Download § 2-1 Numeration Systems

§ 2-1 Numeration Systems
How We Use Numbers
cardinal numbers
How We Use Numbers
cardinal numbers
how many numbers are in a set
How We Use Numbers
cardinal numbers
how many numbers are in a set
ordinal numbers
How We Use Numbers
cardinal numbers
how many numbers are in a set
ordinal numbers
position
which comes first
How We Use Numbers
cardinal numbers
how many numbers are in a set
ordinal numbers
position
which comes first
identification numbers
How We Use Numbers
cardinal numbers
how many numbers are in a set
ordinal numbers
position
which comes first
identification numbers
How We Use Numbers
cardinal numbers
how many numbers are in a set
ordinal numbers
position
which comes first
identification numbers
Numbers are abstract ideas for children to understand, so it is
important to find symbolic ways to represent them to help them grasp
the concept.
Our Number System
What is the base of our number system?
Our Number System
What is the base of our number system?
Why do we have a base 10 number system?
Our Number System
What is the base of our number system?
Why do we have a base 10 number system?
It is useful to talk of other civilizations’ systems when dealing with
children to help them understand the ideas that lead to the
development of these systems.
The Tally System
The Tally System
Properties
most simplistic
The Tally System
Properties
most simplistic
a line represents 1, crossing represents 5
The Tally System
Properties
most simplistic
a line represents 1, crossing represents 5
grouping is to make counting easier
The Tally System
Properties
most simplistic
a line represents 1, crossing represents 5
grouping is to make counting easier
very tedious
The Tally System
Properties
most simplistic
a line represents 1, crossing represents 5
grouping is to make counting easier
very tedious
additive system
The Egyptian Numeration System
The Egyptian Numeration System
The Egyptian Numeration System
Properties
base 10 system
The Egyptian Numeration System
Properties
base 10 system
additive system
The Egyptian Numeration System
Properties
base 10 system
additive system
multiple symbols is an improvement
The Egyptian Numeration System
Properties
base 10 system
additive system
multiple symbols is an improvement
order does not matter
Examples Using Egyptian Hieroglyphics
Example
What decimal number is represented here?
Examples Using Egyptian Hieroglyphics
Example
What decimal number is represented here?
2008
Examples Using Egyptian Hieroglyphics
Example
What decimal number is represented here?
Examples Using Egyptian Hieroglyphics
Example
What decimal number is represented here?
2,320,111
Examples Using Egyptian Hieroglyphics
Example
Write the number 245 using Egyptian hieroglyphics.
Examples Using Egyptian Hieroglyphics
Example
Write the number 245 using Egyptian hieroglyphics.
The Roman Numeration System
Decimal
1
Roman
The Roman Numeration System
Decimal
1
5
Roman
I
The Roman Numeration System
Decimal
1
5
10
Roman
I
V
The Roman Numeration System
Decimal
1
5
10
50
Roman
I
V
X
The Roman Numeration System
Decimal
1
5
10
50
100
Roman
I
V
X
L
The Roman Numeration System
Decimal
1
5
10
50
100
500
Roman
I
V
X
L
C
The Roman Numeration System
Decimal
1
5
10
50
100
500
1000
Roman
I
V
X
L
C
D
The Roman Numeration System
Decimal
1
5
10
50
100
500
1000
Roman
I
V
X
L
C
D
M
The Roman Numeration System
Decimal
1
5
10
50
100
500
1000
Roman
I
V
X
L
C
D
M
Where do we see Roman numerals in modern day?
The Roman Numeration System
Properties
not a base 10 system
The Roman Numeration System
Properties
not a base 10 system
additive system
The Roman Numeration System
Properties
not a base 10 system
additive system
subtractive system
The Roman Numeration System
Properties
not a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000
times that number
The Roman Numeration System
Properties
not a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000
times that number
M = 1, 000, 000
The Roman Numeration System
Properties
not a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000
times that number
M = 1, 000, 000
positional system - where we place the symbols changes the
value
The Roman Numeration System
Properties
not a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000
times that number
M = 1, 000, 000
positional system - where we place the symbols changes the
value
VI=6
IV=4
More Roman Numerals
Example
Write the following using Roman numerals.
1
2
More Roman Numerals
Example
Write the following using Roman numerals.
1
2
2
4
II
More Roman Numerals
Example
Write the following using Roman numerals.
1
2
II
2
4
IV
3
9
More Roman Numerals
Example
Write the following using Roman numerals.
1
2
II
2
4
IV
3
9
IX
4
40
More Roman Numerals
Example
Write the following using Roman numerals.
