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Nossi Ch 1

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Contemporary Math
Chapter 1 notes and assignment

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Nossi Ch 1

  1. 1. Chapter 1 <ul><li>Identification Numbers </li></ul><ul><li>Check Digits </li></ul><ul><li>Codes </li></ul>
  2. 2. Identification Numbers Area Number-Group Number-Serial Number pg 3-5 Where was the mailing address of your SSN?
  3. 3. <ul><li>ISBN - Find the one for our book </li></ul>
  4. 4. <ul><li>VIN </li></ul>
  5. 5. <ul><li>UPC Codes </li></ul><ul><li>First digit - type of item - pg 9 </li></ul><ul><li>Manufacturer code </li></ul><ul><li>Product code </li></ul>
  6. 6. <ul><li>Check Digits </li></ul><ul><li>An additional digit added to an identification number so that a transmission error may be found. </li></ul>
  7. 7. <ul><li>Check Digits </li></ul><ul><li>An additional digit added to an identification number so that a transmission error may be found. </li></ul><ul><li>Before we look at how check digits are used we need to look at some special rules. </li></ul>
  8. 8. <ul><li>Did you know that each of these numbers is divisible by 9? </li></ul><ul><li>81 </li></ul><ul><li>135 </li></ul><ul><li>23616 </li></ul>
  9. 9. <ul><li>Did you know that each of these numbers is divisible by 9? </li></ul><ul><li>81 </li></ul><ul><li>135 </li></ul><ul><li>23616 </li></ul><ul><li>Do you know why? </li></ul>
  10. 10. <ul><li>If the sum of the digits of a number is divisible by 9, then the number is divisible by 9. </li></ul><ul><li>81: 8+1= </li></ul><ul><li>135: 1+3+5= </li></ul><ul><li>23616: 2+3+6+1+6= </li></ul><ul><li>Name a number that is divisible by 9. </li></ul>
  11. 11. <ul><li>Ex. 1.3 </li></ul><ul><li>A biology professor has nearly 1000 students in his class. For reasons of confidentiality he wants to assign each student an identification number. </li></ul><ul><li>Suggest some ways to assign these identification numbers. </li></ul>
  12. 12. <ul><li>The professor could assign a fourth digit as a check digit. One way to do this is to assign the fourth digit so that the the identification number is divisible by 9. </li></ul>
  13. 13. <ul><li>A 3 digit number that does not contain a 9 could be assigned and then the 4th digit makes the sum of the 4 digits divisible by 9. </li></ul><ul><li>If the sum of the 3 digits is already divisible by 9 the 4th digit will be a 0. </li></ul>
  14. 14. <ul><li>This is one example of a check digit. </li></ul><ul><li>We will look at more check digits in the next section. </li></ul><ul><li>1.1 Assignment </li></ul><ul><li>Pg 12 (3,9,11,13,15,19,25,26,33) </li></ul>
  15. 15. <ul><li>Section 1.2 </li></ul><ul><li>Modular Arithmetic and Check-Digit Schemes </li></ul>
  16. 16. <ul><li>Modular Arithmetic and Check- </li></ul><ul><li>Digit Schemes </li></ul><ul><li>How do we know if this is a legitimate VIN number? </li></ul>
  17. 17. <ul><li>Before looking at how to check a VIN for legitimacy, lets look at a simpler example. </li></ul><ul><li>Divide 4 by 13 </li></ul>
  18. 18. <ul><li>Before looking at how to check a VIN for legitimacy, lets look at a simpler example. </li></ul><ul><li>Divide 13 by 4 </li></ul><ul><li>The quotient is: </li></ul><ul><li>3 with remainder 1 </li></ul>
  19. 19. <ul><li>Before looking at how to check a VIN for legitimacy, lets look at a simpler example. </li></ul><ul><li>Divide 13 by 4 </li></ul><ul><li>The quotient is 3 with remainder 1 </li></ul><ul><li>To check: 4 X 3 + 1 = 13 </li></ul>
