Chapter 4 Systems of Numeration 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration. 4.2 Place-Va

1. 1 Chapter 4 � Systems of Numeration 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration. 4.2 Place-Value or Positional-Value Numeration Systems 4.3 Other Bases 4.4 Computation in Other Bases 4.5 Early Computational Methods

2. 2 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration This section covers a brief history of numeration, as well as three different types of numeration systems.

3. 3 A Brief History of Numeration: Mathematics began when people needed a way to count herds, the passage of days, and objects to barter. Initially, physical objects (stones, fingers, etc.) were used to actually represent the objects being counted. Later, these physical objects were used to represent the object and the quantity.

4. 4 A Brief History of Numeration: Probably the first record keeping system was the tally system. In a tally system, marks are made in a one-to-one correspondence with the object being counted. No physical objects (stones, fingers, etc.) were needed. As civilization developed, the tally system was inadequate, and societies developed symbols to replace the tally system.

5. 5 A Brief History of Numeration: Number � is the quantity. It answers the question �How Many�? Numeral � is a symbol used to represent a number. Algorithms � are the rules used for combining the numbers. System of Numeration � a set of numerals and the rules for combining the numerals to represent numbers.

6. 6 A Brief History of Numeration: We will be studying six different numeration systems in this chapter. You do not need to memorize the symbols of each system. You need to understand the principles used to combine the symbols to represent numbers. The symbols will be supplied for you when you take an exam on this chapter. We use the Hindu-Arabic system today. After completing this chapter, you will better understand the Hindu-Arabic system.

7. 7 An Overview of Numeration Systems: Types of Numeration Systems 1. Additive Systems Egyptian Hieroglyphics Roman Numerals 2. Multiplicative Systems Traditional Chinese Numerals 3. Ciphered Systems Ionic Greek Numerals 4. Place-Value Systems

8. 8 Additive Systems: Additive systems are one of the oldest types of numeration systems. In an additive system of numeration, the number being represented is the sum of the value of the numerals. There is no symbol for zero.

9. 9 Egyptian Hieroglyphics: The Egyptian system was one of the first additive systems, dating back to around 3400 B.C. The symbols for this system can be found in Table 4.1 on page 149 in the text. Notice that the system had symbols for the powers of ten. To write the number 40, in this system, you would repeat the symbol for 10 (Heel bone) four times. To write the number 43, we would sketch three 1s (Staff) after the four Heel bones.

10. 10 Egyptian Hieroglyphics - Example #1: Write the following numeral as a Hindu-Arabic numeral.

11. 11 Egyptian Hieroglyphics - Example #1: Write the following numeral as a Hindu-Arabic numeral.

12. 12 Egyptian Hieroglyphics - Example #2: Write 205,632 as an Egyptian numeral.

13. 13 Egyptian Hieroglyphics - Example #2: Write 205,632 as an Egyptian numeral. 205,632 = 200,000 + 5,000 + 600 + 30 + 2 So we would need 2 tadpoles (100,000 each), 5 lotus flowers (1000 each), 6 scrolls (100 each), 3 heel bones (10 each), and 2 staffs (1 each).

14. 14 Egyptian Hieroglyphics - Example #2: Write 205,632 as an Egyptian numeral. 205,632 = 200,000 + 5,000 + 600 + 30 + 2 So we would need 2 tadpoles (100,000 each), 5 lotus flowers (1000 each), 6 scrolls (100 each), 3 heel bones (10 each), and 2 staffs (1 each).

15. 15 Egyptian Hieroglyphic Examples: It is important to note that the ordering of the symbols is not important, and that the symbols do not need to be written in a line. Study Examples 1 and 2 on page 149 in the text to see more examples on how to convert from Egyptian to Hindu-Arabic, and Hindu-Arabic to Egyptian.

16. 16 Egyptian Hieroglyphic Examples: One last thing to note: to write the number 33,888 would require 30 symbols! Can you figure out why? The point, in an additive system, it takes a lot of symbols to represent large numbers. Large numbers cannot be compactly written.

17. 17 Roman Numerals: The Roman numeration system is another example of an additive system. It was developed after the Egyptian system. It was used in most of the European countries until the 18th century. Roman numerals can still be found today. Can you think of where? A list of the Roman numerals can be found on table 4.2 on page 149 in the text.