1
2
II
2
4
IV
3
9
IX
4
40
5
49
XL
More Roman Numerals
Example
Write the following using Roman numerals.
1
2
II
2
4
IV
3
9
IX
4
40
XL
5
49
XLIX
6
99
More Roman Numerals
Example
Write the following using Roman numerals.
1
2
II
2
4
IV
3
9
IX
4
40
XL
5
49
XLIX
6
99
XCIX
7
1489
More Roman Numerals
Example
Write the following using Roman numerals.
1
2
II
2
4
IV
3
9
IX
4
40
XL
5
49
XLIX
6
99
XCIX
7
1489
MCDLXXXIX
More Roman Numerals
Example
Write the following as decimal numbers.
1
CDXXIII
More Roman Numerals
Example
Write the following as decimal numbers.
1
CDXXIII
423
More Roman Numerals
Example
Write the following as decimal numbers.
1
CDXXIII
2
DCXIV
423
More Roman Numerals
Example
Write the following as decimal numbers.
1
CDXXIII
2
DCXIV
423
614
More Roman Numerals
Example
Write the following as decimal numbers.
1
CDXXIII
2
DCXIV
3
CCII
423
614
More Roman Numerals
Example
Write the following as decimal numbers.
1
CDXXIII
2
DCXIV
3
CCII
423
614
200,002
The Babylonian Numeration System
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like the
following:
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like the
following:
Properties
base 60 system
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like the
following:
Properties
base 60 system
additive system
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like the
following:
Properties
base 60 system
additive system
multiplicative
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like the
following:
Properties
base 60 system
additive system
multiplicative
place valued system
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like the
following:
Properties
base 60 system
additive system
multiplicative
place valued system
positional
The Babylonian Numeration System
Base 60
603
602
601
600
The Babylonian Numeration System
Base 60
603
602
Example
Write 53 using Babylonian cuneiform.
601
600
The Babylonian Numeration System
Base 60
603
602
Example
Write 53 using Babylonian cuneiform.
601
600
The Babylonian Numeration System
Example
What decimal number is represented here?
The Babylonian Numeration System
Example
What decimal number is represented here?
Example
What decimal number is represented here?
The Babylonian Numeration System
Example
What decimal number is represented here?
The Babylonian Numeration System
Example
What decimal number is represented here?
The Babylonians realized they needed a place holder to represent that
they had a place with nothing in it when writing their numbers.
The Babylonian Numeration System
Example
Write 3821 using Babylonian cuneiform.
The Babylonian Numeration System
Example
Write 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is the
largest power of 60 that goes into 3821?
The Babylonian Numeration System
Example
Write 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is the
largest power of 60 that goes into 3821?
602 ( ) + 601 ( ) + 600 ( )
The Babylonian Numeration System
Example
Write 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is the
largest power of 60 that goes into 3821?
602 ( ) + 601 ( ) + 600 ( )
602 (1) + 601 ( ) + 600 ( )
The Babylonian Numeration System
Example
Write 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is the
largest power of 60 that goes into 3821?
602 ( ) + 601 ( ) + 600 ( )
602 (1) + 601 ( ) + 600 ( )
602 (1) + 601 (3) + 600 ( )
The Babylonian Numeration System
Example
Write 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is the
largest power of 60 that goes into 3821?
602 ( ) + 601 ( ) + 600 ( )
602 (1) + 601 ( ) + 600 ( )
602 (1) + 601 (3) + 600 ( )
602 (1) + 601 (3) + 600 (41)
The Babylonian Numeration System
Example
Write 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is the
largest power of 60 that goes into 3821?
602 ( ) + 601 ( ) + 600 ( )
602 (1) + 601 ( ) + 600 ( )
602 (1) + 601 (3) + 600 ( )
602 (1) + 601 (3) + 600 (41)
The Babylonian Numeration System
Example
Write 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is the
largest power of 60 that goes into 3821?
602 ( ) + 601 ( ) + 600 ( )
602 (1) + 601 ( ) + 600 ( )
602 (1) + 601 (3) + 600 ( )
602 (1) + 601 (3) + 600 (41)
The Babylonian Numeration System
Example
Write 7205 using Babylonian cuneiform.
The Babylonian Numeration System
Example
Write 7205 using Babylonian cuneiform.
What is the largest power of 60 that goes into 7205?
7205 = 602 ( ) + 601 ( ) + 600 ( )
The Babylonian Numeration System
Example
Write 7205 using Babylonian cuneiform.
What is the largest power of 60 that goes into 7205?
7205 = 602 ( ) + 601 ( ) + 600 ( )
= 602 (2) + 601 ( ) + 600 ( )
The Babylonian Numeration System
Example
Write 7205 using Babylonian cuneiform.
