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
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
2.4.2.10.
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. system called sexagesimal
2.4.3.4.6. see fig. 2.37 and 2.38 p. 108-109
2.4.3.5.
Mayan
numeration system
2.4.3.5.1 based on multiples of 20
2.4.3.5.2 utilized two symbols and a zero symbol
2.4.3.5.3 dot symbol value of 1, bar symbol value of 5, zero
symbol like a boat
2.4.3.5.4 system numerals written vertically vs. horizontally
2.4.3.5.5 system called vigesimal
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
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