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Numbers and Number Systems
Numbers (as we know them)
How many smileys are there?
Thirteen
13One group of ten Three singles
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
Egyptian Numeration
Egyptian System•Modified tally system•No way to represent zero
Page 166
1324 =
Egyptian Numbers from the Temple of Edfu
1,333,331
Babylonian Numeration
Babylonian System•Place value system based on groups of 60•No way to represent zero
Page 67
Cuneiform tablet from approximately 1800 B.C.
______ ______ ______
= 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
42 =
______ ______ ______
160602
Or 3600
203 = 23
180
3
20360
3 · 60 + 23
______ ______ ______
160602
Or 3600
4282 =
22
60
82
600
11
68260
1 · 3600
682
3600
4282
+ 11 · 60 + 22
Mayan Numeration
Mayan System•Place value system•Based on groups of 20 (with one exception)•Places are arranged vertically•Did have a symbol for zero
Page 68
1
20
20 · 18 or 360
20 · 20 · 18 or 7200
15
0 · 20
2 · 360= 14 720
+ 0
+ 15
735
5
7 · 20
0 · 360 + 0
+ 140
+ 5
7345
1 · 7200 7200
16 ·
536 =
1
20
360
7200
1 · 176
360
536
8 ·
16
160
176
Roman Numeration
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
Page 69
DCCCXLIV
500 + 300 + 40 + 4= 844
Section 52 at the Coliseum in Rome
Natural (Counting) Numbers:{1, 2, 3, 4, …}
Whole Numbers:{0, 1, 2, 3, …}
The Natural Numbers are a subset of the Whole Numbers.
Whole Number addition
2 + 3 = 5
addends sum
Models for whole-number addition
2 + 3 = 5
Set Model
Models for whole-number addition
2 + 3 = 5
Number Line Model
2 3
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
Whole Number subtraction
5 – 2 = 3
minuend differencesubtrahend
Subtraction is related to addition:
5 – 2 = 3 if 5 = 2 + 3
Models for whole-number subtraction
5 – 2 = 3
Take-away Model Start with 5 starsTake away 23 stars left
Models for whole-number subtraction
5 – 2 = 3
Comparison Model Start with 5 starsCompare with 2 circles3 stars left
Models for whole-number subtraction
5 – 2 = 3
Missing Addend Model
5 – 2 = ?5 = 2 + ?? = 3