1. Identification Numbers
Rebecca Jacobs
Identification Numbers

2. Warm UP What is the remainder when dividing 416 by 7?
What number would you need to add to 37 to make it evenly divisible by 9?

3. Introduction A code is a symbolic way to represent information
A revolution in information management took place at a grocery store in Ohio in June, 1974, when a 10 pack of Wrigley’s Juicy Fruit gum was the first bar-coded retail product purchased.

4. Introduction A code is a symbolic way to represent information
A revolution in information management took place at a grocery store in Ohio in June, 1974, when a 10 pack of Wrigley’s Juicy Fruit gum was the first bar-coded retail product purchased.
All kinds of things have assigned identification numbers: money orders, airline tickets, bank checks, books, and cars

5. Check Digits
A check digit is a number in an identification code used for error detection.

6. Check Digits
A check digit is a number in an identification code used for error detection.
This method is used in several different identification schemes.

7. Check Digits
A check digit is a number in an identification code used for error detection.
This method is used in several different identification schemes.
Error detecting mechanisms have to be used in identification numbers in order to ensure that the appropriate item is being identified.

8. Division by 9 Schemes Example:
Consider the U.S. Postal Service money order with the number

9. Division by 9 Schemes Example:
Consider the U.S. Postal Service money order with the number
The last digit, 6 (the 11th digit) serves as an error detecting mechanism

10. Division by 9 Schemes Example:
Consider the U.S. Postal Service money order with the number
The last digit, 6 (the 11th digit) serves as an error detecting mechanism
This 11th digit is the remainder obtained when the sum of the first 10 digits is divided by 9.

11. Division by 9 Schemes Example:
Consider the U.S. Postal Service money order with the number
The last digit, 6 (the 11th digit) serves as an error detecting mechanism
This 11th digit is the remainder obtained when the sum of the first 10 digits is divided by 9.
=42

12. Division by 9 Schemes Example:
Consider the U.S. Postal Service money order with the number
The last digit, 6 (the 11th digit) serves as an error detecting mechanism
This 11th digit is the remainder obtained when the sum of the first 10 digits is divided by 9.
=42
42 divided by 9 is 36, so the remainder is 6.

13. Division by 9 Schemes
Suppose instead of the correct number, the number (an error in the 4th position—simply an incorrect number) was entered into the computer.

14. Division by 9 Schemes
Suppose instead of the correct number, the number (an error in the 4th position—simply an incorrect number) was entered into the computer.
The sum would be 44, and the remainder, 8. Therefore, the error would be detected.

15. Division by 9 Schemes Try it:
1. Suppose that a money order has ID number what is the check digit?
2. Suppose that a money order has ID number 719_ Can you determine the missing digit?

16. Problems with Division by 9
This method will not detect the mistake of replacing a 0 with a 9, or vice versa.
Because the value of a sum doesn’t depend on the order in which the numbers are added, this method does not detect the transposition (switching) of digits.

17. Example
American Express Travelers Cheques use a similar method where the check digit is the smallest nonnegative integer such that the sum of the digits, including the check digit, is evenly divisible by 9.

18. Example
American Express Travelers Checks use a similar method where the check digit is the smallest nonnegative integer such that the sum of the digits, including the check digit, is evenly divisible by 9.
The American Express Travelers Cheque with the identification number has what check digit?

19. Example
American Express Travelers Checks use a similar method where the check digit is the smallest nonnegative integer such that the sum of the digits, including the check digit, is evenly divisible by 9.
The American Express Travelers Check with the identification number has what check digit?
It has the check digit 7 because =38 and 38+7 is evenly divisible by 9.

20. Try it:
Determine the check digit for the AE Travelers check number

21. Division by 7 Schemes
This is used on airline tickets and for Avis and National rental cars.

22. Division by 7 Schemes
This is used on airline tickets and for Avis and National rental cars.
It assigns the remainder after division by 7 of the number itself as the check digit rather than dividing the sum of the digits by 7.

