Section 2.4
Numeration
Mathematics for Elementary School Teachers - 4th Edition
O’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK
A symbol is different from what it represents
The word symbol for cat is different than the actual
cat
Numeration Systems
Just as the written symbol 2 is not itself a
number.
The written symbol,
2, that represents a
number is called a
numeral.
Here is another familiar
numeral (or name) for
the number two
Definition of Numeration System
An accepted collection of properties and
symbols that enables people to systematically
write numerals to represent numbers. (p. 106,
text)
Egyptian Numeration System
Babylonian Numeration System
Roman Numeration System
Mayan Numeration System
Hindu-Arabic Numeration System
Hindu-Arabic Numeration System
• Developed by Indian and Arabic cultures
• It is our most familiar example of a numeration system
• Group by tens: base ten system
•10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Place value - Yes! The value of the digit is
determined by its position in a numeral
•Uses a zero in its numeration system
Definition of Place Value
In a numeration system with place value, the
position of a symbol in a numeral determines that
symbol’s value in that particular numeral. For
example, in the Hindu-Arabic numeral 220, the
first 2 represents two hundred and the second 2
represents twenty.
Models of Base-Ten Place Value
Base-Ten Blocks - proportional model for place value
Thousands cube, Hundreds square, Tens stick,
Ones cube
or
block, flat, long, unit
text, p. 110
2,345
Models of Base-Ten Place Value
Colored-chip model:
nonproportional model for place
value
One Hundred
One Thousand
Ten
One
3,462
chips from text, p. 110
Expressing Numerals with Different
Bases:
Show why the quantity of tiles shown can be
expressed as (a) 27 in base ten and (b)102 in
base five, written 102five
we can group
(a) form groups of 10
these tiles into two
27
groups of ten with
7 tiles left over
(b) form groups of 5
102five
we can group these tiles
into groups of 5 and
have enough of these
groups of 5 to make one
larger group of 5 fives,
with 2 tiles left over.
No group of 5 is left over, so we need to
use a 0 in that position in the numeral:
Expressing Numerals with Different
Bases:
Find the base-ten representation for 1324five
1324five = (1×53) + (3×52) + (2×51) + (4×50)
= 1(125) + 3(25) + 2(5) + 4(1)
= 125 + 75 + 10 + 4
= 214ten
Find the base-ten representation for 344six
Find the base-ten representation for 110011two
Expressing Numerals with Different
Bases:
Find the representation of the number 256 in base six
256
- 216
40
-36
4
1(63) + 1(62) + 0(61) + 4(60)
1(216) + 1(36) + 0(6) + 4(1) = 1104six
0
6
=1
1
6 =6
62 = 36
3
6 = 216
64 = 1296
Expressing Numerals with Different
Bases:
Change 42seven to base five
First change to base 10
42seven =
1
4(7 )
+
0
2(7 )
= 30ten
Then change to base five
50 = 1
51 = 5
52 = 25
3
5 = 125
30
- 25
5
-5
0
30ten = 1(52) + 1(51) + 0(50) = 110five
Expanded Notation:
This is a way of writing numbers to show place
value, by multiplying each digit in the numeral
by its matching place value.
Example (using base
10):
1324 = (1×103) + (3×102) + (2×101) + (4×100)
or
1324 = (1×1000) + (3×100) + (2×10) + (4×1)
Egyptian Numeration System
Developed: 3400 B.C.E
reed
Group by tens
New symbols would
be needed as
system grows
No place value
heel bone
One
Ten
coiled rope
One Hundred
lotus flower
One Thousand
bent finger
Ten Thousand
burbot fish
One Hundred Thousand
No use of zero
kneeling figure
or
astonished man
One Million
Babylonian Numeration System
Developed between 3000 and 2000 B.C.E
There are two symbols in the Babylonian Numeration System
Base 60
Place value
one
ten
Zero came later
Write the Hindu-Arabic numerals for the numbers represented
by the following numerals from the Babylonian system:
1
42(60 )
+
0
34(60 )
= 2520 + 34 = 2,554
Roman Numeration System
Developed between 500 B.C.E and 100 C.E.
•Group partially by
fives
•Would need to add new symbols
•Position indicates
when to add or subtract
•No use of zero
Write the Hindu-Arabic
numerals for the numbers
represented by the Roman
Numerals:
ⅭⅯⅩⅭⅼⅩ
900 + 90 + 9 = 999
ⅼ (one)
Ⅴ (five)
Ⅹ (ten)
Ⅼ (fifty)
Ⅽ (one hundred)
Ⅾ (five hundred)
Ⅿ(one thousand)
Mayan Numeration System
Developed between 300 C.E and 900 C.E
•Base - mostly by 20
•Number of symbols: 3
•Place value - vertical
•Use of Zero
Symbols
=0
=1
=5
Write the Hindu-Arabic numerals for the numbers
represented by the following numerals from the Mayan
system:
8(20 ×18) = 2880
6(201) = 120
0(200) = 0
2880 + 120 + 0 = 3000
Summary of Numeration System Characteristics
System Grouping Symbols
Place
Value
Use of
Zero
No
No
Egyptian
By tens
Infinitely
many
possibly
needed
Babylonia
n
By sixties
Two
Yes
Not at first
Roman
Partially
by fives
Infinitely
many
possibly
needed
Position
indicates when
to add or
subtract
No
Mayan
Mostly
by twenties
Three
Yes,
Vertically
Yes
By tens
Ten
Yes
Yes
Hindu-
The End
Section 2.4
Linda Roper