In this chapter we look at some examples of how identification numbers such as Social Security Numbers, Zip Codes and Bank Account Numbers are made up,  and how and why they are used. We see these numbers all the time. For example,  when you go to a supermarket, the string of black lines printed on a packet, called a barcode, represents a string of numbers. The cash register scans the bars and converts them into a string of numbers; the corresponding prices are kept in a file and are looked up automatically.

Many other things in everyday life are also represented by codewords—strings of letters or numbers which identify a bank account, a book, an airline ticket, and so on. When these are copied or transmitted there is always the possibility of errors.

The simplest method to avoid errors is a check digit. For example, a Visa or American Express traveler’s check has a 10-digit identifying code, such as 2411903043. The first nine digits identify the check; the tenth is chosen so that the sum of all ten digits is divisible by 9. In the example, 2 + 4 + 1 + 1 + 9 + 0 + 3+0+4+3 = 27 = 3×9. If the first nine digits were 524135324, we observe that 5+2+4+1+3+5+3+2+4= 29; the next multiple of 9 is 29+7= 36 = 4×9, so the check digit would have to be 7, and the code is 5241353247. If the check sum is wrong, the number was wrong. This simple method detects most errors consisting of one wrong symbol (but not when a 0 is substituted for a 9 or vice versa; in some systems of this sort, no 9s are ever used). Of course, if you get two digits wrong, the check will not always work, and it never helps when you interchange two digits.

A modification of this method is used in US Postal Service money orders. The money order has an 11-digit number; the first ten digits identify the money order,  and the 11th is a check digit. Again, divisors on division by 9 are involved. But in this case, after the sum of the first ten digits is found and divided by 9, the check digit equals the remainder. For example, if the identification number started 2411403043,  then 2+4+1+1+4+0+3+0+4+3= 22 = 2×9+4, so the last digit would be 4.

Sample Problem 1.1 A USPS money order has number 5_2413532404, where the second digit has been rubbed off. What was the second digit?

Solution. Represent the second digit by x. The sum of the first ten digits is 29 + x. When this is divided by 9, the remainder must be 4, so 29 + x is one of 13,22,31,40,... But 0 ≤ x ≤ 9, so the only possibility is 31, and x must equal 2.

These systems will detect some errors, but by no means all. For example,  one of the most common errors we make is to write digits in the wrong order,  especially to exchange two adjacent numbers. If you were to swap two digits on a traveler’s check number, the error would not be noticed. The money order number system is slightly better; if you were to exchange the last digit with another, the error would probably be caught. But these methods do not detect all errors by any means.