23. Division by 7 Schemes
This is used on airline tickets and for Avis and National rental cars.
It assigns the remainder after division by 7 of the number itself as the check digit rather than dividing the sum of the digits by 7.
Example:

24. Division by 7 Schemes
This is used on airline tickets and for Avis and National rental cars.
It assigns the remainder after division by 7 of the number itself as the check digit rather than dividing the sum of the digits by 7.
Example:
The check digit for the number is 4 because =7 x

25. Division by 7 Schemes
This is used on airline tickets and for Avis and National rental cars.
It assigns the remainder after division by 7 of the number itself as the check digit rather than dividing the sum of the digits by 7.
Example:
The check digit for the number is 4 because =7 x
This method will not detect the substitution of 0 for a 7, 1 for an 8, or 2 for a 9, or vice versa.

26. Try it:
Is the number a legitimate airline ticket number?

27. Division by 7 Schemes
Unlike division by 9 schemes, this will detect transpositions of adjacent digits with the exceptions of the aforementioned pairs.

28. Division by 7 Schemes
Unlike division by 9 schemes, this will detect transpositions of adjacent digits with the exceptions of the aforementioned pairs.
Example:
If were entered in as , the machine would determine that the appropriate check digit should be 3.

29. Division by 7 Schemes
Unlike division by 9 schemes, this will detect transpositions of adjacent digits with the exceptions of the aforementioned pairs.
Example:
If were entered in as , the machine would determine that the appropriate check digit should be 3.
You can use Google to determine the check digits that require division by 7 or 9 by entering “ mod 7” (or mod 9).

30. Universal Product Code (UPC)
This is often used on grocery
products.
The first digit identifies a broad category of goods, the next 5 identify the manufacturer, the next 5 identify the product, and the last is a check digit.

32. Universal Product Code (UPC)
For any UPC number
(always 12 digits long) we can write it as:
a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,

33. Universal Product Code (UPC)
For any UPC number a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,
the computer is programmed to carry out the following computation:
3a1+a2+3a3+a4+3a5+a6+a7+a8+3a9+a10+3a11+a12

34. Universal Product Code (UPC)
3a1+a2+3a3+a4+3a5+a6+a7+a8+3a9+a10+3a11+a12
If the answer doesn’t end in zero, the computer knows the entered number is incorrect.
This method can detect all single-position errors in UPC codes

35. Universal Product Code (UPC)
Returning to the Kellogg’s UPC, if we entered (a 5 instead of a 3), would the computer find an error?

36. Universal Product Code (UPC)
Returning to the Kellogg’s UPC, if we entered (a 5 instead of a 3), would the computer find an error?
YES! The sum is 62, which does not end in a zero, so the computer knows something is incorrect.
This simple scheme detects all single-position errors and about 89% of all other kinds of errors.

37. Error Detection in UPC
Check digit schemes that use weighted sums can be used to find a digit that has been corrupted in some way.
Example: If the packaging for a product with UPC was damaged or defective in such a way that the second digit was unintelligible, how would we know that it is supposed to be 6?

38. Error Detection in UPC
Example continued:

39. Error Detection in UPC Example continued:
Lets call the missing digit X

40. Error Detection in UPC Example continued:
Lets call the missing digit X
1+3X+4+3(0)+0+3(0)+2+3(2)+0+3(2)+0+3(3)+4
= 3X+32

41. Error Detection in UPC Example continued:
Lets call the missing digit X
1+3X+4+3(0)+0+3(0)+2+3(2)+0+3(2)+0+3(3)+4
= 3X+32
It has to end in zero.

42. Error Detection in UPC Example continued:
Lets call the missing digit X
1+3X+4+3(0)+0+3(0)+2+3(2)+0+3(2)+0+3(3)+4
= 3X+32
It has to end in zero.
6 is the only digit that makes this true.

43. Error Detection in UPC
This example also shows why the weight 3 is superior to weight 2. If we used weight 2 there would have been other possibilities: 2X+32; X=4 and X=9.

44. Try it:
If the packaging of a retail item were damaged in such a say that the first digit of a UPC code was scratched off, but the remaining digits were , determine the first digit.