18. 18 Roman Numerals: The Roman numeration system has a few advantages over the Egyptian system. The Roman numeration system has an addition, a subtraction, and a multiplication principle. Starting from the left, we add the value of each numeral to the numeral that follows, unless the numeral that follows is larger. If the numeral that follows is larger, then we subtract the two numerals.

19. 19 Roman Numerals: For example VI = 5 + 1 = 6, since the symbols from left to right are decreasing in value. But, IV = 5 � 1 = 4, since the symbols are increasing in value from left to right. Another advantage to the Roman system is its multiplication principle for numbers over 1000. A bar written above a group of symbols indicates that the symbols are to be multiplied by 1000. This allows larger numbers to be written in a somewhat compact form.

20. 20 Roman Numerals - Example #3: Write as a Hindu-Arabic numeral.

21. 21 Roman Numerals - Example #3: Write as a Hindu-Arabic numeral.

22. 22 Roman Numerals � Example #4: Write 5244 as a Roman numeral.

23. 23 Roman Numerals � Example #4: Write 5244 as a Roman numeral. We need to write 5000 + 200 + 40 + 4. To write 5000, we use the multiplication principle and write 5 X 1000 ( ). 200 is CC, 40 = 50 � 10, so write XL (the smaller value first, to signify subtraction). 4 = 5 � 1, so write IV (the subtraction rule again).

24. 24 Roman Numerals � Example #4: Write 5244 as a Roman numeral. We need to write 5000 + 200 + 40 + 4. To write 5000, we use the multiplication principle and write 5 X 1000 ( ). 200 is CC, 40 = 50 � 10, so write XL (the smaller value first, to signify subtraction). 4 = 5 � 1, so write IV (the subtraction rule again). The answer would be

25. 25 Roman Numerals: So, the ordering of the symbols is important when writing Roman numerals. A symbol does not have to be repeated more than three times, because of the subtraction principle. Study Examples 3, 4, and 5 on page 150 in the text to see more examples on how to convert from Roman numerals to Hindu-Arabic, and from Hindu-Arabic to Roman numerals.

26. 26 Multiplicative Systems: Multiplicative systems use multiplication to represent a number. The best example of a multiplicative system is the Traditional Chinese Numeration System. The traditional Chinese system used symbols for 1 to 9 inclusive, and then symbols for the increasing powers of ten. The numerals of the system can be found in Table 4.3 on page 151 in the text.

27. 27 Traditional Chinese Numerals: Chinese numerals are written vertically. It is easiest to start at the top of the number and group the symbols in twos. The number at the top of each group of two will always be a number from 1 � 9. The number at the bottom of each group of two will be a power of ten. Multiply the two numbers within each group together (multiply the number 1-9 by the power of ten that is below it). After you have multiplied each group, then add all of the separate products together.

28. 28 Traditional Chinese � Example #5: Write as a Hindu-Arabic numeral.

29. 29 Traditional Chinese � Example #5:

30. 30 Traditional Chinese � Example #5:

31. 31 Traditional Chinese � Example #5:

32. 32 Traditional Chinese � Example #5:

33. 33 Traditional Chinese � Example #5:

34. 34 Traditional Chinese � Example #5:

35. 35 Traditional Chinese � Example #5:

36. 36 Traditional Chinese � Example #6: Write 3029 as a Chinese numeral.

37. 37 Traditional Chinese � Example #6: Write 3029 as a Chinese numeral. 3029 = 3000 + 20 + 9 = (3 x 1000) + (2 x 10) + 9

38. 38 Traditional Chinese � Example #6: Write 3029 as a Chinese numeral. 3029 = 3000 + 20 + 9 = (3 x 1000) + (2 x 10) + 9 Now write them vertically:

39. 39 Traditional Chinese � Example #6: Write 3029 as a Chinese numeral. 3029 = 3000 + 20 + 9 = (3 x 1000) + (2 x 10) + 9 Now write them vertically: 3 1000 2 10 9

40. 40 Traditional Chinese � Example #6: Write 3029 as a Chinese numeral. 3029 = 3000 + 20 + 9 = (3 x 1000) + (2 x 10) + 9 Now write them vertically: 3 1000 2 10 9

41. 41 Traditional Chinese Numerals: Study Example 6 on page 151 in the text. Notice that when you start at the top and group the symbols in twos, you have an extra symbol at the bottom. The symbol that is not paired off becomes the ones digit of the number. In effect, we are multiplying the �8� by 1. Add this symbol to the products of the other pairs.

