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