A%20Mathematical%20View%20of%20Our%20World

1
A Mathematical View of Our World

• 1st ed.
• Parks, Musser, Trimpe, Maurer, and Maurer

2
Chapter 1

• Numbers in Our Lives

3
Section 1.1ID Numbers and Check Digits

• Goals
• Study social security numbers
• Study general identification numbers
• Transmission errors
• Check digits
• Study universal product codes

4
1.1 Initial Problem

• Can you be confident that you will be charged the
  correct price if you purchase the item with this
  UPC?
• The solution will be given at the end of the
  section.

5
Social Security Numbers

• A social security number (SSN) is made up of
• A three-digit area number
• A two-digit group number
• A four-digit serial number

6
Area Number

• XXX-XX-XXXX
• Area numbers range from 001 728.
• Before 1973, the area number depended on the
  state in which the office issuing the number was
  located.
• Since 1973, the area number is determined by the
  mailing address of the applicant.

7
Area Number, contd
8
Area Number, contd
9
Group Number

• XXX-XX-XXXX
• Group numbers range from 01 99.
• The group number groups together certain social
  security numbers.
• The group number is not determined by any group
  to which a person belongs.

10
Group Number, contd

• Group numbers are issued as follows
• First, odd numbers from 01 to 09.
• Second, even numbers from 10 to 98.
• Third, even numbers from 02 to 08.
• Fourth, odd numbers from 11 to 99.

11
Serial Number

• XXX-XX-XXXX
• Serial numbers range from 0001 9999.
• Serial numbers are issued in numerical order from
  smallest to largest.

12
Example 1

• Which state was listed in the mailing address of
  the applicant who received the number 501-92-3287
  ?
• Which number was issued first
• 362-13-4158 or 362-14-9725 ?

13
Example 1, contd

1. Solution The area number in 501-92-3287 is 501.
   According to Table 1.1, this social security
   number was issued to someone with a mailing
   address in North Dakota.
2. Solution The group number 14 is issued before
   the group number 13, so 362-14-9725 was issued
   before 362-13-4158.

14
General ID Numbers

• Many items besides people are assigned
  identification numbers.
• For example
• International Standard Book Numbers (ISBN) for
  books
• Vehicle Identification Numbers (VIN) for cars
• Universal Product Codes (UPC) for grocery items

15
General ID Numbers, contd

• Identification numbers are divided into two
  types
• Numeric ID numbers
• Strings of digits
• Alphanumeric ID numbers
• Strings of digits, letters, and/or other symbols

16
General ID Numbers, contd

• All digits, letters, or other symbols in an
  identification number are called characters.
• The length of an identification number is the
  number of characters in the string.
• Spaces, dashes, or other separators are not
  counted in the length of the string.

17
Question

• A numeric identification number can contain
• a. numbers only.
• b. letters and numbers only.
• c. letters, numbers, and other characters.

18
Example 2

• Determine the type and length of each ID number.
• SSN 876-87-6543
• ISBN 0-07-231821-X
• VIN GHN5UC265518G
• UPC 0 51000 01031 5

19
Example 2, contd

• Solutions
• The SSN 876-87-6543 is a numeric ID number of
  length 9.
• The ISBN 0-07-231821-X is an alphanumeric ID
  number of length 10.
• The VIN GHN5UC265518G is an alphanumeric ID
  number of length 13.
• The UPC 0 51000 01031 5 is a numeric ID number of
  length 12.

20
Transmission Errors

• An error in recording, reading, or relating an
  identification number is called a transmission
  error.
• Two common ways in which transmission errors
  occur are replacement and transposition.
• One character may accidentally be replaced by a
  different, incorrect, character.
• Two adjacent characters may be interchanged.

21
Example 3

• What type of error occurred in each situation?
• The SSN 123-45-6789 was recorded as 123-45-6798.
• The SSN 123-45-6789 was recorded as 123-45-6788.

