Section 2.4
Numeration
Mathematics for Elementary School Teachers - 4th EditionO’DAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK
The word symbol for cat is different than the actual cat
A symbol is different from what it represents
Here is another familiar numeral (or
name) for the number two
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.
Definition of Numeration SystemAn accepted collection of properties and
symbols that enables people to systematically write numerals to represent numbers. (p. 106, text)
Hindu-Arabic Numeration System
Egyptian Numeration System
Babylonian Numeration System
Roman Numeration System
Mayan 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 ValueIn 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
orblock, flat, long, unit
text, p. 110
2,345
Models of Base-Ten Place ValueColored-chip model:nonproportional model for place value
chips from text, p. 110
OneTenOne HundredOne Thousand
3,462
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
(a) form groups of 10we can group these tiles into two groups of ten with 7 tiles
left over(b) form groups of 5 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.
27
No group of 5 is left over, so we need to use a 0 in that position in
the numeral: 102five
102five
Find the base-ten representation for 1324five
Find the base-ten representation for 344six
Find the base-ten representation for 110011two
= 1(125) + 3(25) + 2(5) + 4(1)
1324five = (1×53) + (3×52) + (2×51) + (4×50)
= 125 + 75 + 10 + 4= 214ten
Expressing Numerals with Different Bases:
Find the representation of the number 256 in base six
64 = 129663 = 21662 = 36
60 = 161 = 6
256- 216
40-36
4
1(216) + 1(36) + 0(6) + 4(1) = 1104six
1(63) + 1(62) + 0(61) + 4(60)
Expressing Numerals with Different Bases:
Change 42seven to base fiveFirst change to base 10
42seven = 4(71) + 2(70) = 30ten
Then change to base five
53 = 12552 = 25
50 = 151 = 5
30- 25
5- 5
0
30ten = 1(52) + 1(51) + 0(50) = 110five
Expressing Numerals with Different Bases:
Expanded Notation:
1324 = (1×1000) + (3×100) + (2×10) + (4×1)
1324 = (1×103) + (3×102) + (2×101) + (4×100)
Example (using base 10):
or
This is a way of writing numbers to show place value, by multiplying each digit in the numeral by its matching place value.
Egyptian Numeration SystemDeveloped: 3400 B.C.E
One
Ten
One Hundred
One Thousand
Ten Thousand
One Hundred Thousand
One Million
reed
heel bone
coiled rope
lotus flower
bent finger
burbot fish
kneeling figureor
astonished man
Group by tens
New symbols would be needed as system grows
No place value
No use of zero
Babylonian Numeration SystemDeveloped between 3000 and 2000 B.C.E
There are two symbols in the Babylonian Numeration System
Base 60Place value one ten
42(601) + 34(600) = 2520 + 34 = 2,554
Zero came later
Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the
Babylonian system:
Roman Numeration SystemDeveloped between 500 B.C.E and 100 C.E.
ⅬⅭⅮⅯ
Ⅹ
ⅼⅤ
(one)
(five)
(ten)
(fifty)
(one hundred)
(five hundred)(one thousand)
•Group partially by fives•Would need to add new symbols
•Position indicates when to add or
subtract•No use of zero
Ⅽ Ⅿ Ⅹ Ⅽ ⅼ Ⅹ
900 + 90 + 9 = 999
Write the Hindu-Arabic numerals for the numbers represented by the Roman
Numerals:
Mayan Numeration SystemDeveloped between 300 C.E and 900 C.E
•Base - mostly by 20•Number of symbols: 3•Place value - vertical•Use of Zero
Symbols
= 1= 5
= 0
Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the
Mayan system:
0(200) = 0
6(201) = 120
8(20 ×18) = 2880
2880 + 120 + 0 = 3000
Summary of Numeration System Characteristics
SystemSystem GroupinGroupingg
SymbolsSymbols Place Place ValueValue
Use of Use of ZeroZero
EgyptiaEgyptiann
By tens
Infinitely many
possibly needed
No No
BabyloniBabylonianan
By sixties Two Yes Not at first
RomanRoman Partiallyby fives
Infinitely many
possibly needed
Position indicates
when to add or subtract
No
MayanMayan Mostlyby twenties
ThreeYes,
VerticallyYes
Hindu-Hindu-ArabicArabic
By tens Ten Yes Yes
The EndSection 2.4
Linda Roper