Your name: _________________________________________

Math 214 – Fall 2020 Project 1: Numeration Systems

Worth 100 points total; each part is worth 5 points. In this project, you will make up your own additive numeration system, using the symbols and base of your choosing. You may type onto this paper in word or pdf form, or print the pages out and write onto them by hand, or copy the table and write everything by hand (no need to rewrite the questions). 1. Choose any base for your number system, between 3 and 18 (but not base 5 or 10).

a. Write the base you will use for your number system, here: ______

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 ONE symbol for each power of your base, starting with the zero power. For example, Egyptians have symbols for each power of ten, starting with 100 = 1.

Powers of your base and what number it is equal to

Your ONE symbol for each

Powers of your base, continued

Your ONE symbol for each

Go up to the 8th power! Make sure you write what each power is equal to. Be careful to use the proper powers of your base. For example, 42 = 4×4= 16, not 4×2 = 8.

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. How many times can you repeat a symbol before you have to instead use the next higher symbol?) Answer:

Explanation:

3. Give one example of how to use several (more than two) symbols together to make a number in your system. Explain in words how you can go 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 your own words how to make that number, above, starting from Hindu Arabic (our system) and going into your system:

4. 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. d. A number of your own choosing. State the number and 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.)

5. What is the largest number you can make in your system?

Hint: Use all the symbols you have. How can you combine all symbols to make a very big number? Things to think about to answer this question: How many times are you allowed to repeat each symbol? How can you use all the symbols the most times you can to get a very large number?

a. Largest number, written using your symbols:

Be sure to say what this largest number is equal to:

b. Your explanation as to why this is the largest number using the rules of your system.

6. Show 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 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.

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 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 ….

8. 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. THIS PAGE IS JUST EXPLANATION, no answers required.

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. 9. I invented a base 7 place value system, with !=1, @=2, #=3, $=4, %=5, and ^=6.

What number could this be? @ # % ^

Hint: in a place value system, each place has its own value. This number has base 7 place values.