42. 42 Ciphered Systems: The last type of system that will be studied in this section is the ciphered numeration system. Ciphered systems allow numbers to be written in a very compact form because there are many symbols. The Ionic Greek system is an example of a ciphered numeration system. The symbols of the system can be found on Table 4.4, on page 152 in the text. (Remember, you do not have to memorize the symbols, they will be provided for you on the test)

43. 43 Ionic Greek Numerals: The Ionic Greek system was developed around 3000 B.C. It used the letters of their alphabet and three symbols taken from the Phoenician alphabet. The symbols are written in decreasing order, and are added together to find the number being represented. Like the Roman system, it also includes a symbol to write large numbers in a compact form. A prime ? placed after a symbol multiplies the value of the symbol by 1000.

44. 44 Ionic Greek � Example #7: Write as a Hindu-Arabic numeral.

45. 45 Ionic Greek � Example #7: Write as a Hindu-Arabic numeral.

46. 46 Ionic Greek � Example #7: Write as a Hindu-Arabic numeral.

47. 47 Ionic Greek � Example #8: Write 1999 as an Ionic Greek numeral.

48. 48 Ionic Greek � Example #8: Write 1999 as an Ionic Greek numeral. 1999 = 1000 + 900 + 90 + 9.

49. 49 Ionic Greek � Example #8: Write 1999 as an Ionic Greek numeral. 1999 = 1000 + 900 + 90 + 9. Write the symbols. Since we do not have a numeral for 1000, take 1 x 1000, so we need to use a prime. There are numerals for all of the other values.

50. 50 Ionic Greek � Example #8: Write 1999 as an Ionic Greek numeral. 1999 = 1000 + 900 + 90 + 9. Write the symbols. Since we do not have a numeral for 1000, take 1 x 1000, so we need to use a prime. There are numerals for all of the other values. The answer is:

51. 51 Ionic Greek Examples: Study Examples 7 and 8 on page 152 in the text for more examples of how to convert from Ionic Greek to Hindu-Arabic, and from Hindu-Arabic to Ionic Greek.

52. 52 4.2 Place-Value or Positional-Value Numeration Systems The Hindu-Arabic numeration system we use in the United States today is a place-value system. This section discusses three different place-value systems: Hindu-Arabic system Babylonian system Mayan system

53. 53 Place-Value Systems: Place-value systems (also called positional-value systems) are the most common type of numeration system used today. In a place-value system, the value of the symbol depends on its position. For example, the 2 in 20 represents 2 tens while the 2 in 200 represents 2 hundreds. The �2� represents different values because the 2 is in different positions.

54. 54 Place-Value Systems: To be a true place-value system, a numeration system must have two things. A base A set of symbols, including zero, and a symbol for each counting number less than the base. For example � the Hindu-Arabic numeration system has 10 for the base, and symbols (digits) 0,1,2,3,4,5,6,7,8,9.

55. 55 Place-Value Systems: The positional values of the Hindu-Arabic system are: . . . (10)3 , (10)2 , 10 , 1 This can also been observed when writing numbers in expanded form. 762 = ( 7 x 100 ) + ( 6 x 10 ) + (2 x 1 )

56. 56 Place-Value Systems: The oldest known numeration system that was similar to a place-value system was the Babylonian numeration system. It was developed around 2500 B.C. The Babylonian system was not a true place-value system since it lacked a symbol for zero. The base in the Babylonian system is 60. Thus, the positional values are . . . (60)3 , (60)2 , 60 , 1 . . . 216,000 , 3600 , 60 , 1

57. 57 Babylonian Numeration System: The symbols for the Babylonian numeration system are in Table 4.5 on page 154 in the text. Notice there is a symbol for 1 and a symbol for 10. In this system, commas are not used. Instead, a space is left between the symbols to indicate the different place-values. There is also a special symbol for subtraction.

58. 58 Babylonian System � Example #1: Write as a Hindu-Arabic numeral.

59. 59 Babylonian System � Example #1: Write as a Hindu-Arabic numeral.

60. 60 Babylonian System � Example #1: Write as a Hindu-Arabic numeral.

61. 61 Babylonian System � Example #1: Write as a Hindu-Arabic numeral.

62. 62 Babylonian System � Example #1: Write as a Hindu-Arabic numeral.

63. 63 Babylonian System � Example #2: Write 86 as a Babylonian numeral.

64. 64 Babylonian System � Example #2: Write 86 as a Babylonian numeral. The Babylonian system has positional values of �603, 602, 601, 1, which are 216000, 3600, 60, 1.

