Identification numbers with check digit algorithms 
Type of errors detected by Euro notes scheme
What is the effectiveness of this system in detecting errors in banknotes serial numbers? Very poor indeed, much worse that the one of BI cards and NIB. In fact, no transposition of two (or more) digits (whether consecutive or not) is detected, due to the commutative property of modular addition; even the singular errors (errors in a single digit) are not all detected: if we write a nine instead of a zero (or vice versa) the check digit remains unchanged. Note that this is a serious practical problem because in most keyboards, the key for \(0\) is next to the one for \(9\). This failure has to do with the fact that in the arithmetic modulus \(9\), \(9\) and \(0\) are equal. Summarizing, this system does not detect all singular errors and all transposition errors, the two most frequent types of errors.
To check the detection rate of these two types of errors (or simply if you want to know if the banknotes that you hold are "true") click here.
To learn more about errors detection in this type of identification systems, click here.
Beyond this example, check digits are used in many other situations like, for example, Identity Card (BI), Barcodes, Bank Account Number (NIB), Visa Card, ...
