1150 day 3
TRANSCRIPT
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Numbers and Number Systems
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Numbers (as we know them)
How many smileys are there?
Thirteen
13One group of ten Three singles
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Hindu-Arabic System (what we use)•Numerals constructed from ten digits
0, 1, 2, 3, 4, 5, 6, 7, 8, 9•Place value based on powers of ten (base ten)
____ ____ ____ ____2 8 6 7
Expanded form2867 = 2 · 1000 + 8 · 100 + 6 · 10 + 7 · 1OR 2 · 103 + 8 · 102 + 6 · 101 + 7 · 100
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Egyptian Numeration
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Egyptian System•Modified tally system•No way to represent zero
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1324 =
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Egyptian Numbers from the Temple of Edfu
1,333,331
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Babylonian Numeration
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Babylonian System•Place value system based on groups of 60•No way to represent zero
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Cuneiform tablet from approximately 1800 B.C.
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______ ______ ______
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= 3 · 10 + 2 = 32
Ones (units)Sixties
= 12 · 60 + 23= 720 + 23= 743
160602
Or 3600
= 1 · 3600 + 11 · 60 + 20= 3600 + 660 + 20= 4280
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42 =
______ ______ ______
160602
Or 3600
203 = 23
180
3
20360
3 · 60 + 23
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______ ______ ______
160602
Or 3600
4282 =
22
60
82
600
11
68260
1 · 3600
682
3600
4282
+ 11 · 60 + 22
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Mayan Numeration
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Mayan System•Place value system•Based on groups of 20 (with one exception)•Places are arranged vertically•Did have a symbol for zero
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1
20
20 · 18 or 360
20 · 20 · 18 or 7200
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15
0 · 20
2 · 360= 14 720
+ 0
+ 15
735
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5
7 · 20
0 · 360 + 0
+ 140
+ 5
7345
1 · 7200 7200
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16 ·
536 =
1
20
360
7200
1 · 176
360
536
8 ·
16
160
176
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Roman Numeration
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Roman System•Additive system•Subtractive property – if a smaller symbol is to the left of a larger symbol, it is subtracted from the larger•No more than three of the same symbol in a row
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DCCCXLIV
500 + 300 + 40 + 4= 844
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Section 52 at the Coliseum in Rome
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Natural (Counting) Numbers:{1, 2, 3, 4, …}
Whole Numbers:{0, 1, 2, 3, …}
The Natural Numbers are a subset of the Whole Numbers.
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Whole Number addition
2 + 3 = 5
addends sum
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Models for whole-number addition
2 + 3 = 5
Set Model
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Models for whole-number addition
2 + 3 = 5
Number Line Model
2 3
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Properties of whole-number addition
Commutative Property of Additiona + b = b + a2 + 1 = 1 + 2
Associative Property of Addition(a + b) + c = a + (b + c)(4 + 3) + 7 = 4 + (3 + 7)
Identity Property of Additiona + 0 = a7 + 0 = 7 0 is the additive identity
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Whole Number subtraction
5 – 2 = 3
minuend differencesubtrahend
Subtraction is related to addition:
5 – 2 = 3 if 5 = 2 + 3
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Models for whole-number subtraction
5 – 2 = 3
Take-away Model Start with 5 starsTake away 23 stars left
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Models for whole-number subtraction
5 – 2 = 3
Comparison Model Start with 5 starsCompare with 2 circles3 stars left
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Models for whole-number subtraction
5 – 2 = 3
Missing Addend Model
5 – 2 = ?5 = 2 + ?? = 3