Chapter 2: Sets and Whole-Number Operations and Properties
2.4 Numeration
2.4.1. Numeration Systems
2.4.1.1. 2 is a symbol, it is NOT a number – numbers are abstract ideas
2.4.1.2. Symbol representing a number is numeral – sometimes called the name of the number
2.4.1.3. Using numerals to communicate all of the numbers requires some systematic method
2.4.1.4. Definition of a numeration system: A numeration system is an accepted collection of
properties and symbols that enables people to systematically write numerals to represent
numbers.
2.4.2. The Hindu-Arabic Place-Value Numeration System
2.4.2.1. Developed by ancient Indian and Arabic cultures
2.4.2.2. In wide use today
2.4.2.3. Most familiar example of a numeration system
2.4.2.4. Grouping by tens and place value are the cornerstones of this system
2.4.2.5. Looking at other systems helps us to better understand our own base ten system
2.4.2.6. See examples of grouping by 3s and 4s on p. 99
2.4.2.7. The group size used determines the base of the numeration system
2.4.2.7.1. Base 2 results in groups of 2
2.4.2.7.2. Base 5 results in groups of 5
2.4.2.8. Base 10 sometimes called the base-ten place-value numeration system
2.4.2.9. Expressing numerals with different bases
2.4.2.9.1. 102 base 5 is written 1025
2.4.2.9.2. Do the Zorandria and Dairy Worksheet in your groups
2.4.2.10. Your turn p. 101: Do the practice and the reflect
2.4.2.11. Models of base-ten place value
2.4.2.11.1. Proportional models for place value actually exhibit the proportional differences
in the values of the digits in the numerals, i.e. base-ten blocks
2.4.2.11.2. Nonproportional models of place value actual quantities are not visible, i.e.
place value is represented by some object or set of objects – colored chips
2.4.2.12. Using expanded notation
2.4.2.12.1. A numeral written to show the sum of its digits times the value of each place
2.4.2.12.2. 1025 = 1 x 52 + 0 x 51 + 2 x 50 = 1 x 25 + 0 x 5 + 2 x 1 = 25 + 0 + 2 = 2710
2.4.2.12.3. 24135 = 2 x 53 + 4 x 52 + 1 x 51 + 3 x 50 = ?
2.4.2.12.4. 12021023 = ?
2.4.2.12.5. If you wrote a number in base 8, what digits could you use?
2.4.2.12.6. How would you convert a number in base ten to another base?
2.4.2.12.6.1.
What is 25710 in base 3?
2.4.2.12.6.2.
Why are we doing this?
2.4.2.13. Your turn p. 103: Do the practice and the reflect
2.4.2.14. Using periods to represent large numbers
2.4.2.14.1. Successive groups of three digits – periods
2.4.2.14.2. See fig. 2.34 p. 104
2.4.2.15. Using place value to compare numbers
2.4.2.15.1. Use the comparative method of subtraction
2.4.2.15.2. Look at example 2.17
2.4.2.15.3. What do you tell the student that says 6 is more than 0, 5 is more than 0, 7 is
less than 8, and 4 is the same as 4, then since 4756 has two numbers greater than
4800, 4756 > 4800?
2.4.2.16. Your turn p. 105: Do the practice and the reflect
2.4.3. Other Early Numeration Systems
2.4.3.1. Token systems and tally systems
2.4.3.1.1. One of the earliest known systems was a token system – symbolic objects
represent quantities of actual objects
2.4.3.1.2. Related to the barter system
2.4.3.1.3. Tally system is based on a one-to-one correspondence between a single mark and
a single object – the marks represent the number of objects
2.4.3.1.4.
2.4.3.1.5. Tally system is still used as a counting aid
2.4.3.2. Egyptian numeration system
2.4.3.2.1. Similar to tally system, but more complicated developed 3400 B.C.
2.4.3.2.2. Used picture symbols called hieroglyphics
2.4.3.2.3. See figure 2.36 p. 106
2.4.3.2.4. See example 2.18 p. 107
2.4.3.3. Your turn p. 107: Do the practice and the reflect
2.4.3.4. Babylonian numeration system
2.4.3.4.1. based on multiples of 60
2.4.3.4.2. utilized only two symbols
2.4.3.4.3. each symbol had multiple meanings
2.4.3.4.4. meaning determined by context or the position in which it appeared
2.4.3.4.5. see fig. 2.37 and 2.38 p. 108-109
2.4.3.5. Your turn p. 109: Do the practice and the reflect
2.4.3.6. Roman numeration system
2.4.3.6.1. developed between 500 B.C. and 100 A.D.
2.4.3.6.2. still used today – super bowl games; year a movie was made; some clocks; etc.
2.4.3.6.3. seven basic symbols
Symbol Whole Number
1
I
5
V
10
X
50
L
100
C
500
D
1000
M
2.4.3.6.3.1.
combined and repeated as necessary to form a number
2.4.3.6.3.2.
NO more than 3 of any symbol are used in a numeral
2.4.3.6.3.3.
A bar over a letter represents multiples of 1000
2.4.3.6.3.4.
A symbol representing a smaller number placed in front of number
representing a larger number reduces the number by that amount, i.e. IV = 4; IX =
9; XL = 40; etc.
2.4.3.7. Your turn p. 110: Do the practice and the reflect
2.4.4. Comparing Numeration Systems
2.4.4.1. Grouping schemes and symbols
2.4.4.1.1. Egyptian by 10s and new symbol for ones, tens, ten tens, etc.
2.4.4.1.2. Egyptian system requires an infinite number of symbols
2.4.4.1.3. Babylonian system only 2 symbols, grouped by 60s
2.4.4.1.4. Roman system modified scheme of grouping by fives and combination of basic
symbols
2.4.4.1.5. Roman system would also require infinite number of symbols
2.4.4.1.6. Hindu-Arabic system uses ten symbols grouped by tens
2.4.4.2. Use of place value
2.4.4.2.1. position of a symbol in a numeral determines its value
2.4.4.2.2. Babylonians first to use place value – but meaning fuzzy, subject to interpretation
2.4.4.2.3. Egyptian system did not use place value
2.4.4.2.4. Roman system uses symbols as tally – placement is important, but the value of I
does not change regardless of its placement
2.4.4.2.5. Hindu-Arabic system makes most efficient use of place value
2.4.4.3. Use of zero
2.4.4.3.1. Babylonian and Hindu-Arabic systems only ones that use zero
2.4.4.4. See table 2.2 p. 112
2.4.5. Problems and Exercises p. 112
2.4.5.1. Home work: 1, 2, 3ab, 4ab, 5, 6ac, 7, 17, 18 (omit verify with colored chips)