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