22
Example 3, contd

• Solutions
• The last two characters 89 were incorrectly
  written as 98. This is a transposition error.
• The last character 9 was incorrectly written as
  an 8. This is a replacement error.

23
Transmission Errors, contd

• Suppose every possible ID number is a valid
  number in the identification system being used.
• Any transmission errors in the number will
  result in the wrong person or item being
  identified.
• It will not be evident that an error was made.
• Most modern ID number systems are designed to
  guard against transmission errors.

24
Check Digits

• Additional digits added to an identification
  number so that errors in transmission can be
  found are called check digits.
• Ideally, a check-digit system should insure that
  any single-digit transmission error will result
  in an invalid ID number so that the error will be
  detected.

25
Question

• Suppose a company assigns each employee a
  four-digit ID number in which the first three
  digits come from 0 through 9 and the last digit
  is a check digit chosen so that the sum of all
  four digits is divisible by 7. Fill in the
  missing digit in the ID number 3_82.
• a. 7 b. 5 c. 3 d. 1

26
Example 4

• Suppose a biology professor assigns a four-digit
  numeric ID number to each of the almost 1000
  students in his class.
• The first three digits are randomly assigned.
• The fourth digit is the smallest number that
  makes the sum of all 4 digits divisible by 9.

27
Example 4, contd

• If a replacement error is made in a single digit,
  it may be detected.
• A 9 changed to a 0 or a 0 changed to a 9 will not
  be detected.
• Other single-digit errors will be detected.
• Transposition errors will not be detected.
• Replacement errors in two or more digits may not
  be detected.

28
Example 5

• Suppose the professor from example 4 assigns a
  four-digit numeric ID number with the same check
  digit scheme, but using only the digits 0 8.
• Any single-digit transmission error will be
  detected.
• If a digit is changed to another digit from 0
  8, the sum changes and is no longer divisible by
  9.
• If a digit is changed to a 9, it will be
  recognized as an invalid digit.

29
Universal Product Codes (UPC)

• Almost every retail product has a 12-digit
  numeric identification number, called a UPC,
  printed on its packaging.
• Examples of UPCs are shown below.

30
UPCs, contd

• The first digit, called the number system
  character, indicates the type of product.

31
UPCs, contd

• The first group of five digits, called the
  manufacturer number, indicates the company that
  makes the product.

32
UPCs, contd

• The second group of five digits, called the
  product number, indicates the specific product
  being sold.
• Items sold by weight are not printed with product
  numbers.
• Different manufacturers may use different or the
  same product numbers. There is no relationship
  between product numbers for various manufacturers.

33
UPCs, contd

• The last digit is the check digit.

• If the digits in a UPC are labeled as

then the check digit is chosen so that the sum
is divisible by 10.
34
UPCs, contd

• The check-digit scheme for UPC numbers is a
  two-weight scheme.
• The weights used are 3 and 1.
• The sum

is called a weighted sum.
35
Example 6

• Answer questions a b about the chicken broth
  UPCs from the example earlier 0 74785 00252 8
  and
• 0 74785 50352 0.
• What is the number system character and what does
  it represent?
• Solution The number system character for both is
  0 and it indicates that the chicken broth is a
  general grocery item.

36
Example 6, contd

• What is the manufacturer number and what does it
  represent?
• Solution The manufacturer number for both is
  74785 and it indicates that the manufacturer is
  Valley Fresh Inc. (not given in the previous
  table of common manufacturers).

37
Example 6, contd

• What is the product number and what does it
  represent?
• Solution The product number for the can of clear
  broth is 00252. The product number for the can
  of fat free broth is 50352.
• Two different products from the same company have
  two different product numbers.

