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§ 3-1 Numeration Systems
§ 3-1 Numeration Systems
How We Use Numbers
cardinal numbers
how many numbers are in a setordinal numbers
positionwhich comes first
identification numbers
Numbers are abstract ideas for children to understand, so it isimportant to find symbolic ways to represent them to help them graspthe concept.
How We Use Numbers
cardinal numbershow many numbers are in a set
ordinal numberspositionwhich comes first
identification numbers
Numbers are abstract ideas for children to understand, so it isimportant to find symbolic ways to represent them to help them graspthe concept.
How We Use Numbers
cardinal numbershow many numbers are in a set
ordinal numbers
positionwhich comes first
identification numbers
Numbers are abstract ideas for children to understand, so it isimportant to find symbolic ways to represent them to help them graspthe concept.
How We Use Numbers
cardinal numbershow many numbers are in a set
ordinal numberspositionwhich comes first
identification numbers
Numbers are abstract ideas for children to understand, so it isimportant to find symbolic ways to represent them to help them graspthe concept.
How We Use Numbers
cardinal numbershow many numbers are in a set
ordinal numberspositionwhich comes first
identification numbers
Numbers are abstract ideas for children to understand, so it isimportant to find symbolic ways to represent them to help them graspthe concept.
How We Use Numbers
cardinal numbershow many numbers are in a set
ordinal numberspositionwhich comes first
identification numbers
Numbers are abstract ideas for children to understand, so it isimportant to find symbolic ways to represent them to help them graspthe concept.
How We Use Numbers
cardinal numbershow many numbers are in a set
ordinal numberspositionwhich comes first
identification numbers
Numbers are abstract ideas for children to understand, so it isimportant to find symbolic ways to represent them to help them graspthe concept.
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 withchildren to help them understand the ideas that lead to thedevelopment of these systems.
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 withchildren to help them understand the ideas that lead to thedevelopment of these systems.
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 withchildren to help them understand the ideas that lead to thedevelopment of these systems.
The Tally System
The Tally System
Propertiesmost simplistic
a line represents 1, crossing represents 5
grouping is to make counting easier
very tedious
additive system
The Tally System
Propertiesmost simplistic
a line represents 1, crossing represents 5
grouping is to make counting easier
very tedious
additive system
The Tally System
Propertiesmost simplistic
a line represents 1, crossing represents 5
grouping is to make counting easier
very tedious
additive system
The Tally System
Propertiesmost simplistic
a line represents 1, crossing represents 5
grouping is to make counting easier
very tedious
additive system
The Tally System
Propertiesmost 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
Propertiesbase 10 system
additive system
multiple symbols is an improvement
order does not matter
The Egyptian Numeration System
Propertiesbase 10 system
additive system
multiple symbols is an improvement
order does not matter
The Egyptian Numeration System
Propertiesbase 10 system
additive system
multiple symbols is an improvement
order does not matter
The Egyptian Numeration System
Propertiesbase 10 system
additive system
multiple symbols is an improvement
order does not matter
Examples Using Egyptian Hieroglyphics
ExampleWhat decimal number is represented here?
2008
Examples Using Egyptian Hieroglyphics
ExampleWhat decimal number is represented here?
2008
Examples Using Egyptian Hieroglyphics
ExampleWhat decimal number is represented here?
2,320,111
Examples Using Egyptian Hieroglyphics
ExampleWhat decimal number is represented here?
2,320,111
Examples Using Egyptian Hieroglyphics
ExampleWrite the number 245 using Egyptian hieroglyphics.
Examples Using Egyptian Hieroglyphics
ExampleWrite the number 245 using Egyptian hieroglyphics.
The Roman Numeration System
Decimal Roman1
I5 V10 X50 L100 C500 D
1000 M
Where do we see Roman numerals in modern day?
The Roman Numeration System
Decimal Roman1 I5
V10 X50 L100 C500 D
1000 M
Where do we see Roman numerals in modern day?
