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A Mathematical View of Our World

A Mathematical View of Our World. 1 st ed. Parks, Musser, Trimpe, Maurer, and Maurer. Chapter 1. Numbers in Our Lives. Section 1.1 ID Numbers and Check Digits. Goals Study social security numbers Study general identification numbers Transmission errors Check digits

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A Mathematical View of Our World

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  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, cont’d

  8. Area Number, cont’d

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

  31. UPCs, cont’d • The first group of five digits, called the manufacturer number, indicates the company that makes the product.

  32. UPCs, cont’d • 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, cont’d • 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, cont’d • 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, cont’d • 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, cont’d • 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, cont’d • 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, cont’d • 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, cont’d • 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, cont’d • 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, cont’d • 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 mdividesa or write m|a.

  50. Example 1 • Use the division algorithm to find the quotient and remainder for the divisor 5 and the dividend 21.

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