?

topic=/com.ibm.printers.afpproducts/

com.ibm.printers.ppfaug/ib6p8mst334.htm IBM Printing

Systems Information Center - Check Digit Calculation Method], IBM.



1 uses the Luhn algorithm.



1 2



1 This example is using the IBM modulo 11 algorithm with a weighting pattern of

(2,3,4,5,6,7)



1 Let X = the final product of the string

to encode.



1 3. Mod the sum by 11, subtract the result from 11, and then apply the mod 11 function

again.


MSI Barcode - Mod 1010 check digit

1 Simply calculate the Mod 10 check digit the first time and then calculate it again with the previous result and append the result of the second Mod

10 Calculation to the string to be encoded.


MSI Barcode - Mod 1110 check digit

1 Same as Mod 1010 but the first calculation should be a Mod 11

Check digit.


International Mobile Equipment Identity - Check digit computation

1 The last number of the IMEI is a check digit calculated using the Luhn algorithm.



1 According to the [http://www.gsma.com/documents/ts-06-6-0-imei-allocation-and-approval-guidelines/20164/ IMEI Allocation and

Approval Guidelines],



1 The Check Digit shall be calculated according to Luhn algorithm|Luhn formula (ISO/IEC 7812). (See GSM 02.16 / 3GPP 22.016). The Check

Digit is a function of all other digits in the IMEI. The Software Version

Number (SVN) of a mobile is not included in the calculation.



1 The purpose of the Check Digit is to help guard against the possibility of incorrect entries to the CEIR and EIR

equipment.



1 The presentation of the Check Digit both electronically and in printed

form on the label and packaging is very important. Logistics (using bar-

code reader) and EIR/CEIR administration cannot use the Check

Digit unless it is printed outside of the packaging, and on the ME IMEI/Type Accreditation label.



1 The check digit is not transmitted over the radio interface, nor is it stored in the EIR database at any

point. Therefore, all references to the last three or six digits of an IMEI refer to the actual IMEI number, to which

the check digit does not belong.



1 # Starting from the right, double every other

digit (e.g., 7 → 14).



1 Conversely, one can calculate the IMEI by choosing the check digit that would give a sum divisible by 10. For the example IMEI 49015420323751?,



1 To make the sum divisible by 10, we set ? = 8, so the IMEI is 490154203237518.


Check digit

1 A 'check digit' is a form of redundancy check used for Error

detection and correction|error detection on identification numbers (e.g. bank account numbers) which

have been input manually. It is analogous to a binary parity bit used

to check for errors in computer-generated data. It consists of a single

digit (sometimes more than one) computed by an algorithm from the

other digits (or letters) in the sequence input.


Check digit

1 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.


Check digit - Design

1 Check digit algorithms are generally designed to capture human

transcription errors. In order of complexity, these include the

following:



1 * single digit errors, such as 1 → 2



1 * transposition errors, such as 12 → 21



1 * twin errors, such as 11 → 22



1 * jump transpositions errors, such as 132 →

231



1 * jump twin errors, such as 131 → 232



1 * phonetic errors, such as 60 → 16 (sixty to sixteen)



1 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.



1 A desirable feature is that left-padding with zeros should not

change the check digit. This allows variable length digits to be used and

the length to be changed.



1 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).



1 A very simple check digit method would be to take the sum of all digits

(digital root|digital sum) modulo operation|modulo 10. This would

catch any single-digit error, as such an error would always change the

sum, but does not catch any transposition errors (switching two

digits) as re-ordering does not change the sum.



1 A slightly more complex method is to take the weighted sum of the digits, modulo 10, with different weights for

each number position.



1 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. 5 modulo 10, and the check digit

would be 5, giving 48715.



1 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.https://store.theartofservice.com/the-check-digit-toolkit.html


1 The code instead uses modulo 11, which is prime, and all the number

positions have different weights 1,2,\dots,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.



1 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.


Check digit - UPC

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


Check digit - UPC

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.


Check digit - UPC

1 # Add the digits (up to but not including the check digit) in the

even-numbered positions (second, fourth, sixth, etc.) to the result.


Check digit - UPC

1 # Take the remainder of the result divided by 10 (modulo operation)

and subtract this from 10 to derive the check digit.


Check digit - UPC

1 For instance, the UPC-A 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.


Check digit - UPC

1 # Add the odd number digits:

0+6+0+2+1+5 = 14


Check digit - UPC

1 # To calculate the check digit, take the remainder of (53 / 10), which is also known as (53 modulo 10), and

subtract from 10. Therefore, the check digit value is 7.


