Day 10, MAT115

Last Time

Next Time

Announcements:
- Your fifth assignment is due.
- Your sixth homework is due Tuesday.
- We have a test on Thursday, one week from today. It will cover everything up to bases (the stuff we're doing today).
Question of the Day: Come on, doesn't anyone think this is funny?

There are 10 kinds of people in the world:
those who understand binary and those who don't.
Last time we saw that ${2_{10}=10_2}$ ! That is, 2 base 10 is equal to 10 base 2. No joke!
Let's just review bases a little, but we'll start in an historical context.
- Take a look at the Mayan number system. What do you notice?
  - Zero!
  - What's the base?
  - How would the Mayans write 1729₁₀?
  - I'll write something in Mayan, and you tell me what it is base 10....
- Let's review Tom Lehrer's problem:
  
  342₈
  
  - 173₈

Today's topic: More Bases

Background

Bases are about powers and places.

The base is used to create the powers: e.g. ${8^0,8^1,8^2,8^3,\ldots}$ in base 8.
Place value -- each power has its place. And we actually reverse the order from that given above: the places go from largest to smallest power as we move from left to right, so ${\ldots,8^3,8^2,8^1,8^0}$ .

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. Any number can be written in any base.
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
Now let's play Detective in the times of Babylon....

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