Before the era of ethernet, TCP/IP, and packet verification, electronic data was transmitted over telephone wires. (Think back to the days of AOL & modems, and that screeching noise when you connected.) A big problem with this method was the risk of external interference garbling your signal. What if a bird flew into the phone wire? Or if someone picked up the other line? Or it started raining? Any external interference could result in the information being sent at that moment to be garbled. So, if you were transmitting something like a credit card number, how would the receiver know that it wasn’t garbled? (For example, what if a 3 was garbled into a 4?) This was a problem solved at IBM back in the 1950s. Of course, the same issues arise when human error is introduced. (What if you are reading the credit card aloud over the phone, and the other person types in one of the digits incorrectly?)
For the rest of this post, I am merely introducing a lecture I attended called “Identification Numbers and Check Digit Schemes”, by Joe Kirtland. Thanks to Joe for sharing his full PPT slides with me (link below). It talks about the checkdigit systems used to validate credit cards, bar-codes, ISBN numbers, currency serial numbers, etc. It’s a great real-world application of Algebra, Geometry, and algorithms.
Here is a very crude example that illustrates the concept: Let’s say you want to transmit the number “34515″. One validation algorithm requires that the sum of the individual digits be divisible by 10 (mod 10 = 0). Currently, the sum of the digits is 18 (3+4+5+1+5=18) So, to make the sum divisible by 10, you just tack on a 2 at the end , and transmit “345152″. The receiver of the data is told to ignore that last number, which is called a check digit. The receiver then adds the numbers up and sees if the sum is divisible by 10. If even one number was changed, it won’t work*. If the final number he gets doesn’t check out, something went wrong, and you need to resend.
*Question: This method is not foolproof, can you think of some reasons why?
If this topic interests you, check out the following slides, where Joe explored check digits as they relate to UPC codes, ISBN numbers, credit cards, and serial numbers on various currency.