1. Chapter 4 Numeration and Mathematical Systems
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2. Chapter 4: Numeration and Mathematical Systems
4.1 Historical Numeration Systems 4.2 Arithmetic in the Hindu-Arabic System 4.3 Conversion Between Number Bases 4.4 Clock Arithmetic and Modular Systems 4.5 Properties of Mathematical Systems 4.6 Groups
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3. Section 4-1 Chapter 1 Historical Numeration Systems
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4. Historical Numeration Systems
Mathematical and Numeration Systems
Ancient Egyptian Numeration – Simple Grouping
Traditional Chinese Numeration – Multiplicative Grouping
Hindu-Arabic Numeration - Positional
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5. Mathematical and Numeration Systems
A mathematical system is made up of
three components:
1. a set of elements;
one or more operations for combining
the elements;
3. one or more relations for comparing the
elements.
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6. Mathematical and Numeration Systems
The various ways of symbolizing and working with the counting numbers are called numeration systems. The symbols of a numeration system are called numerals.
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7. Example: Counting by Tallying
Tally sticks and tally marks have been used for a long time. Each mark represents one item. For example, eight items are tallied by writing
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Counting by Grouping
Counting by grouping allows for less repetition of symbols and makes numerals easier to interpret. The size of the group is called the base (usually ten) of the number system.
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9. Ancient Egyptian Numeration – Simple Grouping
The ancient Egyptian system is an example of a simple grouping system. It used ten as its base and the various symbols are shown on the next slide.
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10. Ancient Egyptian Numeration
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11. Example: Egyptian Numeral
Write the number below in our system.
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12. Traditional Chinese Numeration – Multiplicative Grouping
A multiplicative grouping system involves pairs of symbols, each pair containing a multiplier and then a power of the base. The symbols for a Chinese version are shown on the next slide.
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Chinese Numeration
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14. Example: Chinese Numeral
Interpret each Chinese numeral.
a) b)
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15. Positional Numeration
A positional system is one where the various powers of the base require no separate symbols. The power associated with each multiplier can be understood by the position that the multiplier occupies in the numeral.
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16. Positional Numeration
In a positional numeral, each symbol (called a
digit) conveys two things:
1. Face value – the inherent value of the
symbol.
2. Place value – the power of the base which
is associated with the position that the digit
occupies in the numeral.
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17. Positional Numeration
To work successfully, a positional system must have a symbol for zero to serve as a placeholder in case one or more powers of the base are not needed.
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18. Hindu-Arabic Numeration – Positional
One such system that uses positional form is our system, the Hindu-Arabic system.
The place values in a Hindu-Arabic numeral, from right to left, are 1, 10, 100, 1000, and so on. The three 4s in the number 45,414 all have the same face value but different place values.
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19. Hindu-Arabic Numeration
Hundred thousands
Millions
Ten thousands
Thousands
Decimal point
Units (Ones)
Hundreds
Tens
7, ,
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20. Section 4.1: Historical Numerical Systems
A mathematical system has
a) Elements
b) Operations
c) Relations
d) All of the above

21. Section 4.1: Historical Numerical Systems
2. Our numeration system is an example of a
simple grouping system
multiplicative grouping system
positional system
d) complex grouping

22. Arithmetic in the Hindu-Arabic System
Chapter 1
Section 4-2
Arithmetic in the Hindu-Arabic System

23. Arithmetic in the Hindu-Arabic System
Expanded Form
Historical Calculation Devices

24. Expanded Form
By using exponents, numbers can be written in expanded form in which the value of the digit in each position is made clear.

25. Example: Expanded Form
Write the number 23,671 in expanded form.

26. Distributive Property
For all real numbers a, b, and c,
For example,

27. Example: Expanded Form
Use expanded notation to add 34 and 45.

28. Decimal System
Because our numeration system is based on powers of ten, it is called the decimal system, from the Latin word decem, meaning ten.

