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 20New 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 • 1or 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