Check digit - UPC

1 Another example: to calculate the check digit for the following food item 01010101010.


Check digit - UPC

1 # To calculate the check digit, take the remainder of (5 / 10), which is also known as (5 modulo 10), and

subtract from 10 i.e. (10 - 5 modulo 10) = 5. Therefore, the check digit

value is 5.


Check digit - UPC

1 # If the remainder is 0, subtracting from 10 would give 10. In that case, use 0 as the

check digit.


Check digit - ISBN 10

1 The digit the farthest to the right (which is multiplied by 1) is the

check digit, chosen to make the sum correct



1 While this may seem more complicated than the first scheme, it can be validated simply

by adding all the products together then dividing by 11. The sum can be computed without any multiplications by initializing

two variables, t and sum, to 0 and repeatedly performing t = t + digit; sum =

sum + t; (which can be expressed in C (programming language)|C as sum += t +=

digit;). If the final sum is a multiple of 11, the ISBN is valid.



1 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 that the even digits are multiplied by 3 instead of

the odd digits.


Check digit - EAN (GLN,GTIN, EAN numbers administered by GS1)

1 EAN (European Article Number) check digits (administered by GS1) are calculated by summing the odd position numbers and multiplying 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 as-is if already zero).



1 A GS1 check digit calculator and detailed documentation is online at GS1's website.



1 Another official calculator page shows that the mechanism for GTIN-13 is the same for Global Location

Number/GLN.


Check digit - International

1 * The International SEDOL number.



1 * The final digit of an ISSN code or IMO

Number.



1 * The International Securities

Identifying Number (ISIN).



1 * The International CAS registry

number's final digit.



1 * Modulo 10 check digits in credit card account numbers, calculated by the Luhn

algorithm.



1 **Also used in the Norwegian KID (customer identification number)

numbers used in bank giros (credit transfer).



1 * Last check digit in EAN/UPC serialisation of Global Trade

Identification Number (GTIN). It applies to GTIN-8, GTIN-12, GTIN-13

and GTIN-14.



1 * The final digit of a Data Universal Numbering System|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).



1 * The third and fourth digits in an International Bank Account Number (Modulo

97 check).



1 * The final character encoded in a magnetic stripe card is a computed

Longitudinal redundancy check.


Check digit - In the USA

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



1 * The North American CUSIP number.



1 * The final (ninth) digit of the routing transit number, a bank code used in the United

States.



1 * The ninth digit of a Vehicle identification number|Vehicle Identification Number (VIN).



1 * Mayo Clinic patient identification numbers used in Arizona and Florida

include a trailing check digit.


Check digit - In Central America

1 * The Guatemalan Tax Number (NIT - Número de Identificación Tributaria) based on modulo operator|modulo

11.


Check digit - In Eurasia

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



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



1 * The 13th digit of the Serbian and SFRY|Former Yugoslav Unique Master

Citizen Number|Unique Master Citizen Number (JMBG).



1 * The last two digits of the 11-digit

Turkish Identification Number ().



1 * The ninth character in the 14-character European Union|EU cattle

passport number (cycles from 1 to 7: see British Cattle Movement

Service#Ear tag number|British Cattle Movement Service).



1 * The ninth digit in an Icelandic

Kennitala (national ID number).



1 * Modulo 97 check digits in a Belgium|Belgian and Serbian bank account numbers.



1 * The ninth digit in a Hungary|Hungarian TAJ number (social insurance number).



1 * For the residents of India, the unique identity number named AADHAAR#Salient features of

AADHAAR|Aadhaar has a trailing 12th digit that is calculated with the

Verhoeff algorithm.



1 * The Intellectual Property Office of Singapore|Intellectual Property Office of Singapore (IPOS) has confirmed a new format for application numbers of registrable Intellectual Property (IP, e.g., Trademark|trade marks, patents, Industrial design right|

registered designs). It will include a check character calculated with the

Damm algorithm.https://store.theartofservice.com/the-check-digit-toolkit.html

Check digit - In Oceania

1 * The Australian Tax File Number (based on modulo operator|

modulo 11).



1 * The seventh character of a New Zealand NHI

Number.



1 * The last digit in a Locomotives of New Zealand|New Zealand

locomotive's Traffic Monitoring System (TMS) number.


