1. Chapter 16: Check Digit Systems
MAT 105 Spring 2008
Chapter 16: Check Digit Systems

2. Examples of Check Digit Systems
The check digit systems we will study are used for:
US Postal Service money orders
Airline tickets
UPC (Universal Product Code)
US Bank routing numbers
Credit card numbers
ISBN (International Standard Book Number)

3. US Postal Service Money Orders
The ID number is listed here
The ID number is also listed here in machine-readable numbers (magnetic ink)

4. USPS Money Order Check Digit System
The ID number on a USPS money order is an 11-digit number, and the 11th digit is the check digit
The 11th digit is the remainder when the sum of the first 10 digits is divided by 9

5. Examples In our sample money order, the ID number is 02543750594
If we add up the first 10 digits, we get 40, and the remainder when 40 is divided by 9 is 4, so the check digit is correct
Another example:

6. Finding Remainders on the Calculator
Since many of the check digit systems involve finding remainders, it is useful to know how to find them on your calculator
There are many different methods, but this one is simple
For example, suppose you need to know the remainder when 59 is divided by 7

7. Finding Remainders on the Calculator
To find the remainder when 59 is divided by 7, just type 59 divided by 7 in your calculator
Take the digits appearing after the decimal and multiply them by 7 (the number you divided by)
The result will be the remainder
In this example, the remainder is 3

8. Detecting Errors
Suppose we receive a suspicious money order with ID number
If we add up the first 10 digits and divide by 9, we get remainder 8, which does not match the check digit
So we know this ID number is invalid

9. Types of Errors Look at what happened:
Valid ID number
Invalid ID number
This is a substitution error: an incorrect digit was substituted for the correct one
This error was detected because we were able to tell that the new number is invalid

10. Undetected Errors Let’s look at another example
Correct ID number
Incorrect number
Notice that the incorrect number is actually still a valid ID number, so this error goes undetected by the check digit system
In fact, this system can never detect a substitution of a 0 for a 9 (or vice versa)

11. More Undetected Errors
Since we just add up the first 10 digits, this system is also unable to detect transposition errors
Correct ID number
Incorrect number
Once again, the incorrect number is still valid

12. Another System: Airline Tickets
This is the ticket ID number. The last digit (colored in yellow) is the check digit.

13. Computing the Check Digit
The check digit is the remainder when the ID number (without the check digit) is divided by 7
It is difficult for us to find these remainders on our calculators when the ID numbers are very large, like they are on airline tickets
For our examples, we will use ID numbers that are shorter than normal, just to illustrate how the process works

14. An Example Is the airline ticket ID number 5208162 valid?
Remember, is the ID number, and 2 is the check digit
On our calculators, we divide by 7 and get remainder 2

15. Detecting Errors
This method detects all single substitution errors except 0  7, 1  8, and 2  9
In addition, this system can detect transpositions as long as the two digits are not 0 & 7, 1 & 8, or 2 & 9
Examples
 detected
 not detected
 detected
 not detected

16. Reminder Remember how error detection works:
If we change the ID number and now the check digit is wrong, the error is detected
If we change the ID number and the check digit is still correct, the error is not detected
Make sure you understand the difference between “incorrect” and “invalid”