38
Example 6, contd

• Verify that the check digits are correct.
• Solution For the can of clear broth, the
  calculation is

The weighted sum is divisible by 10, so the check
digit is correct.
39
Example 6, contd

• Solution For the can of fat free broth, the
  calculation is

The weighted sum is divisible by 10, so the check
digit is correct.
40
Example 7

• If the first 11 digits of a UPC are 2 13576
  05341, what must the check digit be?

41
Example 7, contd

• Solution

• Adding a 1 to 69 would make the weighted sum
  divisible by 10. The check digit must be a 1.

42
Example 8

• The first digit of a UPC is missing. If the
  remaining code is 01947 12513 3, what was the
  missing digit?

43
Example 8, contd

• Solution

• The missing digit must be a 2, so that 6 64
  70 will be divisible by 10.

44
1.1 Initial Problem Solution

• Can you be confident that you will be charged the
  correct price if you purchase the item with this
  UPC?

45
Initial Problem Solution, contd

• The UPC is 2 26080 80291 8.
• The initial digit of 2 indicates this item is
  sold by weight.
• When the item was weighed a machine produced a
  label with the cost 2.91 in positions 9 through
  11 of the UPC.
• The check digit is 8. Using the weighted sum
  check digit scheme will verify that this is
  correct.
• The UPC is correct and you will be charged the
  right amount.

46
Section 1.2Modular Arithmetic andCheck Digit
Schemes

• Goals
• Study the division algorithm
• Study congruence modulo m
• Study modular check digit schemes

47
1.2 Initial Problem

• Suppose you want to buy a car. The vehicle
  identification number is 1G4HP54C5KH410030.
• Is this number legitimate?
• The solution will be given at the end of the
  section.

48
Numbers

• Whole numbers are represented by the numerals 0,
  1, 2, 3, 4,
• Integers are represented by the numerals , -3,
  -2, -1, 0, 1, 2, 3,

49
The Division Algorithm for Whole Numbers

• The constants a and m must be whole numbers with
  m not equal to zero.
• There are unique whole numbers q and r such that
  a mq r.
• The constant r, the remainder, is less than m and
  greater than or equal to zero.
• The constant a is the dividend, m is the divisor,
  and q is the quotient.
• If r 0, we say that m divides a or write ma.

50
Example 1

• Use the division algorithm to find the quotient
  and remainder for the divisor 5 and the dividend
  21.

51
Example 1, contd

• Solution We see that 21 5(4) 1, so the
  quotient is q 4 and the remainder is r 1.
• Since the remainder is not 0, 5 does not divide
  21.

52
Example 2

• Use the division algorithm to find the quotient
  and remainder for the divisor 6 and the dividend
  108.

53
Example 2, contd

• Solution We see that 108 6(18) 0, so the
  quotient is q 18 and the remainder is r 0
• Since the remainder is 0, 6 divides 108.

54
Division Algorithm, contd

• Many check digit schemes are based on using the
  division algorithm.
• Usually it is the remainder from a division that
  is used as a check digit.
• The quotient and remainder can be found by doing
  long division or by dividing on a calculator.

55
Division Algorithm, contd

• To find the quotient and remainder for a whole
  number division using a calculator
• Perform the division a/m on the calculator. The
  whole number portion of the result is the
  quotient q.
• Multiply the decimal portion of the result from
  Step 1 by the divisor m to get an approximation
  of the remainder r.
• Round r to the nearest whole number to find the
  remainder.

56
Example 3

• Use a calculator to find the quotient and
  remainder for the divisor 13 and the dividend 543.

57
Example 3, contd

• Solution
• Divide 543 by 13. The calculator shows a result
  of 41.7692307692. The quotient is 41.
• Multiply 0.7692307692 by 13. The result is
  9.9999999996.
• Round 9.9999999996 to the nearest whole number to
  find the remainder. The remainder is 10. Check
  that 13(41) 10 543.

