Your name: _________________________________________ Group Members:

Math 214 Project 1: Numeration Systems

Worth 100 points total; points are listed next to each question.

• You may write directly on this paper. • Each person in the group must submit their own project, written in their own words.

Computer copies of projects from other group members will not be accepted. In this project, you will make up your own numeration system, using the symbols and base of your choosing. 10 points 1. Decide the following with your group: Write the base your group will use here: ______ You can choose any base except base 5, 10, 20, or 60. Write the type of system your group will use here: ______________

Your system can be place value, additive, or some combination.

Symbols your system will use (do not copy any of the systems we have studied). If your system is place value, you will need symbols from 1 to one less than your

base. For example, in Hindu Arabic, with a base of ten, we have symbols for 1 to 9.

If your system is additive, you will need symbols for every power of your base. For example, Egyptians have symbols for every power of ten.

4 points each part 2. Describe your system:

a.) What principle or design connects your symbols? For example, the Mayan system is made from a pattern of lines and dots; the Egyptian system uses pictures of objects that had significance to them. Where did you get your symbols and why?

b.) If your system is additive, how many times are you allowed to repeat a symbol? If

your system is place value, how high a number can you have in any one place? c.) Is there a zero in your system? Is there anything unusual in how the system works –

for example, is it a combination of two types of systems? d.) Give one example of how to make a number (using several symbols) in your system

and explain in words how you converted from Hindu Arabic (our system) into your system. Your explanation should include specifically what you have to do with each of our numbers to figure out how to write them into your number system. Write as if you are helping another person who does not know your system to see what they should do to create a number in your system. You will be giving

this explanation to another group at the end of the unit so they can use it to make a number.

e.) Give another example of how to make a number (using several symbols) in your

system and explain in words how you converted from Hindu Arabic (our system) into your system. Your explanation should include specifically what you have to do with each of our numbers to figure out how to write them into your number system. Write as if you are helping another person who does not know your system to see what they should do to create a number in your system. You will be giving this explanation to another group at the end of the unit so they can use it to make a number.

3 points each part 3. Show how to write the following numbers in your system, and clearly show how you

know that you have the correct answer (does not have to be in words, but show the math): a.) 32 Show your work. b.) 148 Show your work. c.) 12,437 Show your work. CAUTION: show how you got the value of each number or you will not get credit for this part.

3 points each part 4. At home, make three more numbers of your own choosing. They should not be the same

three numbers that anyone else in the group has made. When you are back in class, have another person in your group check that you have done the work correctly to make the number. a.) First number of your own choosing. Show your work.

b.) Second number of your own choosing. Show your work.

c.) Third number of your own choosing. Show your work.

CAUTION: show how you got the value of each number or you will not get credit for this part.

3 points each part 5. Using all the symbols you have made up, what is the largest number you can make

in your system? Explain how you know. Things to think about to answer this question: Do you have an infinite number of symbols you can put down, or a limited number of symbols you can put down? How many times can you repeat each symbol?

a.) Largest number:

b.) Your explanation, using the rules of your system:

3 points each part 6. Show how would you add two large numbers in your system, without translating into

Hindu Arabic (our system). If your system is additive, you can use the way Egyptian numbers were added as your guide; if your system is place value, think of adding in your base. Either way, do not convert into base ten. This example can be the same as the other members in your group. This example must show at least two carries/trades. Circle and use

arrows to show the trades. a.) Two large numbers being added: +

b.) Describe in words each carry or trade you had to do, and why you had to do it

c.) What is the general principle behind how you trade to add in your system? Write something like this: Generally, I have to trade ____ objects for _____. I know I have to trade whenever ______________.

Tip: instead of translating the numbers into base ten, just write down a bunch of your symbols, plus another bunch of your symbols. Think in your system, not in base ten.

3 points each part 7. Show how would you subtract two large numbers in your system, without translating

into Hindu Arabic (our system). If your system is additive, you can use the way Egyptian numbers were subtracted as your guide; if your system is place value, think of subtracting in your base. Either way, do not convert into base ten. This example can be the same as the other members in your group. This example must show at least two borrows/trades. Circle and use arrows to show

the trades. a.) Two large numbers being subtracted: _

b.) Describe in words each trade you had to do, and why you had to do it.

c.) What is the general principle behind how you trade to add in your system? Write something like this: Generally, I have to trade ____ objects for _____. I know I have to trade whenever ______________.

12 points 8. At home,

a) Make up another example of two numbers being added. This example should not be the same as anyone else’s in the group. The example must include two trades. Show how the symbols would combine and carry or trade. You do not need words here, but the trades should be very clear.

b) Make up another example of two numbers being subtracted. This example should

not be the same as anyone else’s in the group. The example must include two trades. Show how the symbols would need to be borrowed or traded. You do not need words here, but the trades should be very clear.

9. Decoding a multiplicative number. This is a description of a multiplicative system, created by math 214 student Catherine Sayaman.

4 points each part a.) Decode the number on the right using Catherine’s explanations of how to do

it.

b.) Write the number 162 in Catherine’s system. Show your work.

3 points for parts b and c 10. Trading with another group – we will do this in class on the day the project is

due. a.) Your group will together make one number in your system using your symbols and

give it to one of the other groups, along with a description of how your system works (a copy of your answer to #1 and 2).

b.) Your group will get a number from one of the other groups and decode it, using the group’s description of the system. Pick a group whose number system is as different from yours as possible (for example, if you made up a positional system, pick a group that used an additive or one). Show your work.

c.) Together, write the number 162 in the other group’s system. Show your work. Each group member should include parts b and c in their own project. If you are absent, contact your group to get the other group’s number. If they do not respond, contact me.