Day 09, MAT115

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Announcements:
- You have an assignment due today (on Egyptian multiplication and division).
- RE your Babylonian and Mayan homework:
  - Babylonians
    - grouped in groups of three. You should too.
    - had to distinguish between the places (1s, 60s, 3600s, etc.) carefully. You should too. Provide a little separation (but not so much that one might assume that there's an "empty place" in there -- unless there should be).
  - Mayans
    - Didn't use a purely "20-based" system: their places were 1s, 20s, 360s, 360*20s, 360*20*20, etc.
    - They had no separate "fives-place". They used the five bar and the dot (and the shell) to build up their 20 distinct digits
  - The Lunar calendar:
    - Some of you didn't actually use the Mayan numbers to get your answers -- you used the pattern that you thought existed (i.e. increasing by 177). It only works for awhile, then there are occasional differences of 178.
    - While mathematicians look for patterns, they should check the patterns.
- Students seem to be able to access the website from all devices when on a secured (but not the public) connection. If you're using public wi-fi the content seems to be forbidden.
  Is that true?
- You have a new assignment today. Please keep an eye on the assignment page.
Here's our "Question of the day":

How do different numbers appear in different bases, and how can we use trees to write them?
Today we're going to
- review the Egyptian multiplication and division -- what questions do you have?
- Then we'll find out how bases bring all of the material we've studied together.

Today's new topic: Bases

You had a reading: on-line introduction to bases, so I'm going to assume that you've read that.
I might even hope that you went out and found Little Twelvetoes, so let's start there....
Base Background
So far we've been busting up numbers in different ways, in different cultures.
We might take a look now at our own culture. Consider the number 1729, and its base 10 representation.
When we say "Seventeen twenty nine", what do we mean?
What does each of the digits in the number 1729 mean?
- The story of base 10.
- Place value -- each power has its place.
Other cultures use other bases (and ours uses other bases, as we will see):
- What's the story of base 2 (Egyptians!)? Of base 8 (us!)? Of base 20 (Mayans!)? Of base 60 (Babylonians!)?
- One of my favorite early introductions to bases was Tom Lehrer's New Math, which informs us about base 8.

The Great Fraudini writes numbers in base 2 -- that is, using powers of two.

How would Fraudini write these numbers?
47	=		32		+8	+4	+2	+1
89	=	64		+16	+8			+1
How would we write these numbers in base 2?
47	=	0	1	0	1	1	1	1
89	=	1	0	1	1	0	0	1

How can we understand this base-2 representation? In terms of powers and their places, obviously!

Now: what do you get if you count using partitions the numbers
- 47?
- 89?
Amazing! My colleague Don Krug has recently given me a way of visualizing the same process of "counting by partition", or "primitive counting", and I want to share it with you. We might call it "exploding sheep"...
Let's do a few base 8 calculations. How about writing 173 (base 10) in base 8?
Other bases are used constantly -- for example, in this web page. How so?
- This page uses HTML -- Hypertext Markup Language -- to control its appearance. For example, the background color for this page is given by the command
  <body bgcolor="#FFF8DC">
  What does #FFF8DC have to do with color?
- What are some other bases that we use in our lives? Where?
Translating between bases.
One strategy to is convert from the starting base to base 10, then to do partition with a tree.
Here's an example: Convert 625₇ to base 3.
1. First convert 625₇ to base 10: 313₁₀
2. Now convert 313₁₀ to base 3: 102121₃
3. Finally your answer is
  
  625₇= 102121₃
Some more to try:
1. Write 537 (written here in base 10) in base 2
2. Write 537 (written here in base 10) in base 8
3. Write 537 (written here in base 10) in base 16
4. Write 537 (written here in base 10) in base 7
(these are just another kind of partition, which we can do with trees).
Now in the other direction:
1. Write 100101 (written here in base 2) in base 10
2. Write 100101 (written here in base 8) in base 10
3. Write 100101 (written here in base 16) in base 10
4. Write 100101 (written here in base 7) in base 10
5. Write 22803 (written here in base 9) in base 10

Links:

My unit fraction table handout
Another unit fraction table, with other interesting links.
A great Egyptian math source book

Website maintained by Andy Long. Comments appreciated.