29. Historical Calculation Devices
One of the oldest devices used in calculations is the abacus. It has a series of rods with sliding beads and a dividing bar. The abacus is pictured on the next slide.

30. Abacus
Reading from right to left, the rods have values of 1, 10, 100, 1000, and so on. The bead above the bar has five times the value of those below. Beads moved towards the bar are in “active” position.

31. Example: Abacus
Which number is shown below?

32. Lattice Method
The Lattice Method was an early form of a paper-and-pencil method of calculation. This method arranged products of single digits into a diagonalized lattice.
The method is shown in the next example.

33. Example: Lattice Method
Find the product by the lattice method. 3 8

34. Napier’s Rods (Napier’s Bones)
John Napier’s invention, based on the lattice method of multiplication, is often acknowledged as an early forerunner to modern computers.
The rods are pictured on the next slide.

35. Napier’s Rods
See figure 2 on page 174

36. Russian Peasant Method
Method of multiplication which works by expanding one of the numbers to be multiplied in base two.

37. Nines Complement Method
Step 1 Align the digits as in the standard subtraction algorithm.
Step 2 Add leading zeros, if necessary, in the subtrahend so that both numbers have the same number of digits.
Step 3 Replace each digit in the subtrahend with its nines complement, and then add.
Step 4 Delete the leading (1) and add 1 to the remaining part of the sum.

38. Example: Nines Complement Method
Use the nines complement method to subtract 2803 – 647.
Solution
Step Step 2 Step 3 Step 4

39. Section 4.2: Arithmetic in the Hindu-Arabic System
1. Which of the following is an example of expanded form?
205 b)
c) 5(40 + 1)

40. Section 4.2: Arithmetic in the Hindu-Arabic System
2. Which of the following is an example of the distributive property? a) b)
c) 5(3 + 4) = 5(7)

41. Conversion Between Number Bases
Chapter 1
Section 4-3
Conversion Between Number Bases

42. Conversion Between Number Bases
General Base Conversions
Computer Mathematics

43. General Base Conversions
We consider bases other than ten. Bases other than ten will have a spelled-out subscript as in the numeral 54eight. When a number appears without a subscript assume it is base ten. Note that 54eight is read “five four base eight.” Do not read it as “fifty-four.”

44. Powers of Alternative Bases
Fourth Power
Third power
Second Power
First Power
Zero Power
Base two 16 8 4 2 1
Base five
625
125 25 5
Base seven
2401
343 49 7
Base eight
4096
512 64
Base sixteen
65,536
256

45. Example: Converting Bases
Convert 2134five to decimal form.

46. Calculator Shortcut for Base Conversion
To convert from another base to decimal form: Start with the first digit on the left and multiply by the base. Then add the next digit, multiply again by the base, and so on. The last step is to add the last digit on the right. Do not multiply it by the base.

47. Example:
Use the calculator shortcut to convert five to decimal form.

48. Example: Converting Bases
Convert 7508 to base seven.

49. Converting Between Two Bases Other Than Ten
Many people feel the most comfortable handling conversions between arbitrary bases (where neither is ten) by going from the given base to base ten and then to the desired base.

50. Computer Mathematics
There are three alternative base systems that are most useful in computer applications. These are binary (base two), octal (base eight), and hexadecimal (base sixteen) systems.
Computers and handheld calculators use the binary system.