58
Question

• Find the quotient and remainder for the divisor
  39 and the dividend 217.
• a. q 22, r 5
• b. q 0, r 39
• c. q 5, r 22
• d. q 3, r 0

59
The Division Algorithm for Integers

• The constants a and m must be integers with m
  greater than or equal to one.
• There are unique integers q and r such that a
  mq r.
• The constant r, the remainder, is less than m and
  greater than or equal to zero.
• The constant a is the dividend, m is the divisor,
  and q is the quotient.
• If r 0, we say that m divides a or write ma.

60
Division Algorithm, contd

• To find the quotient and remainder for an integer
  division using a calculator
• Perform the division a/m on the calculator.
• If the result is a whole number, the result is q.
• If the result is not a whole number, q is the
  next smallest negative number.

61
Division Algorithm, contd

• Contd
• Subtract q from the results from Step 1 and then
  multiply this value by the divisor m to get an
  approximation of the remainder r.
• Round r to the nearest whole number to find the
  remainder.

62
Example 4

• Use a calculator to find the quotient and
  remainder for the divisor 7 and the dividend -359.

63
Example 4, contd

• Solution
• Divide -359 by 7. The calculator shows a result
  of -51.2857142857. The quotient is
• -52.
• Subtract -52 from -51.2857142857 to get
  0.7142857143. Multiply 0.7142857143 by 7. The
  result is 5.0000000001.
• Round 5.0000000001 to the nearest whole number to
  find the remainder. The remainder is 5. Check
  that 7(-52) 5
• -359.

64
Congruence Modulo M

• The constants a, b, and m must be integers with m
  greater than or equal to two.
• Then if m evenly divides a b, we say a is
  congruent to b modulo m.
• The constant m is called the modulus, and the
  phrase modulo m is often shortened to mod m.
• The definition above can be written symbolically
  as follows

65
Question

• Choose the number below that is NOT congruent to
  85 mod 3.
• a. 52
• b. 55
• c. 62
• d. 64

66
Example 5

• Verify each of the congruencies.
• Solution 66 38 28 4(7). Since the
  difference of the two integers is a multiple of
  7, they are congruent mod 7.
• Solution 3422 -153 3575 275(13). The two
  integers are congruent mod 13.

67
Modular Arithmetic

• Let modulus m gt 0 be a fixed integer. The
  numbers a, b, and c are integers.
• Modular arithmetic has the following rules

68
Modular Arithmetic, contd

• Where k is any positive integer

69
Modular Check Digit Schemes

• Common choices of a modulus for check digit
  schemes are 7, 9, 10, and 11.
• Modular check digit schemes follow this pattern
• If necessary, replace each non-numeric character
  with a digit according to a standard code.

70
Modular Check Digit Schemes

• Contd
• Treat the string as a whole number or combine the
  digits in some type of weighted sum.
• The check digit is the whole number r between 0
  and m 1 that is congruent mod m to the number
  from step 2. If a check digit is greater than 9,
  it must be replaced by an alphanumeric character.

71
A Mod 9 Check Digit Scheme

• An ID number has k 1 digits, where k is some
  positive integer
• The check digit, which is the last digit, is the
  whole number from 0 to 8 that is congruent modulo
  9 to the number made up of the first k digits.

72
Example 6

• A company uses a mod 9 check digit scheme for
  5-digit ID numbers. What is the check digit for
  5368?

73
Example 6, contd

• Solution Calculate 5368/9 596.444444444.
• Multiply 0.444444444 by 9 to get 3.99999999996.
• The remainder, which is the check digit, is 4.
  The complete ID number is 53684.

74
Mod 9 Check Digit Scheme, contd

• The check digit in this type of scheme can also
  be found through the shortcut of casting out
  nines.
• Add the first k digits of the ID number.
• This sum must be congruent modulo 9 to the check
  digit
• Example 5368 22, which is congruent to 4
  mod 9. The check digit is again found to be 4.

75
Example 7

• A company uses a mod 9 check digit scheme for
  5-digit ID numbers. What is the missing digit in
  the smudged number?

