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
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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…
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License plate
Driver’s 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?
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What should happen?
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How does the computer know that the
number you typed in isn’t valid?
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There are some quick ways it might be able to
tell
 You typed in some letters or symbols instead of all
numbers
 You didn’t 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?
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Should the computer have to look through all
the millions of credit card numbers to make
sure that what you typed in isn’t on the list?
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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
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This is not just for security, but to detect
errors
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If an ID number is entered incorrectly, it is
important that the error is caught; you don’t
want your account to get charged for
someone else’s order
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Suppose you call customer service to ask a
question about your account
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The representative asks for your account
number, which is 123456
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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
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Transposition: 123546 instead of 123456
 This error accounts for around 10% of all errors
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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 that’s because we
know the real ID number
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Put yourself in the position of the company
representative on the other end of the phone
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That person doesn’t 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
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Check digits are extra digits attached to the ID
number that are only used for checking that the
ID number is valid
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Many different check digit systems are used, and
we will study several of them
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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
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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
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We can use our system to determine whether
or not this is a valid ID number
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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
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It is possible that there was an error that our
system didn’t detect
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Suppose you want to use your account, which
has ID number 153182
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However, when you go to type in your
number, you accidentally type in 156182
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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?
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We try our system: 1+5+6+1+8+2 = 23
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Since the sum isn’t 20, we know the ID
number 156182 is invalid
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We have detected the error
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This time, suppose you type in your ID number
(153182) as 153812
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Again, we can see that an error has occurred, but
the question is whether the ID number system
can detect the error
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In this case, 1+5+3+8+1+2 = 20, which means
that the new ID number is valid
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The error is not detected!
valid
invalid
correct
The ID number is valid and
correct. There are no errors.
Impossible. If the ID number is
correct, then it must be valid.
incorrect
The ID number is valid but
incorrect. There was an error
that was not detected, or a
thief has managed to forge a
valid ID.
The ID number is incorrect and
can be shown to be invalid
using the rules for the ID
number system. Errors are
detected.