1. Numbers
and
Number
Systems

2. Numbers (as we know them)
How many smileys are
there?
Thirteen
13
One group of ten Three singles

3. 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 form
2867 = 2 · 1000 + 8 · 100 + 6 · 10 + 7 · 1
OR 2 · 103 + 8 · 102 + 6 · 101 + 7 · 100

6. Egyptian
Numeration

7. Egyptian System
•Modified tally system
•No way to represent zero
Page 166
1324 =

8. 1,333,331
Egyptian Numbers from the Temple of Edfu

9. Babylonian
Numeration

10. Babylonian System
•Place value system based on groups of 60
•No way to represent zero
Page 67

11. Cuneiform tablet from approximately 1800 B.C.

12. ______ ______ ______

13. = 3 · 10 + 2 = 32
= 12 · 60 + 23
Sixties Ones (units) = 720 + 23
= 743
= 1 · 3600 + 11 · 60 + 20
602 = 3600 + 660 + 20
60 1
Or 3600 = 4280

14. ______ ______ ______
602
60 1
Or 3600 3
60 203
42 =
180
203 = 23
3 · 60 + 23

15. 4282
3600
______ ______ ______ 682
602
60 1
Or 3600 11
60 682
4282 = 600
1 · 3600 + 11 · 60 + 22 82
60
22

16. Mayan
Numeration

17. 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

18. 20 · 20 · 18 or 7200
20 · 18 or 360
20
1

19. = 14 2 · 360 720
0 · 20 +0
15 + 15
735

20. 1 · 7200 7200
0 · 360 +0
7 · 20 + 140
5 +5
7345

21. 536 =
7200 536 176
360 160
1 · 360 176 16
8 · 20
16 · 1

23. Roman
Numeration

24. 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

25. DCCCXLIV
500 + 300 + 40 + 4
= 844

26. Section 52 at the Coliseum in Rome

28. Natural (Counting) Numbers:
{1, 2, 3, 4, …}
Whole Numbers:
{0, 1, 2, 3, …}
The Natural Numbers are a subset of the
Whole Numbers.

29. Whole Number addition
2+3=5
addends sum

30. Models for whole-number addition
2+3=5
Set Model

31. Models for whole-number addition
2+3=5
Number Line Model
2 3

32. Properties of whole-number addition
Commutative Property of Addition
a+b=b+a
2+1=1+2
Associative Property of Addition
(a + b) + c = a + (b + c)
(4 + 3) + 7 = 4 + (3 + 7)
Identity Property of Addition
a+0=a
7+0=7 0 is the additive identity

33. Whole Number subtraction
5–2=3
minuend subtrahend difference
Subtraction is related to addition:
5 – 2 = 3 if 5 = 2 + 3

34. Models for whole-number subtraction
5–2=3
Start with 5 stars
Take-away Model Take away 2
3 stars left

35. Models for whole-number subtraction
5–2=3
Start with 5 stars
Comparison Model Compare with 2 circles
3 stars left

36. Models for whole-number subtraction
5–2=3
Missing Addend Model
5–2=?
5=2+?
?=3