76
Example 7, contd

• Solution Call the missing digit X. The ID
  number is 73X11. Using the procedure of casting
  out nines we find that 7 3 X 1 11 X
  must be congruent mod 9 to the check digit 1. We
  see that 11 8 19, which is 1 more than a
  multiple of 9. The missing digit X must be 8.
• The ID number is 73811.

77
Example 8

• A U.S. Post Office money order has an 11 digit ID
  number with a mod 9 check digit scheme.
• If a money order has an ID number of 2995709918
  what is the check digit that should go in the
  11th place in the string?

78
Example 8, contd

• Solution 2995709918 is congruent mod 9 to its
  check digit. The sum of its digits is 2 9 9
  5 7 0 9 9 1 8 59, which is
  congruent to 5 mod 9.
• The check digit is 5.
• The entire ID number for the money order is
  29957099185.

79
Example 9

• Euro banknotes use a check digit scheme in which
  the check digit is chosen so that the entire
  serial number is divisible by 9.
• The code is alphanumeric, with values assigned to
  the letters as shown in the next slide.

80
Example 9, contd
81
Example 9, contd

• Suppose a 20-euro banknote has a serial number of
  S07090546498
• Verify that the serial number is correct.

82
Example 9, contd

• Solution The serial number is S07090546498. The
  character S has a value of 2.
• The serial number is worth 207090546498.
• The sum of its digits is 2 0 7 0 9 0
  5 4 6 4 9 8 54, which is congruent to
  0 mod 9.

83
A Mod 7 Check Digit Scheme

• An ID number has k 1 digits, where k is some
  positive integer
• The check digit, which is the last digit, is the
  whole number from 0 to 6 that is congruent modulo
  7 to the number made up of the first k digits.

84
Example 10

• An airline ticket has an ID number with a mod 7
  check digit scheme.
• Verify that the number located in the bottom
  center of the ticket below is a valid ID number.

85
Example 10, contd

• Solution The ID number is equal to
  1615042694252.
• For the ID number to be valid, the number made of
  the first 12 digits must be congruent to 2 mod 7.
  
• Check that 7 divides (161504269425 2)
  161504269423, so the congruency holds and the
  number is valid.

86
A Mod 11 Check Digit Scheme

• A check digit scheme for VINs uses congruence
  modulo 11.
• A VIN is a 17-digit alphanumeric ID number with
  values assigned to the characters according to
  the table below.

87
Mod 11 Check Digit Scheme, contd

• The check digit is the 9th character. A check
  digit of 10 is represented by X. A weighted sum
  is used with weights of 8, 7, 6, 5, 4, 3, 2, 10,
  9, 8, 7, 6, 5, 4, 3, and 2 for the characters.

88
1.2 Initial Problem Solution

• The cars VIN is 1G4HP54C5KH410030. Is this
  number legitimate?
• The digit 5 in the 9th position is the check
  digit.
• Convert each letter to a digit.
• The 16 digits, without the check digit, are
  17487543 28410030.

89
Initial Problem Solution, contd

• Calculate the weighted sum
• 8(1) 7(7) 6(4) 5(8) 4(7) 3(5) 2(4)
  10(3) 9(2) 8(8) 7(4) 6(1) 5(0) 4(0)
  3(3) 2(0) 327.
• If the VIN is valid, 327 must be congruent modulo
  11 to the check digit 5. However, this is not
  the case.
• The VIN is not valid. Do not buy the car.

90
Section 1.3Encoding Data

• Goals
• Study binary codes
• Morse code
• UPC bar codes
• Braille code
• ASCII
• Postnet code

91
1.3 Initial Problem

• The envelope below has been damaged. Use the
  Postnet bar code to determine the ZIP4 code.
• The solution will be given at the end of the
  section.

92
Binary Codes

• Coding methods are used to encode numbers and
  other data before they are transmitted.
• A data coding system made up of two states or
  symbols is called a binary code.

