Day 9, MAT115

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Announcements:
- Your fourth assignment is returned.
- You should have read this on-line introduction to bases for today.
- You also have a new assignment.
- Your fifth homework is due Thursday.
Question of the Day: Do understand this joke? There are 10 kinds of people in the world:
those who understand binary and those who don't.
Let's just review Egyptian Division:
- Divide 8 by 17. Whenever I propose one of these, I want you to think about dividing 8 loaves of bread among 17 people. How do we do it so that everyone sees the same thing on their plate (fair division).
  
  1 17
  1/2 8+1/2
  1/4 4+1/4 *
  1/8 2+1/8 *
  1/16 1+1/16 *
  1/17 1
  1/34 1/2 *
  1/68 1/4
  1/136 1/8
  1/272 1/16 *
  
  So the answer is
  
  8/17=1/4+1/8+1/16+1/34+1/272
- Now let's do it by the unit fraction table, which gives
  
  8/17=1/3+1/12+1/34+1/68+1/102
  Notice that the two methods give different results! That's not a problem, but a mathematician would really rather have a unique representation. In the absence of a unique representation, mathematicians look for some good reason for choosing one representation versus the other. What makes one of these better than the other, if anything?

Today's new topic: Bases

Background
So far we've been busting up numbers in different ways. The first way is illustrated by 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.
- One of my favorite early introductions to bases was Tom Lehrer's New Math, which informs us about base 8.
- What's the story of base 2? Of base 8? Of base 20 (we'll get to that!)?

The Great Fraudini writes numbers in base 2 (see p. 11 of our text) -- 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?
For your homework, you should have read this bases reference. Other bases are used constantly -- for example, in this web page. How so?
- This page uses HTML 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

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