Project 1: Numeration Systems

How to write up this project:

· You can print out the project, write on it by hand, then scan or take pictures of each page. Remember that all the pictures/scans must be together in one file. If you get stuck on how to do this, directions are in our files tab.

· You can use the word version of the project, type out parts of your answers into the word doc and handwrite other parts. You can then take pictures of the handwritten parts and paste them into the word doc in the right places.

· You can write the whole thing out on separate paper if you really want to. (You do not have to copy out each question.) But it’s a bit harder to stay organized and make sure you answer every part that way.

Remember: put your name on the first page!

Name your file with your name or you initials in it somewhere, so I know it’s yours!

Your name: _________________________________________

Math 214 – Spring 2021

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 a base for your system and draw the symbols. Make symbols that are significant to you or that have a pattern you like! (Do not copy any of the systems we have studied.)

a. Choose base 6, base 7, base 14 or base 15. Write the base you chose, here: ______

b. Draw the symbols your system will use in the table, below. 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 one symbol 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 in your system? Why?

Hint: in Egyptian, which is base ten, we are allowed to repeat each symbol 9 times, but not more. For example, we can write 9 scrolls, but we don’t write ten of them, because we write a flower instead. 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 in your system before you must instead use the next higher symbol?

Answer:

Explanation:

3. Give one example of how to use at least four of your symbols together to make a number in your system. Explain in words how you know what the number is equal to. Write as if you are helping another person who does not know your system to see what they should do to decode a number in your system. You will give this to another person in the class later on!

Example using your symbols:

You will be giving all of this to another student for question 6!

Explanation, in your own words how you know what that number is equal to:

A different made-up number using your symbols, with no answer (but you should know the answer):

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.

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?

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 in your system the most times possible 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. Decode someone else’s number! Find someone else in class who chose a different base than you did. Give them the example you created in question 3 (without the explanation, just tell them what each symbol is equal to), and take their example 3.

Other student’s name: ____________

Their example number, using their symbols:

Your work figuring out their number (you can check with them to see if you are correct):

7. Show would you add two large numbers in your system, without translating into Hindu Arabic (our system). You can use the way Egyptian numbers were added as your guide. Do not convert into base ten.

This example must show at least two carries/trades. Use your symbols, not numbers! Circle and use arrows to show the trades.

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.

1. Two large numbers being added:

+

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

1. 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. Show how would you subtract two large numbers in your system, without translating into Hindu Arabic (our system). Do not convert into base ten.

This example must show at least two borrows/trades. Use your symbols, not numbers! Circle and use arrows to show the trades.

a.) Two large numbers being subtracted:

_

b.) Describe in words each borrow or trade you had to do, and why you had to do it. 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 subtract in your system? Write something like this:

Generally, I have to trade ____ objects for _____. I know I have to trade whenever …..

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. THIS PAGE IS JUST EXPLANATION, no answers required.

Example of how to make the number 12,000:

4 points each part

1. Decode the multiplicative, base 4 number below using Catherine’s explanations on the previous page of how to do it.

1. Write the number 162 in Catherine’s multiplicative, base 4 system. Show your work.

10. I invented a base 4 place value system, with ! = 1, @ = 2, # = 3.

What number could this be? @ ! # !

Hint: in a place value system, each place has its own value. This number has base 4 place values. Make sure to convert into base ten.