65. 65 Babylonian System � Example #2: Write 86 as a Babylonian numeral. The Babylonian system has positional values of �603, 602, 601, 1, which are 216000, 3600, 60, 1. The largest positional value less than or equal to 86 is 60. There is one group of 60 in 86.

66. 66 Babylonian System � Example #2: Write 86 as a Babylonian numeral. The Babylonian system has positional values of �603, 602, 601, 1, which are 216000, 3600, 60, 1. The largest positional value less than or equal to 86 is 60. There is one group of 60 in 86. That leaves us with 26 remaining or 26 x 1. 86 = (1 x 60) + (26 x 1).

67. 67 Babylonian System � Example #2: Write 86 as a Babylonian numeral. The Babylonian system has positional values of �603, 602, 601, 1, which are 216000, 3600, 60, 1. The largest positional value less than or equal to 86 is 60. There is one group of 60 in 86. That leaves us with 26 remaining or 26 x 1. 86 = (1 x 60) + (26 x 1). Our answer is:

68. 68 Babylonian System � Example #3: Write as a Hindu-Arabic numeral.

69. 69 Babylonian System � Example #3: Write as a Hindu-Arabic numeral.

70. 70 Babylonian System � Example #3: Write as a Hindu-Arabic numeral.

71. 71 Babylonian System � Example #3: Write as a Hindu-Arabic numeral.

72. 72 Babylonian Numeration System: Study Examples 1 and 2 on page 155 in the text. Notice in Example 2, there is a symbol for subtraction. Study Example 4 on page 157 in the text. This example converts a Hindu-Arabic numeral into a Babylonian numeral.

73. 73 Mayan Numeration System: The Mayan numeration system is the last place-value system that is studied in the text. This system was developed by the Mayans who lived on the Yucatan Peninsula. The system was developed anytime between 300 � 900 B.C. depending on what you read. The numerals for the Mayan system are found in Table 4.6 on page 157 in the text. Notice that the Mayan system has a symbol for zero, and a symbol for the numerals 1 � 19.

74. 74 Mayan Numeration System: The positional values for the Mayan system are . . . 18 x (20)3 , 18 x (20)2 , 18 x 20 , 20 , 1 . . . 144,000 , 7200 , 360 , 20 , 1 The Mayan�s sophisticated system conformed to their religious and agricultural calendar which had 360 days. So, initially it looks like a base 20 system, but then the next place-value is 18x20 instead of 202. Another interesting thing about the Mayan system is that numbers were written vertically with the units place on the bottom.

75. 75 Mayan Numeration System: The first three positional values in the Mayan system are: 18x20 20 1

76. 76 Mayan System � Example #4: Write as a Hindu-Arabic numeral.

77. 77 Mayan System � Example #4: Write as a Hindu-Arabic numeral.

78. 78 Mayan System � Example #4: Write as a Hindu-Arabic numeral.

79. 79 Mayan System � Example #4: Write as a Hindu-Arabic numeral.

80. 80 Mayan System � Example #5: Write 257 as a Mayan numeral.

81. 81 Mayan System � Example #5: Write 257 as a Mayan numeral. The Mayan positional values are �7200, 360, 20, 1. The greatest positional value less than or equal to 257 is 20. So divide 257 by 20.

82. 82 Mayan System � Example #5: Write 257 as a Mayan numeral. The Mayan positional values are �7200, 360, 20, 1. The greatest positional value less than or equal to 257 is 20. So divide 257 by 20. You find that 257 has 12 groups of 20 with 17 left over. So, 257 = (12 x 20 ) + 17.

83. 83 Mayan System � Example #5: Write 257 as a Mayan numeral. The Mayan positional values are �7200, 360, 20, 1. The greatest positional value less than or equal to 257 is 20. So divide 257 by 20. You find that 257 has 12 groups of 20 with 17 left over. So, 257 = (12 x 20 ) + 17. Our answer is:

84. 84 Mayan Numeration System: Study Examples 5, 6, and 7 on page 158 in the text for more example of how to convert from Mayan to Hindu-Arabic, and from Hindu-Arabic to Mayan.

85. 85 4.3 Other Bases The Hindu-Arabic system of numeration is a base 10 system. It makes sense to us because we have 10 fingers, 10 toes, and we have used the system our whole life. Base 2 ( binary ) is important because of computers. But what if our numeration system was a base 3 or base 4 system? What would our numbers be like? That is the topic of this section.