The Roman Numeration System
Decimal Roman1 I5 V10
X50 L100 C500 D
1000 M
Where do we see Roman numerals in modern day?
The Roman Numeration System
Decimal Roman1 I5 V10 X50
L100 C500 D
1000 M
Where do we see Roman numerals in modern day?
The Roman Numeration System
Decimal Roman1 I5 V10 X50 L100
C500 D
1000 M
Where do we see Roman numerals in modern day?
The Roman Numeration System
Decimal Roman1 I5 V10 X50 L100 C500
D1000 M
Where do we see Roman numerals in modern day?
The Roman Numeration System
Decimal Roman1 I5 V10 X50 L100 C500 D1000
M
Where do we see Roman numerals in modern day?
The Roman Numeration System
Decimal Roman1 I5 V10 X50 L100 C500 D1000 M
Where do we see Roman numerals in modern day?
The Roman Numeration System
Decimal Roman1 I5 V10 X50 L100 C500 D1000 M
Where do we see Roman numerals in modern day?
The Roman Numeration System
Propertiesnot a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000times that numberM = 1, 000, 000
positional system - where we place the symbols changes thevalue
VI=6 IV=4
The Roman Numeration System
Propertiesnot a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000times that numberM = 1, 000, 000
positional system - where we place the symbols changes thevalue
VI=6 IV=4
The Roman Numeration System
Propertiesnot a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000times that numberM = 1, 000, 000
positional system - where we place the symbols changes thevalue
VI=6 IV=4
The Roman Numeration System
Propertiesnot a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000times that number
M = 1, 000, 000
positional system - where we place the symbols changes thevalue
VI=6 IV=4
The Roman Numeration System
Propertiesnot a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000times that numberM = 1, 000, 000
positional system - where we place the symbols changes thevalue
VI=6 IV=4
The Roman Numeration System
Propertiesnot a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000times that numberM = 1, 000, 000
positional system - where we place the symbols changes thevalue
VI=6 IV=4
The Roman Numeration System
Propertiesnot a base 10 system
additive system
subtractive system
multiplicative - we place a bar over a number to represent 1000times that numberM = 1, 000, 000
positional system - where we place the symbols changes thevalue
VI=6 IV=4
More Roman Numerals
ExampleWrite the following using Roman numerals.
1 2
II2 4 IV3 9 IX4 40 XL5 49 XLIX6 99 XCIX7 1489 MCDLXXXIX
More Roman Numerals
ExampleWrite the following using Roman numerals.
1 2 II2 4
IV3 9 IX4 40 XL5 49 XLIX6 99 XCIX7 1489 MCDLXXXIX
More Roman Numerals
ExampleWrite the following using Roman numerals.
1 2 II2 4 IV3 9
IX4 40 XL5 49 XLIX6 99 XCIX7 1489 MCDLXXXIX
More Roman Numerals
ExampleWrite the following using Roman numerals.
1 2 II2 4 IV3 9 IX4 40
XL5 49 XLIX6 99 XCIX7 1489 MCDLXXXIX
More Roman Numerals
ExampleWrite the following using Roman numerals.
1 2 II2 4 IV3 9 IX4 40 XL5 49
XLIX6 99 XCIX7 1489 MCDLXXXIX
More Roman Numerals
ExampleWrite the following using Roman numerals.
1 2 II2 4 IV3 9 IX4 40 XL5 49 XLIX6 99
XCIX7 1489 MCDLXXXIX
More Roman Numerals
ExampleWrite the following using Roman numerals.
1 2 II2 4 IV3 9 IX4 40 XL5 49 XLIX6 99 XCIX7 1489
MCDLXXXIX
More Roman Numerals
ExampleWrite the following using Roman numerals.
1 2 II2 4 IV3 9 IX4 40 XL5 49 XLIX6 99 XCIX7 1489 MCDLXXXIX
More Roman Numerals
ExampleWrite the following as decimal numbers.