93
Binary Codes, contd

• Many common coding systems are binary codes.
• Morse code
• UPC bar codes
• Braille code
• ASCII
• Postnet code

94
Morse Code

• In Morse code, each character is encoded using
  dots and dashes.
• Morse code is a binary code because it was
  developed for telegraphs which have two states,
  ON and OFF.
• The code is created by leaving the telegraph
  circuit ON or OFF for a certain length of time.

95
Morse Code, contd

• The circuit is ON for one unit of time to create
  a dot.
• The circuit is ON for three units of time to
  create a dash.
• The circuit is OFF for one unit of time between
  any dots or dashes in a character.

96
Morse Code, contd

• The circuit is OFF for three units of time
  between characters.
• The circuit is OFF for six units of time between
  words.

97
Morse Code, contd

• The codes for each character are shown below.

98
Example 1

• The Morse code for the word MATH is illustrated
  in the figure below.
• ON is represented by a black square and OFF by a
  white square.

99
Morse Code, contd

• ON and OFF can also be represented by the binary
  digits 0 and 1.
• ON is represented by 1.
• OFF is represented by 0.
• Example The letter A can be encoded using 0s and
  1s as shown below.

100
Morse Code, contd
101
Example 2

• Convert the message MATH to Morse code using 0s
  and 1s.

102
Example 2, contd

• Solution
• M is encoded as 1110111
• A is encoded as 10111
• T is encoded as 111
• H is encoded as 1010101
• A 0 is inserted between each character.
• The coded message is 11101110001011100011100010
  10101.

103
Example 3

• Convert the Morse code message 101010100010100000
  01110001010101000100010111010001 to English.
• Recall that the code 000 separates characters,
  while 000000 separates words.

104
Example 3, contd

• Solution
• The first word is 1010101000101.
• 1010101 H
• 000 break between characters
• 101 I
• The first word is HI.

105
Example 3, contd

• The second word is 1110001010101000100010111010001
  
• 111 T
• 000 break between characters
• 1010101 H
• 1 E
• 1011101 R
• 1 E
• The decoded message is HI THERE

106
Question

• Translate the message SMILE into Morse code.
• a. 10101011101110101010111010101
• b. 1010100011101110001010001011101010001
• c. 10101000000111011100000010100000010111010100000
  01
• d. 10101000111011100010111000101110101000101110101
  

107
UPC Bar Codes

• Universal Product Code (UPC) numbers on retail
  items are encoded using a binary code.
• The encoding is represented by vertical bars.
• The bars are easily read by the laser scanner.

108
UPC Bar Codes, contd

• Each digit is first replaced with a binary code.

109
Question

• Does the UPC binary code below represent a
  manufacturer number or a product number?
• 1010000101110011101000
• a. manufacturer number
• b. product number
• c. impossible to determine

110
UPC Bar Codes, contd

• Similarly to Morse code, each 0 is represented by
  a white bar and each 1 by a black bar.

111
Example 4

• Convert the manufacturer number 365 into a
  sequence of 0s and 1s.

112
Example 4, contd

• Solution
• The digit 3 is encoded as 0111101.
• The digit 6 is encoded as 0101111.
• The digit 5 is encoded as 0110001.
• The encoding is 011110101011110110001.

113
Example 5

• Convert the manufacturer number 365 into a bar
  code.

114
Example 5, contd

• Solution The individual bar codes for 3, 6, and
  5 are found in the table.

115
Example 5, contd

• Solution, contd The individual bar codes are
  placed in order from left to right, and then
  scaled to look like a UPC.

116
UPC Bar Codes contd

• The laser scanner at the store must be able to
  read
• In which direction a bar code is scanned
• Where a bar code begins and ends
• Various widths of bars used on different products.