86. 86 Base 3, Base 4, Base 5? Really? Not every society in the world uses a base 10 place-value system. Read pages 160 and 161 in the text. It is interesting reading that covers systems of other bases and where they are found today. It also points out a few examples of other bases that we see every day and probably do not even realize.

87. 87 Converting to Base 10: To understand more about base 10, it is helpful to be able to convert between different bases. 1. Convert 1546 to base 10. In base 6, the positional values are . . . 63 , 62 , 6 , 1. Writing 1546 in expanded form: 1546 = (1 x 62 ) + ( 5 x 6 ) + ( 4 x 1 ) = ( 1 x 36 ) + ( 5 x 6 ) + ( 4 x 1 ) = 36 + 30 + 4 = 70

88. 88 Converting to Base 10: Notice that the last problem was base 6, which is lower than our base. What if the number was in base 12 (which is larger than our base)? In base 12, there should be 12 different symbols, starting with 0. So we could list 0,1,2,3,4,5,6,7,8,9, but then what? We cannot use 10 and 11 because they are not different symbols, they are a combination of two symbols. In our text, the following symbols for base 12 are used: 0,1,2,3,4,5,6,7,8,9,T,E. To remember this, think �the T stands for ten� and �the E stands for eleven�.

89. 89 Converting to Base 10: 2. Convert 123E12 to base 10. In base 12, the positional values are . . . 123 , 122 , 12 , 1. Writing 123E12 in expanded form: 123E12 = (1x123 ) + (2x122 ) + (3x12) + (11x1) = (1x1728) + (2x144) + (3x12) + (11x1) = 1728 + 288 + 36 + 11 = 2063

90. 90 Converting to Base 10: Basically, converting to base 10 involved writing a number in expanded form and multiplying to find the value of each digit. Then find the sum of all of the values. To see more examples of converting to base 10, look at page 162 in the text, Examples 2,3, and 4.

91. 91 Converting from Base 10 to another base: What if we want to convert the other way? It would make sense that we would do the opposite operation � division.

92. 92 Converting from Base 10 to another base: 3. Convert 146 ( which is in base 10) to base 5. The positional values in base 5 are . . . 54 , 53 , 52 , 5 , 1 or . . . 625 , 125 , 25 , 5 , 1 The highest power of 5 that is less than or equal to 146 is 53 or 125. Divide 146 by 125. 146/125 = 1 with remainder 21. Therefore, there is 1 group of 125 in 146. ( 1 is the first digit in the answer )

93. 93 Converting from Base 10 to another base: 3 continued: Next divide the remainder, 21, by 25 (the next lower positional-value of 5). 21/25 = 0 with remainder 21 Therefore, there are 0 groups of 25 in 21. (0 is the 2nd digit in the answer) Next divide the remainder, 21, by 5 (the next lower positional-value of 5). 21/5 = 4 with remainder 1 Therefore, there are 4 groups of 5 in 21 with 1 unit remaining. (4 is the 3rd digit in the answer)

94. 94 Converting from Base 10 to another base: 3 continued: The answer is: 146 = (1 x 125) + (0 x 25) + (4 x 5) + (1 x 1) = (1 x 53) + (0 x 52) + (4 x 5) + (1 x 1) = 10415 Note: We place a subscript 5 to the right of 1041 to show that it is a base 5 number.

95. 95 Converting from Base 10 to another base: 4. Convert 1695 to base 12. The positional values in base 12 are . . .123 , 122 , 12 , 1 or . . . 1728 , 144 , 12 , 1 The highest power of 12 that is less than or equal to 1695 is 144. Divide 1695 by 144. 1695/144 = 11 with remainder 111 Therefore, there are 11 groups of 144 in 1695. ( E, the symbol for 11 in base 12, is the first digit in the answer)

96. 96 Converting from Base 10 to another base: Next, divide the remainder, 111, by 12 (the next lower positional-value of 12). 111/12 = 9 with remainder 3 Therefore, there are 9 groups of 12 in 111 and 3 units remaining. (9 is the 2nd digit in the answer) 1695 = (E x 122 ) + ( 9 x 12 ) + ( 3 x 1 ) = E9312

97. 97 Converting from Base 10 to another base: More examples converting from base 10 to another base can be found in the text on pages 162-164, Examples 5-7.

98. 98 4.4 Computation in Other Bases It was mentioned before that computers perform calculations in base 2. This section covers computations in base 2 and other bases.