1 CDXXIII
4232 DCXIV 6143 CCII 200,002
More Roman Numerals
ExampleWrite the following as decimal numbers.
1 CDXXIII 423
2 DCXIV 6143 CCII 200,002
More Roman Numerals
ExampleWrite the following as decimal numbers.
1 CDXXIII 4232 DCXIV
6143 CCII 200,002
More Roman Numerals
ExampleWrite the following as decimal numbers.
1 CDXXIII 4232 DCXIV 614
3 CCII 200,002
More Roman Numerals
ExampleWrite the following as decimal numbers.
1 CDXXIII 4232 DCXIV 6143 CCII
200,002
More Roman Numerals
ExampleWrite the following as decimal numbers.
1 CDXXIII 4232 DCXIV 6143 CCII 200,002
The Babylonian Numeration System
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like thefollowing:
Propertiesbase 60 system
additive system
multiplicative
place valued system
positional
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like thefollowing:
Propertiesbase 60 system
additive system
multiplicative
place valued system
positional
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like thefollowing:
Propertiesbase 60 system
additive system
multiplicative
place valued system
positional
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like thefollowing:
Propertiesbase 60 system
additive system
multiplicative
place valued system
positional
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like thefollowing:
Propertiesbase 60 system
additive system
multiplicative
place valued system
positional
The Babylonian Numeration System
For the sake of simplicity, we will use symbols that look like thefollowing:
Propertiesbase 60 system
additive system
multiplicative
place valued system
positional
The Babylonian Numeration System
Base 60
603 602 601 600
ExampleWrite 53 using Babylonian cuneiform.
The Babylonian Numeration System
Base 60
603 602 601 600
ExampleWrite 53 using Babylonian cuneiform.
The Babylonian Numeration System
Base 60
603 602 601 600
ExampleWrite 53 using Babylonian cuneiform.
The Babylonian Numeration System
ExampleWhat decimal number is represented here?
ExampleWhat decimal number is represented here?
The Babylonian Numeration System
ExampleWhat decimal number is represented here?
ExampleWhat decimal number is represented here?
The Babylonian Numeration System
ExampleWhat decimal number is represented here?
The Babylonians realized they needed a place holder to represent thatthey had a place with nothing in it when writing their numbers.
The Babylonian Numeration System
ExampleWhat decimal number is represented here?
The Babylonians realized they needed a place holder to represent thatthey had a place with nothing in it when writing their numbers.
The Babylonian Numeration System
ExampleWrite 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is thelargest 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
ExampleWrite 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is thelargest 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
ExampleWrite 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is thelargest 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
ExampleWrite 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is thelargest 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
ExampleWrite 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is thelargest 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
ExampleWrite 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is thelargest 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
ExampleWrite 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is thelargest 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
ExampleWrite 3821 using Babylonian cuneiform.
To figure this out, we want to write this in expanded form. What is thelargest 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
ExampleWrite 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
ExampleWrite 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
ExampleWrite 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
ExampleWrite 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
ExampleWrite 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
ExampleWrite 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
ExampleWrite 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
Properties1 additive system2 multiplicative system3 positional system4 place valued system5 symbol for 0
The Mayan Numeration System
Properties1 additive system
2 multiplicative system3 positional system4 place valued system5 symbol for 0
The Mayan Numeration System
Properties1 additive system2 multiplicative system
3 positional system4 place valued system5 symbol for 0
The Mayan Numeration System
Properties1 additive system2 multiplicative system3 positional system
4 place valued system5 symbol for 0
The Mayan Numeration System
Properties1 additive system2 multiplicative system3 positional system4 place valued system
5 symbol for 0
The Mayan Numeration System
Properties1 additive system2 multiplicative system3 positional system4 place valued system5 symbol for 0
The Mayan Numeration System
1’s
20’s
18 · 20’s
18 · 202’s
The Mayan Numeration System
1’s
20’s
18 · 20’s
18 · 202’s
The Mayan Numeration System
1’s
20’s
18 · 20’s
18 · 202’s
The Mayan Numeration System
1’s
20’s
18 · 20’s
18 · 202’s
The Mayan Numeration System
ExampleWrite 7 in Mayan.
