Transcript Example

Chapter
3
Numeration Systems
and Whole Number
Operations
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3-1 Numeration Systems
Students will be able to understand and explain
• Numbers, their origin, and their representation in
numerals and models.
• Different numeration systems including the HinduArabic system.
• Place value and counting in base ten and other
bases.
• Issues in learning with different numeration
systems.
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Slide
2
Definition
Numerals: written symbols to represent cardinal
numbers.
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Slide
3
Definition
Numeration system: a collection of properties
and symbols agreed upon to represent numbers
systematically.
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Slide
4
Hindu-Arabic Numeration System
1. All numerals are constructed from the 10 digits.
2. Place value is based on powers of 10.
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Slide
5
Place value assigns a value to a digit
depending on its placement in a numeral.
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Slide
6
Expanded form
6789  6  10  7  10  8  10  9  1
3
2
1
Factor
If a is any number and n is any natural number,
then
n factors
an  a  a  a 
a
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Slide
7
Base-ten blocks
1 long →101 = 1 row of 10 units
1 flat →102 = 1 row of 10 longs, or 100 units
1 block→103 = 1 row of 10 flats, or 100 longs, or
1000 units
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Slide
8
Example
What is the fewest number of pieces you can
receive in a fair exchange for 11 flats, 17 longs, and
16 units?
11 flats
11 flats
17 longs
1 long
18 longs
16 units (16 units = 1 long
6 units and 6 units)
6 units (after the first trade)
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Slide
9
Example (continued)
11 flats
1 flat
12 flats
18 longs
8 longs
8 longs
6 units (18 longs = 1 flat)
and 8 longs)
6 units (after the second
trade)
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Slide 10
Example(continued)
(12 flats = 1 block and 2 flats)
1 block
1 block
12 flats
2 flats
2 flats
8 longs
6 units
8 longs
6 units
The fewest number of pieces = 1 + 2 + 8 + 6 = 17.
This is analogous to rewriting
as
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Slide 11
Tally Numeration System
Uses single strokes (tally marks) to represent each
object that is counted.
= 13
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Slide 12
Egyptian Numeration System
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Slide 13
Example
Use the Egyptian numeration system to
represent 2,345,123.
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Slide 14
Babylonian Numeration System
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Slide 15
Example
Use the Babylonian numeration system to
represent 305,470.
1  603 + 24  602
216,000 + 86,400
+ 51  60
+ 3060
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+10  1 =
+10
= 305,470
Slide 16
Mayan Numeration System
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Slide 17
Example
Use the Mayan numeration system to represent
305,470.
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Slide 18
Roman Numeration System
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Slide 19
Roman Numeration System
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Slide 20
In the Middle Ages, a bar was placed over a
Roman number to multiply it by 1000.
V represents 5  1000  5000
CDX represents 410  1000  410,000
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Slide 21
Example
Use the Roman numeration system to represent
15,478.
XVCDLXXVIII
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Slide 22
Other Number Base Systems
Quinary (basefive) system
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Slide 23
Example
Convert 11244five to base 10.
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Slide 24
Base Two
Binary system – only two digits
Base two is especially important because of its use
in computers.
One of the two digits is represented by the
presence of an electrical signal and the other by
the absence of an electrical signal.
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Slide 25
Example
Convert 10111two to base ten.
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Slide 26
Example
Convert 27 to base two.
16
–16
8
–8
4
–0
2
–2
1
–1
0
27
1
11
1
3
0
3
1
1
1
or
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Slide 27
Base Twelve
Duodecimal system – twelve digits
Use T to represent a group of 10.
Use E to represent a group of 11.
The base-twelve digits are 0, 1, 2, 3, 4, 5, 6, 7, 8,
9, T, and E.
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Slide 28
Example
Convert E2Ttwelve to base ten.
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Slide 29
Example
Convert 1277 to base twelve.
144 1277
–1152
12 125
–120
1
5
–5
0
8
T
5
1277ten = 8T5twelve
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Slide 30
Example
What is the value of g in g36twelve = 1050ten?
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Slide 31