99. 99 Addition in Other Bases: In some examples, the book uses addition tables from each base to solve addition problems. In this course, we will not be using addition tables. In this course, you will not be asked to use addition tables, and you will not be allowed to use them when taking a test. In the long run, it is faster to learn to add numbers without the tables.

100. 100 Addition in Other Bases: To add in bases other that base 10, we will add as if we were in base 10, and then convert the answer to the given base. Once the sum is converted to the given base, then we can record it under the problem. It is much easier to explain this by showing an example. The following slides will show how to add in bases other than 10.

101. 101 Addition in Other Bases: 1. Add 44 7 + 65 7

102. 102 Addition in Other Bases: 1 1. Add 44 7 + 65 7 2

103. 103 Addition in Other Bases: 1 1. Add 44 7 + 65 7 2

104. 104 Addition in Other Bases: 1 1. Add 44 7 + 65 7 1 42 7

105. 105 Addition in Other Bases: 1 1. Add 44 7 + 65 7 1 42 7

106. 106 Addition in Other Bases: 2. Add 341 5 + 341 5

107. 107 Addition in Other Bases: 2. Add 341 5 + 341 5 2

108. 108 Addition in Other Bases: 2. Add 341 5 + 341 5 2

109. 109 Addition in Other Bases: 1 2. Add 341 5 + 341 5 32

110. 110 Addition in Other Bases: 1 2. Add 341 5 + 341 5 32

111. 111 Addition in Other Bases: 1 2. Add 341 5 + 341 5 1232 5

112. 112 Addition in Other Bases: Study Examples 4 and 5 on pages 168 and 169 in the text. These are more examples of addition in other bases.

113. 113 Subtraction in Other Bases: Subtraction can be performed in other bases relatively easily. Remember that when you have to borrow, you borrow the amount of the base given in the problem. For example, if you are subtracting in base 3, when you borrow, you borrow 3.

114. 114 Subtraction in Other Bases: 3. Subtract 536 7 - 124 7

115. 115 Subtraction in Other Bases: 3. Subtract 536 7 - 124 7

116. 116 Subtraction in Other Bases: 3. Subtract 536 7 - 124 7 2

117. 117 Subtraction in Other Bases: 3. Subtract 536 7 - 124 7 2

118. 118 Subtraction in Other Bases: 3. Subtract 536 7 - 124 7 12

119. 119 Subtraction in Other Bases: 3. Subtract 536 7 - 124 7 12

120. 120 Subtraction in Other Bases: 3. Subtract 536 7 - 124 7 412 7

121. 121 Subtraction in Other Bases: 3. Subtract 536 7 - 124 7 412 7

122. 122 Subtraction in Other Bases: Notice how that was no different than if we would have been calculating in base 10. That is because we did not have to borrow. Let�s look at an example that involves borrowing.

123. 123 Subtraction in Other Bases: 4. Subtract 1221 3 - 202 3

124. 124 Subtraction in Other Bases: 4. Subtract 1221 3 - 202 3

125. 125 Subtraction in Other Bases: 1 4. Subtract 1221 3 - 202 3

126. 126 Subtraction in Other Bases: 1 4 4. Subtract 1221 3 - 202 3

127. 127 Subtraction in Other Bases: 1 4 4. Subtract 1221 3 - 202 3

128. 128 Subtraction in Other Bases: 1 4 4. Subtract 1221 3 - 202 3 2

129. 129 Subtraction in Other Bases: 1 4 4. Subtract 1221 3 - 202 3 2

130. 130 Subtraction in Other Bases: 1 4 4. Subtract 1221 3 - 202 3 12

131. 131 Subtraction in Other Bases: 1 4 4. Subtract 1221 3 - 202 3 12

132. 132 Subtraction in Other Bases: 1 4 4. Subtract 1221 3 - 202 3 012

133. 133 Subtraction in Other Bases: 1 4 4. Subtract 1221 3 - 202 3 012

134. 134 Subtraction in Other Bases: 1 4 4. Subtract 1221 3 - 202 3 1012 3

135. 135 Subtraction in Other Bases: 1 4 4. Subtract 1221 3 - 202 3 1012 3

136. 136 Subtraction in Other Bases: Study examples 6 and 7 on page 169 in the text. These are examples of subtraction in other bases. Just remember to borrow the amount of the base in the problem.

