Alphabitia
• Read the introduction.
• Use the “artifacts”.
Unit
Flat
Long
Presentation
• Use a large white board to present your
group’s numeration system.
• Include the information needed for
someone to understand the system and
be able to use it.
• Select two members of the group to
explain and answer questions.
Show how you would write the number
4,628
in your system.
Alphabitia Numeration
System Proposals
• What characteristics do the systems
have in common and how are they
different?
• What are the pros and cons of your
group’s system and the other groups’
systems?
What makes an efficient
numeration system?
Alphabitia
• A numbering system is only powerful if
it can be reliably continued.
• Ex: 7, 8, 9, … what comes next?
• Ex: 38, 39, … what comes next?
• Ex: 1488, 1489, … what comes next?
The Numeration System we use today:
The Hindu-Arabic System
• Zero is used to represent nothing and as a place
holder.
• Base 10
Why?
• Any number can be represented using only 10
symbols.
• Easy to determine what number comes next or what
number came before.
• Operations are relatively easy to carry out.
In Base 10…
• Digits used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• We can put the digit 9 in the units
place. Can we put the next number
(ten) in the units place?
• Only one digit per place
• Placement of digits is important!
• 341 ≠ 143. Can you explain why not?
Alphabitia
• If we apply the aspects of our system to
the artifacts of the Alphabitians, what
would we come up with?
• Symbols: A, B, C, D, 0
Exploration 2.9
• Different Bases
In another base…
•
•
•
•
•
We need a 0, and some other digits
So, in base 10, we had 0 plus 9 digits
What will the digits be in base 9?
What will the digits be in base 3?
Which base was involved in alphabitia?
So, let’s count in base 6
• Digits allowed: 0, 1, 2, 3, 4, 5
• There is no such thing as 6
• When we read a number such as 2136,
we don’t typically say “two hundred
thirteen.” We say instead “two, one,
three, base 6.”
Count! In base 6
•
•
•
•
•
•
•
1, 2, 3, 4, 5, …
10, 11, 12, 13, 14, 15, …
20, 21, 22, 23, 24, 25, …
30, 31, 32, 33, 34, 35, …
40, 41, 42, 43, 44, 45, …
100, …
100, 101, 102, 103, 104, 105, … 110
Compare base 6 to base
10
• Digits
0,1,2,3,4,5,6,7,8,9
• New place value after 9
in a given place
• Each place is 10 times
as valuable as the one
to the right
• 243 =
2 • (10 • 10) + 4 • 10 +
3•1
• Digits 0, 1, 2, 3, 4, 5
• New place value after 5
in a given place
• Each place is 6 times
as valuable as the one
to the right.
• 243base 6 =
2 • (6 • 6) + 4 • 6 + 3 • 1
or 99 in base 10
Compare Base 6 to Base
10
• 312 =
3 • 100 +
1 • 10 +
2•1
• 312base 6 =
3 • 36 +
1•6+
2•1=
116 in base 10
How to change from Base
10 to Base 6?
• Suppose your number is 325 in base
10.
• We need to know what our place
values will look like.
• _____ _____
_____
_____
6•6•6
6•6
6
1
Now, 6•6•6 = 216. 216 = 1000 in base 6.
Base 10 to Base 6
• ___1__ _____
_____
_____
6•6•6
6•6
6
1
• Now, 325 - 216 = 109. Since 109 is
less than 216, we move to the next
smaller place value: 6 • 6 = 36.
• 109 - 36 = 73. Since 73 is greater than
36, we stay with the same place value.
Base 10 to Base 6
• __1___ ___3__ _____
_____
6•6•6
6•6
6
1
• We had 109: 109 - 36 - 36 - 36 = 1.
We subtracted 36 three times, so 3
goes in the 36ths place.
• We have 1 left. 1 is less than 6, so
there are no 6s. Just a 1 in the units
place.
Base 10 to Base 6
• __1___ ___3__ __0___ __1___
6•6•6
6•6
6
1
• Check: 1 • 216 + 3 • 36 + 1 • 1 = 325
• So 325 = 13016
Homework for Thursday 1/28
For Exploration 2.8, write up the following in an
essay format:
Describe the process your group went through
to come up with a numeration system for
Alphabitia. Explain your system. Describe
your thinking about this project. Turn in your
descriptions, along with the table on p. 41
and your answers to Part 3: #2,3,5
Count! In base 16
1, 2, 3, 4, 5, 6, 7, 8, 9, a,
b, c, d, e, f,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1a,
1b, 1c, 1d, 1e, 1f,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2a,
2b, 2c, 2d, 2e, 2f
Homework for Tuesday 2/2
• Exploration 2.9: Part 1: for Base 6, 2, and
16, do #2; Part 3: #2, 3, Part 4: #1, 2, 4.
For the base 16 section, change all the base
12 to base 16 (typo)
• Read Textbook pp. 109-118
• Do Textbook Problems pp. 120-121:
15b,c, 16b,d, 17a,i, 18b,f, 19, 29