51. Example: Convert Binary to Decimal
Convert two to decimal form.
Solution
111001two

52. Example: Convert Hexadecimal to Binary
Convert 8B4Fsixteen to binary form.

53. Section 4.3: Conversion Between Number Bases
Which of the following is not a valid base 8 number?
a) 456
b) 781
c) 0

54. Section 4.3: Conversion Between Number Bases
The following is a way to convert what base to base 10?
a) 7
b) 5
c) 2

55. Section 4-4 Chapter 1 Clock Arithmetic and Modular Systems 4-4-55 55
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56. Clock Arithmetic and Modular Systems
Finite Systems and Clock Arithmetic
Modular Systems
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Finite Systems
Because the whole numbers are infinite, numeration systems based on them are infinite mathematical systems. Finite mathematical systems are based on finite sets.
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12-Hour Clock System
The 12-hour clock system is based on an ordinary clock face, except that 12 is replaced by 0 so that the finite set of the system is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
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Clock Arithmetic
As an operation for this clock system, addition is defined as follows: add by moving the hour hand in the clockwise direction. 1 11 2 10
5 + 3 = 8 3 9 4 8 5 7 6
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60. Example: Finding Clock Sums by Hand Rotation
Find the sum: in 12-hour clock arithmetic
Plus 7 hours
Solution
Start at 8 and move the hand clockwise through 7 more hours.
Answer: 3 1 11 2 10 3 9 4 8 5 7 6
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61. 12-Hour Clock Addition Table
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62. 12-Hour Clock Addition Properties
Closure The set is closed under addition.
Commutative For elements a and b, a + b = b + a.
Associative For elements a, b, and c,
a + (b + c) = (a + b) + c.
Identity The number 0 is the identity element.
Inverse Every element has an additive inverse.
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63. Inverses for 12-Hour Clock Addition
Clock value a 1 2 3 4 5 6 7 8 9 10 11
Additive Inverse -a
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Subtraction on a Clock
If a and b are elements in clock arithmetic, then the difference, a – b, is defined as
a – b = a + (–b)
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65. Example: Finding Clock Differences
Find the difference 5 – 9.
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66. Example: Finding Clock Products
Find the product
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Modular Systems
In this area the ideas of clock arithmetic are expanded to modular systems in general.
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Congruent Modulo m
The integers a and b are congruent modulo m (where m is a natural number greater than 1 called the modulus) if and only if the difference a – b is divisible by m. Symbolically, this congruence is written
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69. Example: Truth of Modular Equations
Decide whether each statement is true or false.
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70. Criterion for Congruence
if and only if the same remainder is obtained when a and b are divided by m.
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71. Example: Solving Modular Equations
Solve the modular equation below for the whole number solutions.
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72. Section 4.4: Clock Arithmetic and Modular Systems
In 12-hour clock addition, find the additive inverse of 3.
a) 8
b) 9
c) –3

73. Section 4.4: Clock Arithmetic and Modular Systems
2. Is it true that
a) Yes
b) No

74. Properties of Mathematical Systems
Chapter 1
Section 4-5
Properties of Mathematical Systems

75. Properties of Mathematical Systems
An Abstract System
Closure Property
Commutative Property
Associative Property
Identity Property
Inverse Property
Distributive Property

76. An Abstract System
The focus will be on elements and operations that have no implied mathematical significance. We can investigate the properties of the system without notions of what they might be.

77. Operation Table
Consider the mathematical system with elements {a, b, c, d} and an operation denoted by ☺.
The operation table on the next slide shows how operation ☺ combines any two elements. To use the table to find c ☺ d, locate c on the left and d on the top. The row and column intersect at b, so c ☺ d = b.

78. Operation Table for ☺ ☺ a b c d

79. Closure Property
For a system to be closed under an operation, the answer to any possible combination of elements from the system must in the set of elements. ☺ a b c d
This system is closed.

80. Commutative Property ☺ a b c d
For a system to have the commutative property, it must be true that for any elements X and Y from the set, X ☺ Y = Y ☺ X. ☺ a b c d
This system has the commutative property. The symmetry with respect to the diagonal line shows this property

81. Associative Property ☺ a b c d
For a system to have the associative property, it must be true that for any elements X, Y, and Z from the set, X ☺ (Y ☺ Z) = (X ☺ Y) ☺ Z. ☺ a b c d
This system has the associative property. There is no quick check – just work through cases.

82. Identity Property ☺ a b c d
For the identity property to hold, there must be an element E in the set such that any element X in the set, X ☺ E = X and E ☺ X = X. ☺ a b c d
a is the identity element of the set.

