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Math for Liberal Studies

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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

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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…

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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?

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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

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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)

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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

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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

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Suppose you call customer service to ask a question about your account The representative asks for your account number, which is 123456 What kinds of errors can occur when reporting an ID number like this?

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Substitution: 123476 instead of 123456 This error is the most common, occurring almost 80% of the time Transposition: 123546 instead of 123456 This error accounts for around 10% of all errors Jump transposition: 125436 instead of 123456 This error is much rarer, accounting for only about 1% of all errors

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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

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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

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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

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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

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Someone calls in to phone card service and wants to place a call using account number 516224 We can use our system to determine whether or not this is a valid ID number 5+1+6+2+2+4 = 20, so this number is valid

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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

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Suppose you want to use your account, which has ID number 153182 However, when you go to type in your number, you accidentally type in 156182 We can see that an error has occurred because we can see both numbers, but the company only knows the number you typed

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Is the error detected? We try our system: 1+5+6+1+8+2 = 23 Since the sum isnt 20, we know the ID number 156182 is invalid We have detected the error

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This time, suppose you type in your ID number (153182) as 153812 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, 1+5+3+8+1+2 = 20, which means that the new ID number is valid The error is not detected!

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