What is the largest power of 60 that goes into 7205?
7205 = 602 ( ) + 601 ( ) + 600 ( )
= 602 (2) + 601 ( ) + 600 ( )
= 602 (2) + 601 (0) + 600 ( )
The Babylonian Numeration System
Example
Write 7205 using Babylonian cuneiform.
What is the largest power of 60 that goes into 7205?
7205 = 602 ( ) + 601 ( ) + 600 ( )
= 602 (2) + 601 ( ) + 600 ( )
= 602 (2) + 601 (0) + 600 ( )
= 602 (2) + 601 (0) + 600 (5)
The Babylonian Numeration System
Example
Write 7205 using Babylonian cuneiform.
What is the largest power of 60 that goes into 7205?
7205 = 602 ( ) + 601 ( ) + 600 ( )
= 602 (2) + 601 ( ) + 600 ( )
= 602 (2) + 601 (0) + 600 ( )
= 602 (2) + 601 (0) + 600 (5)
The Babylonian Numeration System
Example
Write 7205 using Babylonian cuneiform.
What is the largest power of 60 that goes into 7205?
7205 = 602 ( ) + 601 ( ) + 600 ( )
= 602 (2) + 601 ( ) + 600 ( )
= 602 (2) + 601 (0) + 600 ( )
= 602 (2) + 601 (0) + 600 (5)
The Mayan Numeration System
The Mayan Numeration System
Properties
1
additive system
The Mayan Numeration System
Properties
1
additive system
2
multiplicative system
The Mayan Numeration System
Properties
1
additive system
2
multiplicative system
3
positional system
The Mayan Numeration System
Properties
1
additive system
2
multiplicative system
3
positional system
4
place valued system
The Mayan Numeration System
Properties
1
additive system
2
multiplicative system
3
positional system
4
place valued system
5
symbol for 0
The Mayan Numeration System
1’s
The Mayan Numeration System
20’s
1’s
The Mayan Numeration System
18 · 20’s
20’s
1’s
The Mayan Numeration System
18 · 202 ’s
18 · 20’s
20’s
1’s
The Mayan Numeration System
Example
Write 7 in Mayan.
The Mayan Numeration System
Example
Write 7 in Mayan.
• •
The Mayan Numeration System
Example
Write 7 in Mayan.
• •
Example
Write 45 in Mayan.
The Mayan Numeration System
Example
Write 7 in Mayan.
• •
Example
Write 45 in Mayan.
• •
The Mayan Numeration System
Example
Write 7 in Mayan.
• •
Example
Write 45 in Mayan.
• •
Spacing is very important ...
The Mayan Numeration System
Example
Write 611 in Mayan
The Mayan Numeration System
Example
Write 611 in Mayan
First, we need to write 611 in expanded form.
The Mayan Numeration System
Example
Write 611 in Mayan
First, we need to write 611 in expanded form.
611 =
The Mayan Numeration System
Example
Write 611 in Mayan
First, we need to write 611 in expanded form.
611 = 360 + 251
The Mayan Numeration System
Example
Write 611 in Mayan
First, we need to write 611 in expanded form.
611 = 360 + 251
= 360 + 240 + 11
The Mayan Numeration System
Example
Write 611 in Mayan
First, we need to write 611 in expanded form.
611 = 360 + 251
= 360 + 240 + 11
= 18 · 20 + 12 · 20 + 11
The Mayan Numeration System
Example
Write 611 in Mayan
First, we need to write 611 in expanded form.
611 = 360 + 251
= 360 + 240 + 11
= 18 · 20 + 12 · 20 + 11
•
• •
•
The Hindu-Arabic Numeration System
Properties
1
base 10 system
The Hindu-Arabic Numeration System
Properties
1
base 10 system
2
positional system
The Hindu-Arabic Numeration System
Properties
1
base 10 system
2
positional system
3
additive system (values for various numerals added)
The Hindu-Arabic Numeration System
Properties
1
base 10 system
2
positional system
3
additive system (values for various numerals added)
4
multiplicative
The Hindu-Arabic Numeration System
Properties
1
base 10 system
2
positional system
3
additive system (values for various numerals added)
4
multiplicative
5
10 symbols called digits
The Hindu-Arabic Numeration System
Properties
1
base 10 system
2
positional system
3
additive system (values for various numerals added)
4
multiplicative
5
10 symbols called digits
We can use manipulatives to help children understand how our system
works. A common manipulative is base 10 blocks.
Other Bases
To see the types of troubles children have with the base 10 system, we
will use numeration systems with different bases.
Other Bases
To see the types of troubles children have with the base 10 system, we
will use numeration systems with different bases.