• •
ExampleWrite 45 in Mayan.
• •
Spacing is very important ...
The Mayan Numeration System
ExampleWrite 7 in Mayan.
• •
ExampleWrite 45 in Mayan.
• •
Spacing is very important ...
The Mayan Numeration System
ExampleWrite 7 in Mayan.
• •
ExampleWrite 45 in Mayan.
• •
Spacing is very important ...
The Mayan Numeration System
ExampleWrite 7 in Mayan.
• •
ExampleWrite 45 in Mayan.
• •
Spacing is very important ...
The Mayan Numeration System
ExampleWrite 7 in Mayan.
• •
ExampleWrite 45 in Mayan.
• •
Spacing is very important ...
The Mayan Numeration System
ExampleWrite 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
ExampleWrite 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
ExampleWrite 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
ExampleWrite 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
ExampleWrite 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
ExampleWrite 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
ExampleWrite 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
Properties1 base 10 system
2 positional system3 additive system (values for various numerals added)4 multiplicative5 10 symbols called digits
We can use manipulatives to help children understand how our systemworks. A common manipulative is base 10 blocks.
The Hindu-Arabic Numeration System
Properties1 base 10 system2 positional system
3 additive system (values for various numerals added)4 multiplicative5 10 symbols called digits
We can use manipulatives to help children understand how our systemworks. A common manipulative is base 10 blocks.
The Hindu-Arabic Numeration System
Properties1 base 10 system2 positional system3 additive system (values for various numerals added)
4 multiplicative5 10 symbols called digits
We can use manipulatives to help children understand how our systemworks. A common manipulative is base 10 blocks.
The Hindu-Arabic Numeration System
Properties1 base 10 system2 positional system3 additive system (values for various numerals added)4 multiplicative
5 10 symbols called digits
We can use manipulatives to help children understand how our systemworks. A common manipulative is base 10 blocks.
The Hindu-Arabic Numeration System
Properties1 base 10 system2 positional system3 additive system (values for various numerals added)4 multiplicative5 10 symbols called digits
We can use manipulatives to help children understand how our systemworks. A common manipulative is base 10 blocks.
The Hindu-Arabic Numeration System
Properties1 base 10 system2 positional system3 additive system (values for various numerals added)4 multiplicative5 10 symbols called digits
We can use manipulatives to help children understand how our systemworks. A common manipulative is base 10 blocks.
Other Bases
To see the types of troubles children have with the base 10 system, wewill use numeration systems with different bases.
To do this, we first have to understand what the place values in base10 really mean.We can see this using expanded form.
ExampleWrite 4321 in expanded form.
4321 = 4(1000) + 3(100) + 2(10) + 1(1)
= 4 · 103 + 3 · 102 + 2 · 101 + 1 · 100
Other Bases
To see the types of troubles children have with the base 10 system, wewill use numeration systems with different bases.
To do this, we first have to understand what the place values in base10 really mean.We can see this using expanded form.
ExampleWrite 4321 in expanded form.
4321 = 4(1000) + 3(100) + 2(10) + 1(1)
= 4 · 103 + 3 · 102 + 2 · 101 + 1 · 100
Other Bases
To see the types of troubles children have with the base 10 system, wewill use numeration systems with different bases.
To do this, we first have to understand what the place values in base10 really mean.We can see this using expanded form.
ExampleWrite 4321 in expanded form.