  20. 20. <ul><li>If a number divides evenly and the remainder is zero: </li></ul><ul><li>For example: 15/5 = 3 </li></ul><ul><li>We can say “5 divides 15” </li></ul><ul><li>This can be written 5|15 </li></ul>
  21. 21. <ul><li>The remainder is the important number. </li></ul><ul><li>Division by 7 </li></ul><ul><li>Integer Remainder </li></ul><ul><li>. . . , -21, -14, -7, 0, 7, 14, 21, . . . 0 </li></ul><ul><li>. . . , -20, -13, -6, 1, 8, 15, 22, . . . 1 </li></ul><ul><li>. . . , -19, -12, -5, 2, 9, 16, 23, . . . 2 </li></ul><ul><li>. . . , -18, -11, -4, 3, 10, 17, 24, . . . 3 </li></ul><ul><li>. . . , -17, -10, -3, 4, 11, 18, 25, . . . 4 </li></ul><ul><li>. . . , -16, -9, -2, 5, 12, 19, 26, . . . 5 </li></ul><ul><li>. . . , -15, -8, -1, 6, 13, 20, 27, . . . 6 </li></ul><ul><li>What patterns do you see? </li></ul>
  22. 22. <ul><li>In every row, the difference between the numbers in a row is a multiple of 7 </li></ul><ul><li>Integer Remainder </li></ul><ul><li>. . . , -21, -14, -7, 0, 7, 14, 21, . . . 0 </li></ul><ul><li>. . . , -20, -13, -6, 1, 8, 15, 22, . . . 1 </li></ul><ul><li>. . . , -19, -12, -5, 2, 9, 16, 23, . . . 2 </li></ul><ul><li>. . . , -18, -11, -4, 3, 10, 17, 24, . . . 3 </li></ul><ul><li>. . . , -17, -10, -3, 4, 11, 18, 25, . . . 4 </li></ul><ul><li>. . . , -16, -9, -2, 5, 12, 19, 26, . . . 5 </li></ul><ul><li>. . . , -15, -8, -1, 6, 13, 20, 27, . . . 6 </li></ul>
  23. 23. <ul><li>The language we will use is: </li></ul><ul><li>The two numbers are: </li></ul><ul><li>“ congruent modulo 7” </li></ul><ul><li>Because 7 divides the difference between 29 and 15: </li></ul><ul><li>7|(29-15) and 29=15 mod 7 </li></ul><ul><li>(= should be a symbol with 3 lines) </li></ul>
  24. 24. <ul><li>Integer Remainder </li></ul><ul><li>. . . , -21, -14, -7, 0, 7, 14, 21, . . . 0 </li></ul><ul><li>. . . , -20, -13, -6, 1, 8, 15, 22, . . . 1 </li></ul><ul><li>. . . , -19, -12, -5, 2, 9, 16, 23, . . . 2 </li></ul><ul><li>. . . , -18, -11, -4, 3, 10, 17, 24, . . . 3 </li></ul><ul><li>. . . , -17, -10, -3, 4, 11, 18, 25, . . . 4 </li></ul><ul><li>. . . , -16, -9, -2, 5, 12, 19, 26, . . . 5 </li></ul><ul><li>. . . , -15, -8, -1, 6, 13, 20, 27, . . . 6 </li></ul><ul><li>State 2 more congruence relationships. </li></ul>
  25. 25. <ul><li>See the Example 1.11 on pg 24 </li></ul>
  26. 26. <ul><li>See example 1.12 on pg 24 </li></ul>
  27. 27. <ul><li>Modular Check Digit Schemes </li></ul><ul><li>9 is a popular check digit </li></ul><ul><li>Ex. A company uses a mod 9 check-digit scheme for its 5 digit id number. The 5th digit is the check digit. </li></ul>
  28. 28. <ul><li>Determine the check digit for </li></ul><ul><li>5368 </li></ul>
  29. 29. <ul><li>Determine the check digit for </li></ul><ul><li>5368 </li></ul><ul><li>Add the digits: </li></ul><ul><li>5 + 3 + 6 + 8 = 22 </li></ul><ul><li>9|(22 - ?) </li></ul>
  30. 30. <ul><li>9|18, so </li></ul><ul><li>9|(22 - 4) </li></ul><ul><li>The check digit is 4 </li></ul>
  31. 31. <ul><li>Find the missing digit if the 5th digit is the check-digit using mod 9: </li></ul><ul><li>73?11 </li></ul>
  32. 32. <ul><li>7 + 3 + d 3 + 1 = 1 mod 9 </li></ul><ul><li>Remember, the 5th digit is the check digit </li></ul>
  33. 33. <ul><li>7 + 3 + d 3 + 1 = 1 mod 9 </li></ul><ul><li>11 + d 3 = 1 mod 9 </li></ul><ul><li>What number minus 1 is divisible by 9? </li></ul><ul><li>19=1mod 9 or 9|(19-1) </li></ul>
  34. 34. <ul><li>11 + d 3 = 1 mod 9 </li></ul><ul><li>What number minus 1 is divisible by 9? </li></ul><ul><li>19=1mod 9 or 9|(19-1) </li></ul><ul><li>What does d 3 have to equal? </li></ul>