45. Homework:
Page 526: 2, 4, 5, 6, 8, 10

46. Homework: Page 526: 2, 4, 5, 6, 8, 10 Homework Page 526: 2,4,5,6,8,10
2. If a U.S. Postal Service money order is numbered X, where X indicates that the last digit is obliterated, X is _____. 1
4. The sum of the digits of a correctly coded American Express Travelers Cheque identification number is evenly divisible by _____. 9
5. Is the number a legitimate airline ticket number?
Top of Form
 Yes.
 No, but if the final digit is changed to a 5, the resulting number is legitimate.
 No, but if the final digit is changed to a 3, the resulting number is legitimate. b
Bottom of Form
6. If an American Express Travelers Cheque is numbered X , where X indicates that the first digit is obliterated, X is _____.
8. A correctly coded UPC number has a weighted sum that is evenly divisible by _____. 10
10. The check digit that should be appended to the UPC code is _____.

47. Warm Up
The UPC codes of the following products have a number that is scratched out and they will not scan. They are the last of their type on the shelf and you decide that instead of suing the store you will help the distressed cashier find the missing number.

48. Universal Product Code (UPC)
The 12 digit UPC used in the US paved the way for the 13 digit EAN (European Article Number) used in Europe, which became the world-wide standard. The 12-digit UPCs were converted by adding a 0 to the beginning.

49. Universal Product Code (UPC)
The check digit for a 13-digit EAN number is selected so that
a1+3a2+a3+3a4+a5+3a6+a7+3a8+a9+3a10+a11+3a12+a13
ends with a 0.
The 0 added to the beginning doesn’t affect the check digit of UPC codes.

50. Universal Product Code (UPC)
The coefficient 3 for the terms with even subscripts is called a weight.
Because 1ai = ai, we say that terms with odd subscripts have weight 1.

51. Bank Identification Numbers
The US banking system uses a variation of UPC scheme.

52. Bank Identification Numbers
The US banking system uses a variation of UPC scheme.
Every bank has an 8-digit routing number, together with a 9th check digit.
The first four digits identify the bank’s Federal Reserve District, office, and state, The next four digits are the bank’s identification number. The last digit is the check digit.

53. Bank Identification Numbers
Every bank has an 8-digit routing number, together with a 9th check digit as the last digit of the following:
7a1+3a2+9a3+7a4+3a5+9a6+7a7+3a8

54. Example
The First Chicago Bank has the routing number _ on the bottom of all of its checks. What is the check digit?

55. Example
The First Chicago Bank has the routing number _ on the bottom of all of its checks. What is the check digit?
7(0)+3(7)+9(1)+7(0)+3(0)+9(0)+7(0)+3(1)=33
The check digit is 3 because it is the last digit of the sum of the first 8 digits in the scheme

56. Advantages to Using 3 Weights
It detects 100% of single position errors and some transposition errors involving adjacent digits.

57. Advantages to Using 3 Weights
It detects 100% of single position errors and some transposition errors involving adjacent digits.
This scheme will also detect most transposition errors in the form of abccba, where as the UPC scheme does not.

58. Advantages to Using 3 Weights
It detects 100% of single position errors and some transposition errors involving adjacent digits.
This scheme will also detect most transposition errors in the form of abccba, where as the UPC scheme does not.
Example: look at the number 241, in the UPC scheme and the number 142 in the UPC scheme. Then check it with the bank scheme.

59. Homework
Page 526: 12; Page 527: 6,7,12,13,23

60. Homework
Page 526: 12; Page 527: 6,7,12,13,23
12. The check digit that should be appended to the bank routing number is _____. 9 
6. Determine the check digit for the Avis rental car with identification number 4 
7. Determine the check digit for the UPC number 1
12. Determine the check digit for the bank routing number 9
13. Determine the check digit for the American Express Travelers Cheque with identification number 6

61. Homework
Page 526: 12; Page 527: 6,7,12,13,23
23. The check digit a8 for a UPC Version E identification number a1a2a3a4a5a6a7, where a7 is 0, 1, or 2, is chosen so that
a1 + a2 + 3a3 + 3a4 + a5 + 3a6 + a7 + a8
is divisible by 10. Use this fact to determine the check digit for the following Version E numbers: 7 4 2

62. Codabar
This is one of the most efficient error-detection methods, and is used by all major credit-card companies.
A credit card number is 16 digits long.
The first 15 digits identify the credit card, and the 16th digit is the check digit.