Check digit - Algorithms

1 Notable algorithms include:


Luhn algorithm - Verification of the check digit

1 digits = digits_of(card_numb

er)


Luhn algorithm - Calculation of the check digit

1 The algorithm above checks the validity of an input with a check digit. Calculating the check digit requires

only a slight adaptation of the algorithm—namely:


Luhn algorithm - Calculation of the check digit

1 # Append a zero check digit to the partial number and

calculate checksum


ISBN - ISBN-10 check digits

1 The 2001 edition of the official manual of the [http://www.isbn-

international.org/ International ISBN Agency] says that the ISBN-10 check

digit– which is the last digit of the ten-digit ISBN– must range from 0 to 10 (the symbol X is used for 10), and must be such that the sum of all the

ten digits, each multiplied by its (integer) weight, descending from 10 to 1, is a multiple of 11 (number)|11.



1 The two most common errors in handling an ISBN (e.g., typing or writing it) are a single

altered digit or the transposition of adjacent digits. It can be mathematical proof|proved

that all possible valid ISBN-10's have at least two digits different from each other. It can

also be proved that there are no pairs of valid ISBN-10's with eight identical digits and two

transposed digits. (These are true only because the ISBN is less than 11 digits long,

and because 11 is a prime number.)



1 The ISBN check digit method therefore ensures that it will always be possible to detect these two most

common types of error, i.e


ISBN - ISBN-10 check digit calculation

1 The resulting remainder, plus the check digit, must equal 11; therefore,

the check digit is (11 minus the remainder of the sum of the products

modulo 11) modulo 11



1 The value x_ required to satisfy this condition might be 10; if so, an 'X' should be used.



1 The 2005 edition of the International ISBN Agency's official manual

describes how the 13-digit ISBN check digit is calculated. The ISBN-13 check digit, which is the last digit of

the ISBN, must range from 0 to 9 and must be such that the sum of all the thirteen digits, each multiplied by its (integer) weight, alternating between 1 and 3, is a multiple of 10 (number)|

10.https://store.theartofservice.com/the-check-digit-toolkit.html


1 The ISBN-10 formula uses the prime number|prime modulus 11 which

avoids this blind spot, but requires more than the digits 0-9 to express

the check digit.


Routing transit number - Check digit

1 The ninth, check digit provides a checksum test using a position-

weighted sum of each of the digits. High-speed check-sorting equipment will typically verify the checksum and

if it fails, route the item to a reject pocket for manual examination,

repair, and re-sorting. Mis-routings to an incorrect bank are thus greatly

reduced.https://store.theartofservice.com/the-check-digit-toolkit.html


1 : (Modulo operation|Mod or modulo is the

remainder of a division operation.)



1 In terms of weights, this is 371 371 371



1 As an example, consider 111000025 (which is a valid routing number of

Bank of America in Virginia). Applying the formula, we get:



1 The following formula can be used to generate the ninth digit in the checksum:



1 This is just moving all terms other than d_9 to the right hand side of the

equation, which inverts the coefficients with respect to 10 (3 \

mapsto (10-3) = 7; 7 \mapsto (10-7) = 3; 1 \mapsto (10-1) = 9).



1 Following the above example for the Bank of America routing number 111000025,



1 This checksum is very easy to represent in computer programming

languages. The following Python (programming language)|Python example will print True when the

checksum is valid:


Damm algorithm - Validating a number against the included check digit

1 #Set up an interim digit and initialize it

to 0.



1 #Process the number digit by digit: Use the number's digit as column index and the interim digit as row

index, take the table entry and replace the interim digit with it.



1 #The number is valid if and only if the resulting interim digit has the value of 0.


Damm algorithm - Calculating the check digit

1 The resulting interim digit is '4'. This is the calculated check digit. We

append it to the number and obtain '5724'.


UPC code - Check digits

1 # Adding the odd-numbered digits (0+6+0+2+1+5 = 14)



1 # Calculating modulo ten (58mod10 = 8)



1 # Subtracting from ten (10minus;8 = 2)


Base 11 - ISBN check digit

1 The check digit for ISBN|ISBN-10 is found as the result of taking modular

arithmetic|modulo 11. Since this could give 11 possible results, the

digit X, not A, is used in place of 10. Remember that X (disambiguation)|X is the Roman numeral for ten. (The

newer ISBN-13 standard uses modulo 10, so no extra digits are required.)



1 The remainder of the division is the check digit's 'value' which is then

converted into a character (following the instructions given Code

128#Conversion to char|below) and appended to the end of the barcode.


Code 128 - Calculating check digit with multiple variants

1 As Code 128 allows multiple variants, as well as switching between variants within a single barcode, the absolute Code 128 value of a character is completely independent of its value within a given variant. For instance

the Variant C value 33 and the Variant B value A are both considered to be a Code 128 value of 33, and the check digit would

be computed based on the value of 33 times the character's position within the

barcode.https://store.theartofservice.com/the-check-digit-toolkit.html

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Check Digit