1. 3 Chapter Numeration Systems and Whole Number Operations
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

2. 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 Hindu-Arabic system.
• Place value and counting in base ten and other bases.
• Issues in learning with different numeration systems.

3. Definition
Numerals: written symbols to represent cardinal numbers.

4. Definition
Numeration system: a collection of properties and symbols agreed upon to represent numbers systematically.

5. Hindu-Arabic Numeration System
1. All numerals are constructed from the 10 digits.
2. Place value is based on powers of 10.

6. Place value assigns a value to a digit depending on its placement in a numeral.

7. Expanded form
Factor
If a is any number and n is any natural number, then
n factors

8. 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

9. 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 17 longs 16 units (16 units = 1 long
1 long 6 units and 6 units)
11 flats 18 longs 6 units (after the first trade)

10. Example (continued) 11 flats 18 longs 6 units (18 longs = 1 flat)
1 flat 8 longs and 8 longs)
12 flats 8 longs 6 units (after the second trade)

11. Example(continued) 12 flats 8 longs 6 units 1 block 2 flats
1 block 2 flats 8 longs 6 units
The fewest number of pieces = = 17.
(12 flats = 1 block and 2 flats)
This is analogous to rewriting as

12. Tally Numeration System
Uses single strokes (tally marks) to represent each object that is counted.
= 13

13. Egyptian Numeration System

14. Example
Use the Egyptian numeration system to represent 2,345,123.

15. Babylonian Numeration System

16. Example Use the Babylonian numeration system to represent 305,470.
1  603
216,000
24  602
86,400
51  60
3060
10  1 = 10 +
= 305,470

17. Mayan Numeration System

18. Example
Use the Mayan numeration system to represent 305,470.

19. Roman Numeration System

20. Roman Numeration System

21. In the Middle Ages, a bar was placed over a Roman number to multiply it by 1000.

22. Example Use the Roman numeration system to represent 15,478.
XVCDLXXVIII

23. Other Number Base Systems
Quinary (base-five) system

24. Example
Convert 11244five to base 10.

25. 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.

26. Example
Convert 10111two to base ten.

27. Example Convert 27 to base two. 16 27 1 –16 8 11 1 –8 4 3 0 –0 or
–16
8 11 1 –8 –0 –2 –1 or

28. 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.

29. Example
Convert E2Ttwelve to base ten.

30. Example
Convert 1277 to base twelve – T – –5 0
1277ten = 8T5twelve

31. Example
What is the value of g in g36twelve = 1050ten?