Chapter 16: Check Digit Systems

1. 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: 63024383845

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 63054383845 • 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 63024383845 • Invalid ID number 63054383845 • 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 63024383845 • Incorrect number 63924383845 • 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 63024383845 • Incorrect number 63023483845 • 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, 520816 is the ID number, and 2 is the check digit • On our calculators, we divide 520816 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 • 5208162  5204162 detected • 5208162  5201162 not detected • 5208162  5280162 detected • 5208162  5201862 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”