  35. 35. <ul><li>11 + d 3 = 1 mod 9 </li></ul><ul><li>What number minus 1 is divisible by 9? </li></ul><ul><li>19=1mod 9 or 9|(19-1) </li></ul><ul><li>What does d 3 have to equal? </li></ul><ul><li>d 3 = 8 </li></ul>
  36. 36. <ul><li>Example of uses of </li></ul><ul><li>mod 9 check-digits: </li></ul><ul><li>Money Orders </li></ul><ul><li>European Currency </li></ul>
  37. 37. <ul><li>Airline tickets use a mod 7 check digit system. Read ex 1.7 on pg 29. </li></ul>
  38. 38. <ul><li>Assignment for sec. 1.2: </li></ul><ul><li>Pg 32 (9,21,23,25,31,33,35) </li></ul>
  39. 39. <ul><li>Binary Codes </li></ul><ul><li>Morse Code </li></ul><ul><li>UPC </li></ul><ul><li>Braille </li></ul><ul><li>ASCII </li></ul>Section 1.3 Encoding Data
  40. 40. <ul><li>A data coding system made up of two states (on/off) or two symbols </li></ul>Binary Codes
  41. 41. <ul><li>Morse code is one type of a binary code. </li></ul><ul><li>See chart on pg 38 </li></ul>Morse Code
  42. 42. <ul><li>A dot is one unit ON </li></ul><ul><li>A dash is 3 units ON </li></ul><ul><li>The circuit is OFF for 1 unit between dots and dashes </li></ul>Morse Code
  43. 43. Morse Code
  44. 44. <ul><li>The dots and dashes can be converted to black and white squares </li></ul><ul><li>Black square = ON </li></ul><ul><li>White square = OFF </li></ul>Morse Code
  45. 45. <ul><li>The black and white squares can be converted to 1’s and 0’s </li></ul><ul><li>1 = ON </li></ul><ul><li>0 = OFF </li></ul><ul><li>See chart on pg 39 </li></ul>Morse Code
  46. 46. <ul><li>Convert the word MATH to Morse Code using 1’s and 0’s </li></ul><ul><li>Insert 3 0’s between letters. </li></ul>Morse Code
  47. 47. <ul><li>Convert a list of 1’s and 0’s into English. </li></ul><ul><li>See ex. 1.19 on pg 39 </li></ul>Morse Code
  48. 48. UPC Bar Codes
  49. 49. <ul><li>The first 5 digits are the manufacturer code </li></ul><ul><li>The 2nd 5 digits are the product code </li></ul>UPC Bar Codes
  50. 50. <ul><li>The last digit is a check digit chosen according to the following rule: </li></ul><ul><li>3(d 1 +d 3 +d 5 +d 7 +d 9 +d 11 ) + </li></ul><ul><li>1(d 2 +d 4 +d 6 +d 8 +d 10 +d 12 ) = 0mod10 </li></ul><ul><li>*The total is divisible by 10 </li></ul>UPC Bar Codes
  51. 51. Braille
  52. 52. <ul><li>Each letter consists of 6 dots. </li></ul><ul><li>Each dot is either raised or not raised. </li></ul><ul><li>For a combination of 2x2x2x2x2x2 characters </li></ul>Braille
  53. 53. <ul><li>2x2x2x2x2x2 = 2 6 = 64 </li></ul><ul><li>Possible characters </li></ul>Braille
  54. 54. <ul><li>More than 64 characters were needed for computers so the ASCII code was developed. </li></ul>ACSII
  55. 55. <ul><li>A bit is a information unit having one of two states: </li></ul><ul><li>On or Off </li></ul><ul><li>1 or 0 </li></ul>ACSII
  56. 56. <ul><li>A groups of 8 bits is called a byte. </li></ul><ul><li>This would be a set of 8 1’s and 0’s. </li></ul>ACSII
  57. 57. <ul><li>ASCII code is an 8-bit code. </li></ul><ul><li>See Table 1.14 on pg 48. </li></ul>ACSII
  58. 58. <ul><li>How many characters could an 8-bit code represent? </li></ul>ACSII
  59. 59. <ul><li>How many characters could an 8-bit code represent? </li></ul><ul><li>2 8 = 256 </li></ul>ACSII
  60. 60. <ul><li>Pg 51 (1,7,9,11,13,15,17,21,25a,b,29) </li></ul>Section 1.3 Assignment
  61. 61. <ul><li>Send me an e-mail so that I have your address. </li></ul><ul><li>[email_address] </li></ul><ul><li>Pg 12 (3,9,11,13,15,19,25,26,33) </li></ul><ul><li>Pg 32 (9,21,23,25,31,33,35) </li></ul><ul><li>Pg 51 (1,7,9,11,13,15,17,21,25a,b,29) </li></ul>Chapter 1 Assignment

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