63. Codabar Credit cards use a very efficient error-detection method.
They add the digits in position 1, 3, 5, 7, 9, 11, 13, and 15 (The odd-positiions)
Then double the result
Then count the number of digits in those same positions that are higher than 4 and add that to the total
Take the new total and add the numbers in the even positions.
The check digit is whatever is needed to bring the final tally to a number that ends with zero

64. Example Consider the credit card numbered 4128 0012 3456 7890
Is this a valid credit card number?

65. Example
Suppose a bank wants to issue a credit card numbered
What should the check digit be?

66. Example
The sum of the odd-numbered positions is =26.
Doubled is 2(26)=56
Add the total of odd-numbered positions above 4, 56+2=58
Add the even digits, =74
The check digit is chosen so that the total ends in 0, thus is 6

67. Error Detection
This method allows computers to detect 100% of single position errors and about 98% of all other common errors.

68. Error Detection
This method allows computers to detect 100% of single position errors and about 98% of all other common errors.
A check digit offers partial protection against fraudulent numbers. A person who wanted to create a phony card would have to know the appropriate check digit scheme to be successful.

69. Error Detection
This method allows computers to detect 100% of single position errors and about 98% of all other common errors.
A check digit offers partial protection against fraudulent numbers. A person who wanted to create a phony card would have to know the appropriate check digit scheme to be successful.
The algorithm is called the Luhn Algorithm, after IBM scientist Hans Peter Luhn, who created it in It can be applied to identification numbers of lengths other than 15, as well.

70. Homework
Page 527: 15, 21

71. Homework
15. Determine whether the Master Card number is valid.
Since ( )×2=44 and one of the summands exceeds 4, we have 44+1+( )=60. So, the number is valid.

72. Homework
21. Use the credit card scheme to determine the check digit for the number
We begin with ( )×2=50. Adding 2, we obtain 52 and have the 
following.
= 68
So the check digit is 2

73. ISBN
So far, none of these methods can detect 100% of both single errors and transposition errors.

74. ISBN
So far, none of these methods can detect 100% of both single errors and transposition errors.
But—there is a scheme which can detect 100% of both single errors and transposition errors—the 10 digit ISBN (ISBN-10).
(not to be confused with the 13 digit ISBN or ISBN-13)

75. ISBN A correctly coded ten-digit ISBN a1 a2 …a10 has the property that
10a1 + 9a2 + 8a3 + 7a4 + 6a5 + 5a6 + 4a7 + 3a8 + 2a9 + a10
is evenly divisible by 11.

76. ISBN A 10 digit ISBN is made up of ten-digits as follows:
first digit—published country’s language,
next four digits—the publisher,
next four digits—type of book,
last digit—check digit.

77. ISBN If the ISBN method is so good, why don’t we use it more?
Sometimes, a check digit using this method should be 10…but that is obviously not a one digit number, so sometimes ISBNs use an X to denote the check digit 10..which makes not all ISBN numbers solely consist of digits

78. ISBN
13 digit ISBNs were introduced in 2007 to make them compatible with EAN codes.
The 13 digit is the same as the 10 digit ISBN except for the addition of the numbers 978 or 979 and the check digit
Check digits for 13 digit ISBNs are calculated the same was as any other UPC/EAN check digit.

79. VIN Numbers
Automobiles and trucks are given a vehicle identification number (VIN) by the manufacturer.
A typical VIN has 17 alphanumeric characters that code information, such as country where the vehicle was built, manufacturer, make, body style, engine type, plant where the vehicle was built, model year, model, type of restraint, a check digit, and a production sequence number.

80. VIN Numbers
The check digit is calculated by converting the 26 consecutive letters of the alphabet, respectively, to the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9 (note the skipped digit after the second 9)
to obtain a 16-digit number a1a2 a15a16 that is weighted with 8, 7, 6, 5, 4, 3, 2, 10, 9, 8, 7, 6, 5, 4, 3, 2.
The check digit is the remainder when the weighted sum
8 · a1 + 7 · a · a · a16
is divided by 11 unless the remainder is 10, in which case an X is used instead. The check digit is inserted in position 9.