83. Inverse Property
If there is an inverse in the system then for any element X in the system there is an element Y (the inverse of X) in the system such that
X ☺ Y = E and Y ☺ X = E, where E is the identity element of the set. ☺ a b c d
You can inspect the table to see that every element has an inverse.

84. Potential Properties of a Single Operation Symbol
Let a, b, and c be elements from the set of any system, and ◘ represent the operation of the system.
Closure a ◘ b is in the set
Commutative a ◘ b = a ◘ b.
Associative a ◘ (b ◘ c) = (a ◘ b) ◘ c
Identity The system has an element e such that a ◘ e = a and e ◘ a = a.
Inverse there exists an element x in the set such that a ◘ x = e and x ◘ a = e.

85. Example: Identifying Properties
Consider the system shown with elements {0, 1, 2, 3, 4} and operation Which properties are satisfied by this system? 1 2 3 4

86. Distributive Property
Let ☺ and ◘ be two operations defined for elements in the same set. Then ☺ is distributive over ◘ if
a ☺ (b ◘ c) = (a ☺ b) ◘ (a ☺ c)
for every choice of elements a, b, and c from the set.

87. Example: Testing for the Distributive Property
Is addition distributive over multiplication on the set of whole numbers?

88. Section 4.5: Properties of Mathematical Systems
For elements a and e and the operation
represents which property?
a) Commutative
b) Inverse
c) Identity

89. Section 4.5: Properties of Mathematical Systems
2. When a system has two operations, which property would we look for?
a) Commutative
b) Distributive
c) Associative

90. Chapter 1
Section 4-6
Groups

91. Groups
Groups
Symmetry Groups
Permutation Groups

92. Group
A mathematical system is called a group if, under its operation, it satisfies the closure, associative, identity, and inverse properties.

93. Example: Checking Group Properties
Does the set {–1, 1} under the operation of multiplication form a group?

94. Example: Checking Group Properties
Does the set {–1, 1} under the operation of addition form a group?

95. Symmetry Groups
A group can be built on sets of objects other than numbers. Consider the group of symmetries of a square. Start with a square labeled below.
Front
Back 4 1 1' 4' 3 2 2' 3'

96. Symmetries - Rotational
M rotate 90°
N rotate 180° 3 4 2 3 2 1 1 4
P rotate 270°
Q original 1 2 4 1 4 3 3 2

97. Symmetries - Flip Flip about horizontal line through middle. R 4 1 3'2' 3 2 4' 1'
Flip about vertical line through middle. S 4 1 1' 4' 3 2 2' 3'

98. Symmetries - Flip Flip about diagonal line upper left to lower right.4 1 4' 3' 3 2 1' 2'
Flip about diagonal line upper right to lower left. V 4 1 2' 1' 3 2 3' 4'

99. Symmetries of the Square□ M N P Q R S T V

100. Example: Verifying a Subgroup
Form a mathematical system by using only the set {M, N, P, Q} from the group of symmetries of a square. Is this new system a subgroup?
Solution □ M N P Q
The operational table is given and the system is a group. The new group is a subgroup of the original group.

101. Permutation Groups
A group comes from studying the arrangements, or permutations, of a list of numbers.
The next slide shows the possible permutations of the numbers

102. Arrangements of 1-2-3 A*: 1-2-3 2-3-1 B*: 1-2-3 2-1-3 C*: 1-2-3 1-2-3
D*: 1-2-3
1-3-2
E*: 1-2-3
3-1-2
F*: 1-2-3
3-2-1

103. Example: Combining Arrangements
Find D*E*.
Solution
1-2-3
Rearrange according to D*.
3 E* replaces 1 with 3.
E* replaces 2 with 1.
E* replaces 3 with 2.

104. Section 4.6: Groups
1. A mathematical group does not have to satisfy which property?
a) Commutative
b) Closure
c) Associative

105. Section 4.6: Groups
Does {a, b} satisfy a group under the operation shown below?
Yes No a b