Your name: _________________________________________ Group Members:

Math 214 Project 1: Numeration Systems

Worth 100 points total; points are listed next to each question. In this project, you will make up your own additive numeration system, using the symbols and base of your choosing.

• 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. 10 points 1. Decide the following with your group:

a. Write the base your group will use here: ______ You can choose any base between 3 and 18 (but not base 5 or 10).

b. Draw the symbols your system will use (do not copy any of the systems we have studied). Make symbols that are significant to you or that have a pattern you like! Since your system is additive, you will need symbols for every power of your base, starting with the zero power. For example, Egyptians have symbols for every power of ten, starting with 100 = 1.

Powers of your base

Your symbol for each

Powers of your base, continued

Your symbol for each

Go up to the 7th power! Make sure you write what each power is equal to.

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. How many times are you allowed to repeat a symbol? Why? Hint: in Egyptian, which is base ten, we are allowed to repeat each symbol 9 times. In Mayan, base 20, we are allowed to go up to 19 in a given place value. In Babylonian, base 60, you can have up to 59 in a place value. Answer:

Explanation:

c. Give one example of how to use several symbols together to make a number in your system. Explain in words how you went from Hindu Arabic (our system) into your 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. Example:

Explanation, in words: (Remember to explain this in your own words, not the same as what your group members have said.)

4 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. (You do not have to explain in words, but show all work.)

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

numbers that anyone else in your 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. Be sure to show how you got each number (but you do not have to explain in words). a. First number of your own choosing. Show your work.

b. Second 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. (You do not have to explain in words, but show all work.)

4 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: How many times care you allowed to repeat each symbol? How can you use all the symbols to get a very large number?

a. Largest number, written using your symbols:

b. Your explanation as to why this is the largest number using the rules of your system. Remember to explain this in your own words, not the same as what your group members have said.

4 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.

Here, be specific about each trade and why you did it and what was left.

Trade 1 with explanation/description of the trade and why you had to do it, what you wrote down for an answer and why:

Trade 2 with explanation/description of the trade and why you had to do it, what you wrote down for an answer and why:

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.

4 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. Here, be

specific about each trade and why you did it and what was left. Trade 1 with explanation/description of the trade and why you had to do it,

what you wrote down for an answer and why:

Trade 2 with explanation/description of the trade and why you had to do it, what you wrote down for an answer and why:

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 ….

10 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. Use circles and arrows and a different color if possible to show each trade.

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. Use circles and arrows and a different color if possible to show each trade.

9. This is a description of a multiplicative base 4 system, created by math 214 student Catherine Sayaman. On the next page, you will have a number in her system to try to decode.

Example of how to make the number 12,000:

4 points each part a.) Decode the number below using Catherine’s explanations on the previous page of

how to do it.

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

4 points for parts b and c 10. Trading with another group

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 copy of page one of this project.

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. 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.