4321 = 4(1000) + 3(100) + 2(10) + 1(1)
= 4 · 103 + 3 · 102 + 2 · 101 + 1 · 100
Other Bases
To see the types of troubles children have with the base 10 system, wewill use numeration systems with different bases.
To do this, we first have to understand what the place values in base10 really mean.We can see this using expanded form.
ExampleWrite 4321 in expanded form.
4321 = 4(1000) + 3(100) + 2(10) + 1(1)
= 4 · 103 + 3 · 102 + 2 · 101 + 1 · 100
Other Bases
To see the types of troubles children have with the base 10 system, wewill use numeration systems with different bases.
To do this, we first have to understand what the place values in base10 really mean.We can see this using expanded form.
ExampleWrite 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 placevalue quickly.
ExampleConvert 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
Binary is the language of computers and also helps children see placevalue quickly.
ExampleConvert 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
Binary is the language of computers and also helps children see placevalue quickly.
ExampleConvert 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
Binary is the language of computers and also helps children see placevalue quickly.
ExampleConvert 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 =
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 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 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 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 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 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 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 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 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 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?
. . . 73 72 71 70
343 49 7 1
ExampleConvert 321 to base 7.
321 = 49( ) + 7( ) + 1( )
= 49(6) + 7( ) + 1( )
= 49(6) + 7(3) + 6
So, 321 = 636SEVEN .
Base 7
If we are in base 7, what are the first 4 place values?
. . . 73 72 71 70
343 49 7 1
ExampleConvert 321 to base 7.
321 = 49( ) + 7( ) + 1( )
= 49(6) + 7( ) + 1( )
= 49(6) + 7(3) + 6
So, 321 = 636SEVEN .
Base 7
If we are in base 7, what are the first 4 place values?
. . . 73 72 71 70
343 49 7 1
ExampleConvert 321 to base 7.
321 = 49( ) + 7( ) + 1( )
= 49(6) + 7( ) + 1( )
= 49(6) + 7(3) + 6
So, 321 = 636SEVEN .
Base 7
If we are in base 7, what are the first 4 place values?
. . . 73 72 71 70
343 49 7 1
ExampleConvert 321 to base 7.
321 = 49( ) + 7( ) + 1( )
= 49(6) + 7( ) + 1( )
= 49(6) + 7(3) + 6
So, 321 = 636SEVEN .
Base 7
If we are in base 7, what are the first 4 place values?
. . . 73 72 71 70
343 49 7 1
ExampleConvert 321 to base 7.
321 =
49( ) + 7( ) + 1( )
= 49(6) + 7( ) + 1( )
= 49(6) + 7(3) + 6
So, 321 = 636SEVEN .
Base 7
If we are in base 7, what are the first 4 place values?
. . . 73 72 71 70
343 49 7 1
ExampleConvert 321 to base 7.
321 = 49( ) + 7( ) + 1( )
= 49(6) + 7( ) + 1( )
= 49(6) + 7(3) + 6
So, 321 = 636SEVEN .
Base 7
If we are in base 7, what are the first 4 place values?
. . . 73 72 71 70
343 49 7 1
ExampleConvert 321 to base 7.
321 = 49( ) + 7( ) + 1( )
= 49(6) + 7( ) + 1( )
= 49(6) + 7(3) + 6
So, 321 = 636SEVEN .
Base 7
If we are in base 7, what are the first 4 place values?
. . . 73 72 71 70
343 49 7 1
ExampleConvert 321 to base 7.
321 = 49( ) + 7( ) + 1( )
= 49(6) + 7( ) + 1( )
= 49(6) + 7(3) + 6
So, 321 = 636SEVEN .
Base 7
ExampleConvert 219 to base 7.
219 = 432SEVEN
ExampleConvert 693SEVEN to base 10.
DNE ... why?
Base 7
ExampleConvert 219 to base 7.
219 = 432SEVEN
ExampleConvert 693SEVEN to base 10.
DNE ... why?
Base 7
ExampleConvert 219 to base 7.
