Math for Liberal Studies

An identification number is a sequence of letters and/or numbers that identifies an object, person, place, or concept The number should unambiguously identify something: no two things should have the same ID number

Think about the different ways you encounter ID numbers in your daily life… License plate Drivers license number Student ID number Social Security number Phone numbers Credit card numbers ISBN (International Standard Book Number) UPC (Universal Product Code) Many, many more…

What happens if you go to Amazon.com and type in something random when it asks you for a credit card number? What should happen?

How does the computer know that the number you typed in isnt valid? There are some quick ways it might be able to tell You typed in some letters or symbols instead of all numbers You didnt type in 16 digits

ID numbers typically have a common format Standard PA license plates have a format of three letters followed by four digits Social security numbers are 9 digits long Phone numbers are 10 digits long (with area code)

What if you type in a number with the correct format? Should the computer have to look through all the millions of credit card numbers to make sure that what you typed in isnt on the list? There are billions of credit cards in the world, even with fast computers this would take too long

Most ID numbers have a quick way to check to make sure that the number is valid This is not just for security, but to detect errors If an ID number is entered incorrectly, it is important that the error is caught; you dont want your account to get charged for someone elses order

Suppose you call customer service to ask a question about your account The representative asks for your account number, which is What kinds of errors can occur when reporting an ID number like this?

Substitution: instead of This error is the most common, occurring almost 80% of the time Transposition: instead of This error accounts for around 10% of all errors Jump transposition: instead of This error is much rarer, accounting for only about 1% of all errors

We can immediately see that any of those ID numbers are incorrect, but thats because we know the real ID number Put yourself in the position of the company representative on the other end of the phone That person doesnt know what the real ID number is, they only know what you tell them

Most errors are detected using check digits Check digits are extra digits attached to the ID number that are only used for checking that the ID number is valid Many different check digit systems are used, and we will study several of them Some check digit systems are secret so that they cannot be used to generate fake (but valid) ID numbers

When an ID number is communicated, errors can occur, and we can sometimes detect those errors by determining that the erroneous number is invalid However, sometimes even ID numbers with errors turn out to be valid. In these cases there is no way to know that the ID number is incorrect

Suppose we run a company that sells phone cards, and each card comes with an account number The format of the account number is a 6-digit number Validity is determined using this rule: if the sum of the digits of the account number is 20, then the number is valid. If not, then the number is invalid

Someone calls in to phone card service and wants to place a call using account number We can use our system to determine whether or not this is a valid ID number = 20, so this number is valid

However, just because the number is valid does not mean that it is correct It is possible that there was an error that our system didnt detect

Suppose you want to use your account, which has ID number However, when you go to type in your number, you accidentally type in We can see that an error has occurred because we can see both numbers, but the company only knows the number you typed

Is the error detected? We try our system: = 23 Since the sum isnt 20, we know the ID number is invalid We have detected the error

This time, suppose you type in your ID number (153182) as Again, we can see that an error has occurred, but the question is whether the ID number system can detect the error In this case, = 20, which means that the new ID number is valid The error is not detected!