81. VIN Numbers

82. The ZIP Code
This system was established in The numbers begin with 0s (farthest east, Adjuntas, Puerto Rico) and work up to 9s (farthest west, Ketchikan, Alaska).
The first digit represents one of the 10 geographic areas. The second two digits identify a central mailing distribution point known as a sectional center. The last two digits indicate the town or local post office.

83. The ZIP Code
The first digit represents one of the 10 geographic areas. The second two digits identify a central mailing distribution point known as a sectional center. The last two digits indicate the town or local post office.
Example: Duluth, Minnesota: 
55812
5- represents one of the 10 geographic areas
58—identify the central mail-distribution point
12—town or local post office. Often alphabetical for towns within a delivery area (A usually has a low number)

84. The Zip Code

85. The ZIP Code
1983: Four more digits were added after a dash, called the ZIP + 4 code.
The extra digits represent the delivery sector, which can be a group of city blocks or a small geographic area.

86. Bar Codes
A bar code is a series of dark bars and light spaces that represent characters.

87. Bar Codes
A bar code is a series of dark bars and light spaces that represent characters.
They are decoded by scanning with a light. The dark bars reflect very little back, and the white reflects a lot. They are converted into 0s and 1s. This is called binary coding.

88. Bar Codes
A bar code is a series of dark bars and light spaces that represent characters.
They are decoded by scanning with a light. The dark bars reflect very little back, and the white reflects a lot. They are converted into 0s and 1s. This is called binary coding.

89. Bar Codes
The simplest bar code is the Postnet code, used by the USPS and are commonly found on Business Reply Cards.
There are 52 vertical bars of two possible lengths (long and short). The long bars at the beginning and end are called guard bars and provide a frame for the remaining 50 bars.
In blocks of five, the 50 bars within the guard bars represent the ZIP+4 code and a tenth digit for error correction. Each block of 5 is composed to exactly two long bars and three short bars according to the pattern

90. Postnet Codes

91. Postnet Codes
The tenth digit of a Postnet code number is a check digit chosen so that the sum of the nine digits of the ZIP+4 code and the tenth one is evenly divisible by 10.
What is the check digit for the ZIP+4 code

92. Postnet Codes
Because each group of 5 bars in a Postnet code has a series of 2 long bars and 3 short bars, an incorrectly printed bar code is easily detectable.

93. Postnet Codes
Because each group of 5 bars in a Postnet code has a series of 2 long bars and 3 short bars, an incorrectly printed bar code is easily detectable.
Obviously the 6th digit is incorrect because it has only one long bar. To correct the error, the computer sums the remaining 9 digits and figures out what the missing digit must be.

94. Homework
Page 527 #18, Page ,51

95. Homework
Page 527 #18, Page ,51
18) if the sixth digit of the Postnet code is incorrect, the correct Postnet code is _____.
49. Determine the ZIP + 4 code and check digit for each of the following Postnet bar codes:
(a) ; 2
(b) ; 1
(c) ; 1

96. Homework
Page 527 #18, Page ,51
(that is, a long bar appears instead of a short one, or vice versa). Determine the correct ZIP code.
(a)
(b)
(c)

97. Encoding Personal Data
The social security number has no personal data encoded in the number. It is entirely determined by the place and time that it is issued.
In contrast, some states driver’s license numbers are determined entirely by personal information about the holders.

98. Encoding Personal Data
The social security number has no personal data encoded in the number. It is entirely determined by the place and time that it is issued.
In contrast, some states driver’s license numbers are determined entirely by personal information about the holders.
Coding license numbers solely from personal data enables automobile insurers, government entities, and law enforcement agencies to determine the number from the personal data.

99. Personal Information in Driver’s License Numbers
The last five digits of Illinois and Florida driver’s license numbers capture the year and date of birth as well as the sex.
Illinois: each day of the year is assigned a three digit number in sequence beginning with 001 for January 1. All months are assumed to have 31 days, so March 1 is 063. For females the scheme is identical but 600 is added to the number.

100. Personal Information in Driver’s License Numbers
The last two digits of the birth year, separated by a dash are listed in the fifth and fourth positions from the end of the driver’s license number. Thus, a male born October 13, 1940 would have the last five digits , whereas a female would have