219 = 432SEVEN
ExampleConvert 693SEVEN to base 10.
DNE ... why?
Base 7
ExampleConvert 219 to base 7.
219 = 432SEVEN
ExampleConvert 693SEVEN to base 10.
DNE ... why?
Base 7
ExampleConvert 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
Base 7
ExampleConvert 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
Base 7
ExampleConvert 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
Base 7
ExampleConvert 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
Base 7
ExampleConvert 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
Base 7
ExampleConvert 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
Hexadecimal (Base 16)
How can we deal with number systems where we may need to exceed9 in any one position?
10 11 12 13 14 15A B C D E F
Hexadecimal (Base 16)
How can we deal with number systems where we may need to exceed9 in any one position?
10 11 12 13 14 15A B C D E F
Hexadecimal
ExampleConvert 352 to hexadecimal.
How do we begin?
. . . 162 161 160
256 16 1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
= 256(1) + 16(6) + 0
So, 352 = 160HEX .
Hexadecimal
ExampleConvert 352 to hexadecimal.
How do we begin?
. . . 162 161 160
256 16 1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
= 256(1) + 16(6) + 0
So, 352 = 160HEX .
Hexadecimal
ExampleConvert 352 to hexadecimal.
How do we begin?
. . . 162 161 160
256 16 1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
= 256(1) + 16(6) + 0
So, 352 = 160HEX .
Hexadecimal
ExampleConvert 352 to hexadecimal.
How do we begin?
. . . 162 161 160
256 16 1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
= 256(1) + 16(6) + 0
So, 352 = 160HEX .
Hexadecimal
ExampleConvert 352 to hexadecimal.
How do we begin?
. . . 162 161 160
256 16 1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
= 256(1) + 16(6) + 0
So, 352 = 160HEX .
Hexadecimal
ExampleConvert 352 to hexadecimal.
How do we begin?
. . . 162 161 160
256 16 1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
= 256(1) + 16(6) + 0
So, 352 = 160HEX .
Hexadecimal
ExampleConvert 352 to hexadecimal.
How do we begin?
. . . 162 161 160
256 16 1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
= 256(1) + 16(6) + 0
So, 352 = 160HEX .
Hexadecimal
ExampleConvert 352 to hexadecimal.
How do we begin?
. . . 162 161 160
256 16 1
352 = 256( ) + 16( ) + 1( )
= 256(1) + 16( ) + 1( )
= 256(1) + 16(6) + 0
So, 352 = 160HEX .
Hexadecimal
ExampleConvert 792 to hexadecimal.
792 = 318HEX
Hexadecimal
ExampleConvert 792 to hexadecimal.
792 = 318HEX
Hexadecimal
ExampleConvert 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
Hexadecimal
ExampleConvert 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
Hexadecimal
ExampleConvert 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
Hexadecimal
ExampleConvert 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
Hexadecimal
ExampleConvert 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
Hexadecimal
ExampleConvert 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
Hexadecimal
ExampleConvert 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
Hexadecimal
ExampleConvert 3EF1HEX to a decimal number.
3EF1HEX = 3(163) + E(162) + F(16) + 1(1)
= 3(4096) + 14(256) + 15(16) + 1
= 16113
Hexadecimal
ExampleConvert 3EF1HEX to a decimal number.
3EF1HEX = 3(163) + E(162) + F(16) + 1(1)
= 3(4096) + 14(256) + 15(16) + 1
= 16113
Hexadecimal
ExampleConvert 3EF1HEX to a decimal number.
3EF1HEX = 3(163) + E(162) + F(16) + 1(1)
= 3(4096) + 14(256) + 15(16) + 1
= 16113
Hexadecimal
ExampleConvert 3EF1HEX to a decimal number.
3EF1HEX = 3(163) + E(162) + F(16) + 1(1)
= 3(4096) + 14(256) + 15(16) + 1
= 16113
