Check digit

A check digit is a form of redundancy check used for error-detection in a series of numbers. It is calculated by a sequence of operations on the other digits in the number. When the check digit is calculated, it can be compared with the original number to check for accuracy. If the numbers match, it is an indication that the number is valid.

Check digits are often used in identification or serial numbers, such as those found on credit cards, bank cards, and product barcodes. They are also used in ISBN numbers and International Standard Book Numbers (ISBNs).

The check digit is usually the last digit in a series of numbers. Its purpose is to provide a verification that the other digits in the number are correct. The check digit is calculated using a mathematical formula, which is dependent on the number of digits in the number.

The purpose of the check digit is to provide a way to check if the other digits in the number are correct. It can also be used to prevent numbers from being entered incorrectly. By using a check digit, any errors in the number can be detected and corrected before it is entered into a system.

Example of Check digit

Let's take a look at an example of a check digit. Suppose we have the number 1234. The check digit for this number would be calculated using the formula above:

Failed to parse (syntax error): {\displaystyle \begin{equation*} Check\ digit=10-(1+3\times2+3+3\times4) \mod 10 = 10-(15) \mod 10 = 5 \end{equation*}}

The check digit for this number is 5, so the complete number is 12345. To check the accuracy of the number, we can calculate the check digit again using the same formula.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation*} Check\ digit=10-(1+3\times2+3+3\times4) \mod 10 = 10-(15) \mod 10 = 5 \end{equation*}}

Since the check digit matches the original number, we know that the number is valid.

In conclusion, a check digit is a form of redundancy check used for error-detection in a series of numbers. It is calculated using a mathematical formula and is usually the last digit in the number. Its purpose is to provide a way to verify that the other digits in the number are correct. By using a check digit, any errors in the number can be detected and corrected before it is entered into a system.

Formula of Check digit

Check digits are calculated using a mathematical formula, which is dependent on the number of digits in the number. The following formula can be used to calculate a check digit for a number with four digits:

Failed to parse (syntax error): {\displaystyle \begin{equation*} Check\ digit=10-(d_1+3d_2+d_3+3d_4) \mod 10 \end{equation*}}

Where d_1 is the first digit of the number, d_2 is the second digit, and so on.

When to use Check digit

Check digits are used in a variety of applications, including:

• Identification numbers: These can be used to verify the accuracy of any number, such as credit card numbers, bank account numbers, and product barcodes.
• ISBNs and ISSNs: Check digits are used in International Standard Book Numbers (ISBNs) and International Standard Serial Numbers (ISSNs). This is to ensure that the numbers are accurately entered in records and databases.
• Serial numbers: Serial numbers are used to track products and goods. Adding a check digit to the serial number can ensure accuracy and prevent fraud.

Types of Check digit

• Modulus 10: The most common type of check digit is Modulus 10. This type of check digit uses a formula to calculate the check digit from the other digits in the number.
• Modulus 11: This type of check digit also uses a formula to calculate the check digit from the other digits in the number. However, it is slightly more complex than Modulus 10, as it uses different weights for different digits.
• Luhn Algorithm: The Luhn algorithm is a type of check digit that is used in credit and debit cards. It is a more complex formula that uses the other digits in the number to calculate the check digit.
• Double-digit: This approach is similar to modulo 10, but instead of performing a modulo 10 operation, the check digit is calculated by adding the digits in the number together and subtracting the result from the next multiple of 10.

Steps of calculating Check digit

• The first step of calculating a check digit is to determine the number of digits in the number. This number will be used in the formula for calculating the check digit.
• Next, the digits in the number are multiplied by a weighting factor. The weighting factor is usually 2, 3, or 4.
• The multiplied digits are then added together to get a sum.
• The sum is then divided by 10 to get the remainder.
• Finally, the remainder is subtracted from 10 to get the check digit.

Advantages of Check digit

• It can detect any human errors in entering a number into a system.
• It can detect any number transposition errors, such as swapping two numbers in a sequence.
• It can detect any incorrect insertion or deletion of a digit.
• It can detect any errors in a sequence of numbers that has been printed or written down.

Limitations of Check digit

Despite its benefits, the check digit also has some limitations. It cannot detect all errors in a number, as some errors may not be detected by the formula used to calculate the check digit. Also, it cannot detect any data that is missing or incorrect.

In addition, the check digit may not be able to detect errors that occur due to the wrong number of digits being entered. For example, if a number is entered with one digit too many, the check digit will not be able to detect this error.

Lastly, the check digit may not be able to detect errors in a number if the wrong algorithm is used to calculate the check digit. For example, if the wrong formula is used to calculate the check digit, the wrong result may be produced.

Check digit — recommended articles

C chart — Confidence level — Harmonic mean — Aggregate function — Reliability of information — Overfitting — Measurement uncertainty — Precision and recall — Systematic error

References

• Shaheen, R., & Winterhof, A. (2010). Permutations of finite fields for check digit systems. Designs, Codes and Cryptography, 57(3), 361-371.
• Schulz, R. H. (2000). On check digit systems using anti-symmetric mappings (pp. 295-310). Springer US.
• Wheeler, M. L. (1994). Check-digit schemes. The Mathematics Teacher, 87(4), 228-230.