137. 137 Multiplication in Other Bases: To multiply in bases other that base 10, we will multiply as if we were in base 10, and then convert the answer to the given base. Once the product is converted to the given base, then we can record it under the problem. It is much easier to explain this by showing an example. The following slides will show how to multiply in bases other than 10.

138. 138 Multiplication in Other Bases: 5. Multiply 123 5 x 4 5

139. 139 Multiplication in Other Bases: 5. Multiply 123 5 x 4 5

140. 140 Multiplication in Other Bases: 2 5. Multiply 123 5 x 4 5 2

141. 141 Multiplication in Other Bases: 2 5. Multiply 123 5 x 4 5 2

142. 142 Multiplication in Other Bases: 22 5. Multiply 123 5 x 4 5 02

143. 143 Multiplication in Other Bases: 22 5. Multiply 123 5 x 4 5 02

144. 144 Multiplication in Other Bases: 22 5. Multiply 123 5 x 4 5 1102 5

145. 145 Multiplication in Other Bases: 22 5. Multiply 123 5 x 4 5 1102 5

146. 146 Multiplication in Other Bases: Study Example 9 on page 171 in the text. Notice how similar this is to multiplying in base 10. Notice how the columns line up.

147. 147 Division in Other Bases: Look at Examples 10 and 11 on pages 171-173 in the text. These are examples of division in bases other than 10. Try to follow along and see how a quotient is calculated. In this course, we will not be studying division in bases other than 10, but it is nice to see that it is possible.

148. 148 4.5 Early Computational Methods The algorithm that we use today to multiply is not the only way to find the product of two numbers. Early civilizations had some very interesting ways to calculate products. This section covers four such methods: Duplation and Mediation, Egyptian Multiplication, Galley Method, and Napier Rods. You will be amazed at some of the following methods, and you will be surprised that they work!

149. 149 Duplation and Mediation: This method of multiplication was used in Medieval Europe from the 5th � 15th centuries. Another name for this method could be doubling and halving. It consists of making two columns (one for each factor to be multiplied) and taking half of one number (dropping any remainders), and doubling the other number.

150. 150 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation.

151. 151 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35

152. 152 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6

153. 153 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70

154. 154 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3

155. 155 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3 � 140

156. 156 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3 � 140

157. 157 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3 � 140 1

158. 158 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3 � 140 1 � 280

159. 159 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3 � 140 1 � 280

160. 160 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3 � 140 1 � 280

161. 161 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3 � 140 1 � 280

162. 162 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3 � 140 1 � 280

163. 163 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3 � 140 1 � 280

164. 164 Duplation and Mediation: 1. Multiply 12 x 35 by duplation and mediation. 12 � 35 6 � 70 3 � 140 1 � 280 420

165. 165 Duplation and Mediation: This method of multiplication was used in Medieval Europe from the 5th � 15th centuries. Here is another example.

166. 166 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71

167. 167 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9

168. 168 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142

169. 169 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4

170. 170 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4 � 284

171. 171 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4 � 284 2

172. 172 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4 � 284 2 � 568

173. 173 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4 � 284 2 � 568 1

174. 174 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4 � 284 2 � 568 1 � 1136

175. 175 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4 � 284 2 � 568 1 � 1136

176. 176 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4 � 284 2 � 568 1 � 1136

177. 177 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4 � 284 2 � 568 1 � 1136

178. 178 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4 � 284 2 � 568 1 � 1136

179. 179 Duplation and Mediation: 2. Multiply 19 x 71 using duplation and mediation. 19 � 71 9 � 142 4 � 284 2 � 568 1 � 1136 1349

180. 180 Duplation and Mediation: Isn�t that amazing? Another example of duplation and mediation can be found in the text on page 175. One thing to note: multiplication is commutative, and so is this method of multiplication. If you reverse 19 x 71 to 71 x 19, and take half of 71 and double 19, you will still get the same answer. Amazing.

181. 181 Egyptian Multiplication Algorithm: The Egyptian multiplication algorithm is not covered in the text, but you will want to learn it. The Egyptian multiplication algorithm is based on the principle that every positive integer can be expressed as a sum of powers of 2 ( 1, 2, 4, 8, 16, 32, etc.). For example, if you choose the number 12, 12 = 4 + 8 (notice both 4 and 8 are powers of 2). The number 19 = 1 + 2 + 16 (all are powers of 2). Following are examples of Egyptian Multiplication.