117
UPC Bar Codes contd

• Every UPC bar code begins and ends with a guard
  bar pattern.
• The pattern consists of a white strip between two
  black strips, all the same width.
• The width of each strip is called a module and is
  the unit of width for that particular bar code.

118
UPC Bar Codes contd

• Every UPC bar code also has a center bar pattern.
• The pattern consists of five alternating strips,
  starting with white.
• Each strip is one module wide.

119
UPC Bar Codes contd

• A bar code from a can of Campbells Cream of
  Chicken Soup is shown below.

120
Example 6

• Check this bar code for Campbells Cream of
  Chicken Soup. The manufacturer number is 51000
  and the product number is 01031.
• Recall that a UPC consists of a single digit,
  followed by the manufacturer number, the product
  number, and finally a check digit.
• For general grocery items the initial digit is 0.

121
Example 6, contd

• Solution The first 11 digits of the UPC number
  are 05100001031. The check digit, d, must be
  found.
• We know that the weighted sum
• 3(0 1 0 0 0 1) 1(5 0 0 1 3
  d) must be a multiple of 10. So 6 9 d 15
  d indicates that d must be a 5.
• The UPC number is 051000010315.

122
Example 6, contd

• Solution contd The bar code for the UPC
  051000010315 must be created.
• A guard bar pattern begins the code.
• The bar code for each digit in the manufacturer
  number and the product number follows.
• Recall that the center bar pattern goes between
  the manufacturer and product codes.
• A guard bar pattern ends the code.

123
Example 6, contd

• Solution contd The bar code is shown below, in
  its initial state and its final scaled version.

124
Example 6, contd

• Solution contd Compare our bar code for the
  soup to a bar code from an actual can of soup.

125
Braille Code

• The Braille system of writing is a binary code.
• Every Braille symbol is a pattern of 6 dots, each
  of which is raised or not raised.

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Braille Code, contd

• The Braille code for the alphabet is shown below.

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Example 7

• Decode the Braille message shown below.

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Example 7, contd

• Solution Find each character in the Braille
  alphabet.

• The message is decoded as NEVERMORE.

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ASCII

• The American Standard Code for Information
  Interchange (ASCII) is used by most computers.
• ASCII is pronounced ask-key.
• Each character is encoded using 8 binary digits,
  called a byte.

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ASCII, contd
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ASCII, contd

• A second table below gives the ASCII code for 32
  non-printing control characters.

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Example 8

• Convert the message from ASCII to English.
• 01001000 01000101 01001100 01001100 01001111.

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Example 8, contd

• Solution
• 01001000 H
• 01000101 E
• 01001100 L
• There are two of these.
• 01001111 E
• The message reads HELLO.

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Postnet Code

• The U.S. Postal Service uses a code called
  Postnet to encode ZIP codes as bars.
• Postnet encodes
• 9-digit ZIP4 codes used on business reply forms
• 11-digit ZIP4 plus delivery-point code used on
  reduced-rate business mail.
• A delivery-point code consists of the last two
  digits of the street address or box number.

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Postnet Code, contd

• Postnet uses 5 bits to allow for error detection.
  The code for each digit is shown in the table
  below.

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Postnet Code, contd

• On pieces of mail, the Postnet code is
  represented as a series of bars.
• The 1s are represented by tall bars.
• The 0s are represented by short bars.
• The Postnet bar codes are shown below.

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Postnet Code, contd

• Complete Postnet codes include
• Guard bars, consisting of a single tall bar on
  each end.
• An extra check digit before the last guard bar.
• The check digit is chosen so that the sum of all
  the digits is divisible by 10.

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1.3 Initial Problem Solution

• The envelope below has been damaged. Use the
  Postnet bar code to determine the ZIP4 code.

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Initial Problem Solution, contd

• Each set of 5 bars inside the guard bars
  represents a digit. Compare the code on the
  envelope to the decoding table.

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Initial Problem Solution, contd

• The final 0 is the check digit. The ZIP4 code
  is 80323-8510.