To do this, we first have to understand what the place values in base
10 really mean. We can see this usign expanded form.
Other Bases
To see the types of troubles children have with the base 10 system, we
will use numeration systems with different bases.
To do this, we first have to understand what the place values in base
10 really mean. We can see this usign expanded form.
Example
Write 4321 in expanded form.
Other Bases
To see the types of troubles children have with the base 10 system, we
will use numeration systems with different bases.
To do this, we first have to understand what the place values in base
10 really mean. We can see this usign expanded form.
Example
Write 4321 in expanded form.
4321 = 4(1000) + 3(100) + 2(10) + 1(1)
Other Bases
To see the types of troubles children have with the base 10 system, we
will use numeration systems with different bases.
To do this, we first have to understand what the place values in base
10 really mean. We can see this usign expanded form.
Example
Write 4321 in expanded form.
4321 = 4(1000) + 3(100) + 2(10) + 1(1)
= 4 · 103 + 3 · 102 + 2 · 101 + 1 · 100
Base 2
Binary is the language of computers and also helps children see place
value quickly.
Base 2
Binary is the language of computers and also helps children see place
value quickly.
Example
Convert 73 to base 2
Base 2
Binary is the language of computers and also helps children see place
value quickly.
Example
Convert 73 to base 2
The place values for binary are
...
27
26
25
24
23
22
21
20
Base 2
Binary is the language of computers and also helps children see place
value quickly.
Example
Convert 73 to base 2
The place values for binary are
...
27
26
25
24
23
22
21
20
We need to determine what is the highest power of 2 that goes into 73.
Base 2
73 =
Base 2
73 = 1(64) + 9
Base 2
73 = 1(64) + 9
= 1(64) + 0(32) + 9
Base 2
73 = 1(64) + 9
= 1(64) + 0(32) + 9
= 1(64) + 0(32) + 0(16) + 9
Base 2
73 = 1(64) + 9
= 1(64) + 0(32) + 9
= 1(64) + 0(32) + 0(16) + 9
= 1(64) + 0(32) + 0(16) + 1(8) + 1
Base 2
73 = 1(64) + 9
= 1(64) + 0(32) + 9
= 1(64) + 0(32) + 0(16) + 9
= 1(64) + 0(32) + 0(16) + 1(8) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 1
Base 2
73 = 1(64) + 9
= 1(64) + 0(32) + 9
= 1(64) + 0(32) + 0(16) + 9
= 1(64) + 0(32) + 0(16) + 1(8) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 0(2) + 1
Base 2
73 = 1(64) + 9
= 1(64) + 0(32) + 9
= 1(64) + 0(32) + 0(16) + 9
= 1(64) + 0(32) + 0(16) + 1(8) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 0(2) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 0(2) + 1(1)
Base 2
73 = 1(64) + 9
= 1(64) + 0(32) + 9
= 1(64) + 0(32) + 0(16) + 9
= 1(64) + 0(32) + 0(16) + 1(8) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 0(2) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 0(2) + 1(1)
So, we have 73 = 1001001TWO .
Base 2
73 = 1(64) + 9
= 1(64) + 0(32) + 9
= 1(64) + 0(32) + 0(16) + 9
= 1(64) + 0(32) + 0(16) + 1(8) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 0(2) + 1
= 1(64) + 0(32) + 0(16) + 1(8) + 0(4) + 0(2) + 1(1)
So, we have 73 = 1001001TWO .
Why do we need to put the subscript ‘TWO’ but not ‘TEN’?
Base 7
If we are in base 7, what are the first 4 place values?
Base 7
If we are in base 7, what are the first 4 place values?
...
73
72
71
70
Base 7
If we are in base 7, what are the first 4 place values?
...
73
343
72
49
71
7
70
1
Base 7
If we are in base 7, what are the first 4 place values?
...
Example
Convert 321 to base 7.
73
343
72
49
71
7
70
1
Base 7
If we are in base 7, what are the first 4 place values?
...
Example
Convert 321 to base 7.
321 =
73
343
72
49
71
7
70
1
Base 7
If we are in base 7, what are the first 4 place values?
...
73
343
72
49
71
7
70
1
Example
Convert 321 to base 7.
321 = 49( ) + 7( ) + 1( )
Base 7
If we are in base 7, what are the first 4 place values?
...
73
343
72
49
71
7
70
1
Example
Convert 321 to base 7.
321 = 49( ) + 7( ) + 1( )
= 49(6) + 7( ) + 1( )
Base 7
If we are in base 7, what are the first 4 place values?
...
73
343
72
49
71
7
70
1
Example
Convert 321 to base 7.
