Transcript Week 2--Glad you came back! - University of Arizona Math

Numeration System Presentation
On a white board, prepare your
presentation of your group’s system.
When you present, all members of the
group must talk.
While watching the other groups present
their system, make notes on them and
the advantages and disadvantages.
Include in your presentation:
How would you write the number 834 in
your system?
Show how you would add two numbers in
your system.
Alphabitia Numeration
System Proposals
• What did you come up with in your
group?
• 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?
Mayan Numerals
• Used the concept of zero, but only for
place holders
• Used three symbols:
•
--1
5
0
• Wrote their numbers vertically:
••• is 3 + 5 = 8, --- is 5 + 5 = 10
Mayan Numeration
Uses base 20
New place value… left a vertical gap.
•
is one 20, and 0 ones = 20.
••
•
is two ____ + 5 + 1 = _____
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?
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 Tuesday, Sept 7th
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
Use the Alphabitian system we developed together in
class to answer 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 2/3
• 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