182. 182 Egyptian Multiplication Algorithm: 3. Multiply 12 x 35 using Egyptian Multiplication

183. 183 Egyptian Multiplication Algorithm: 3. Multiply 12 x 35 using Egyptian Multiplication 1 � 35

184. 184 Egyptian Multiplication Algorithm: 3. Multiply 12 x 35 using Egyptian Multiplication 1 � 35 2 � 70

185. 185 Egyptian Multiplication Algorithm: 3. Multiply 12 x 35 using Egyptian Multiplication 1 � 35 2 � 70 4 � 140

186. 186 Egyptian Multiplication Algorithm: 3. Multiply 12 x 35 using Egyptian Multiplication 1 � 35 2 � 70 4 � 140 8 � 280

187. 187 Egyptian Multiplication Algorithm: 3. Multiply 12 x 35 using Egyptian Multiplication 1 � 35 2 � 70 4 � 140 8 � 280 16 � 560

188. 188 Egyptian Multiplication Algorithm: 3. Multiply 12 x 35 using Egyptian Multiplication 1 � 35 2 � 70 4 � 140 8 � 280 16 � 560

189. 189 Egyptian Multiplication Algorithm: 3. Multiply 12 x 35 using Egyptian Multiplication 1 � 35 2 � 70 4 � 140 8 � 280 16 � 560

190. 190 Egyptian Multiplication Algorithm: 3. Multiply 12 x 35 using Egyptian Multiplication 1 � 35 2 � 70 4 � 140 8 � 280 16 � 560

191. 191 Egyptian Multiplication Algorithm: Isn�t that interesting? Did you notice that was the same problem (12 x 35 ) as example 1 for duplation and mediation? In approximately 3400 B.C., this method was used in Egypt. What makes is more amazing is that while using this algorithm, the Egyptians would have been using their hieroglyphics (the symbols we learned about in section 4.1)! Following is another example of the Egyptian algorithm.

192. 192 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication.

193. 193 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71

194. 194 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142

195. 195 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142 4 � 284

196. 196 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142 4 � 284 8 � 568

197. 197 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142 4 � 284 8 � 568 16 � 1136

198. 198 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142 4 � 284 8 � 568 16 � 1136 32 � 2272

199. 199 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142 4 � 284 8 � 568 16 � 1136 32 � 2272

200. 200 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142 4 � 284 8 � 568 16 � 1136 32 � 2272

201. 201 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142 4 � 284 8 � 568 16 � 1136 32 � 2272

202. 202 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142 4 � 284 8 � 568 16 � 1136 32 � 2272

203. 203 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142 4 � 284 8 � 568 16 � 1136 32 � 2272

204. 204 Egyptian Multiplication Algorithm: 4. Multiply 19 x 71 using Egyptian multiplication. 1 � 71 2 � 142 4 � 284 8 � 568 16 � 1136 32 � 2272

205. 205 Egyptian Multiplication Algorithm: Egyptian multiplication is not covered in the text. To practice, complete the homework exercises for duplation and mediation (page 177 # 5 � 11 odd), and then solve the same problems using the Egyptian method of multiplication. If you work the two methods side by side for the same multiplication problem, you can observe the similarities and differences of the two methods.

206. 206 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

207. 207 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

208. 208 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

209. 209 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

210. 210 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

211. 211 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

212. 212 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

213. 213 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

214. 214 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

215. 215 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

216. 216 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

217. 217 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

218. 218 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

219. 219 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

220. 220 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

221. 221 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

222. 222 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

223. 223 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

224. 224 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

225. 225 The Galley Method of Multiplication: 5. Multiply 251 x 34 using the galley method.

226. 226 The Galley Method of Multiplication: Another example of galley multiplication can be found in the text, on pages 175 and 176. An interesting note: some students today are taught to multiply using the galley method. It is actually quite fast, once you have drawn the grid.

227. 227 Napier Rods: In the 17th century, John Napier developed a simple calculating machine based on the galley method of multiplication. The �machine� had eleven separate wooden or ivory rods which were numbered 0-9 and �index�. Read about Napier rods on page 176 and 177 in the text. Note that once the rods are arranged for a certain multiplication problem, the rods form a filled in �grid� from the galley method. To solve, all that is needed is to sum along the diagonals.

228. 228 Napier Rods: It is interesting to see how Napier multiplication is related to galley multiplication. In this course, it is important to see the connections between these two methods of multiplication, but you will not have homework or test questions on Napier multiplication.

229. 229 Congratulations! You have now completed the PowerPoint slides for Chapter 4 !

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