321 = 49( ) + 7( ) + 1( )
= 49(6) + 7( ) + 1( )
= 49(6) + 7(3) + 6
So, 321 = 636SEVEN .
Base 7
Example
Convert 219 to base 7.
Base 7
Example
Convert 219 to base 7.
219 = 432SEVEN
Base 7
Example
Convert 219 to base 7.
219 = 432SEVEN
Example
Convert 693SEVEN to base 10.
Base 7
Example
Convert 219 to base 7.
219 = 432SEVEN
Example
Convert 693SEVEN to base 10.
DNE ... why?
Base 7
Example
Convert 623SEVEN to base 10.
Base 7
Example
Convert 623SEVEN to base 10.
We again have to think about what the place values represent.
Base 7
Example
Convert 623SEVEN to base 10.
We again have to think about what the place values represent.
623SEVEN =
Base 7
Example
Convert 623SEVEN to base 10.
We again have to think about what the place values represent.
623SEVEN = 6(72 ) + 2(7) + 3(1)
Base 7
Example
Convert 623SEVEN to base 10.
We again have to think about what the place values represent.
623SEVEN = 6(72 ) + 2(7) + 3(1)
= 294 + 14 + 3
Base 7
Example
Convert 623SEVEN to base 10.
We again have to think about what the place values represent.
623SEVEN = 6(72 ) + 2(7) + 3(1)
= 294 + 14 + 3
= 311
Hexidecimal (Base 16)
How can we deal with number systems where we may need to exceed
9 in any one position?
Hexidecimal (Base 16)
How can we deal with number systems where we may need to exceed
9 in any one position?
10
A
11
B
12
C
13
D
14
E
15
F
Hexidecimal
Example
Convert 352 to hexidecimal.
Hexidecimal
Example
Convert 352 to hexidecimal.
How do we begin?
Hexidecimal
Example
Convert 352 to hexidecimal.
How do we begin?
...
162
161
160
Hexidecimal
Example
Convert 352 to hexidecimal.
How do we begin?
...
162
256
161
16
160
1
Hexidecimal
Example
Convert 352 to hexidecimal.
How do we begin?
...
162
256
161
16
160
1
352 = 256( ) + 16( ) + 1( )
Hexidecimal
Example
Convert 352 to hexidecimal.
How do we begin?
...
162
256
161
16
160
1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
Hexidecimal
Example
Convert 352 to hexidecimal.
How do we begin?
...
162
256
161
16
160
1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
= 256(1) + 16(6) + 0
Hexidecimal
Example
Convert 352 to hexidecimal.
How do we begin?
...
162
256
161
16
160
1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
= 256(1) + 16(6) + 0
So, 352 = 160HEX .
Hexidecimal
Example
Convert 792 to hexidecimal.
Hexidecimal
Example
Convert 792 to hexidecimal.
792 = 318HEX
Hexidecimal
Example
Convert ABCHEX to a decimal number.
Hexidecimal
Example
Convert ABCHEX to a decimal number.
Now we are going the other way ...
Hexidecimal
Example
Convert ABCHEX to a decimal number.
Now we are going the other way ...
ABCHEX =
Hexidecimal
Example
Convert ABCHEX to a decimal number.
Now we are going the other way ...
ABCHEX = A(162 ) + B(16) + C
Hexidecimal
Example
Convert ABCHEX to a decimal number.
Now we are going the other way ...
ABCHEX = A(162 ) + B(16) + C
= 10(162 ) + 11(16) + 12
Hexidecimal
Example
Convert ABCHEX to a decimal number.
Now we are going the other way ...
ABCHEX = A(162 ) + B(16) + C
= 10(162 ) + 11(16) + 12
= 2560 + 176 + 12
Hexidecimal
Example
Convert ABCHEX to a decimal number.
Now we are going the other way ...
ABCHEX = A(162 ) + B(16) + C
= 10(162 ) + 11(16) + 12
= 2560 + 176 + 12
= 2748
Hexidecimal
Example
Convert 3EF1HEX to a decimal number.
Hexidecimal
Example
Convert 3EF1HEX to a decimal number.
3EF1HEX = 3(163 ) + E(162 ) + F(16) + 1(1)
Hexidecimal
Example
Convert 3EF1HEX to a decimal number.
3EF1HEX = 3(163 ) + E(162 ) + F(16) + 1(1)
= 3(4096) + 14(256) + 15(16) + 1
Hexidecimal
Example
Convert 3EF1HEX to a decimal number.
3EF1HEX = 3(163 ) + E(162 ) + F(16) + 1(1)
= 3(4096) + 14(256) + 15(16) + 1
= 16113