Day 10: Mat115

Last time: Practice RSA encryption; cardinality

Next time: Exam 1

Today:
- Warnings:
  - Exam 1 is coming up next class, Wednesday!
    - It will cover chapters 1 and 2.
    - Some of the special topics we've covered:
      - Pidgeonhole principle
      - Estimation
      - Fibonacci numbers
      - Fibonacci nim
      - Prime numbers and factorizations
      - UPC codes
      - RSA -- an application of Prime numbers
  - Your project ideas and partners are due in a week (next Wednesday, 2/27). Please think about those. I need preliminary information. If you haven't identified a partner, but want one, please make a note of it on your proposal. I'll try to match you up.
- Hand in problems 2.5 (these will be graded and posted by tomorrow morning)
- Problems 2.4 returned: graded
  - II.6 -- the clock now reads 10:45
    - 96 = 0 mod 24, so in 96 hours it will read 10:45
    - 1063 = 7 mod 24, so the clock will read 5:45
    - If it now reads 7:10, it read 8:10 23 hours ago
    - If it now reads 7:10, it read 7:10 108 hours ago (since 108 = 12 mod 24)
  - II.10
```
"051000025624" 
(0 5 1 0 0 0 0 2 5 6 2) 
(3 1 3 1 3 1 3 1 3 1 3) 
congruent to 3, mod 10
check digit: 7

"051000025264" 
(0 5 1 0 0 0 0 2 5 2 6) 
(3 1 3 1 3 1 3 1 3 1 3) 
congruent to 5, mod 10
check digit: 5
 
Correct UPC:
"051000025265" 
(0 5 1 0 0 0 0 2 5 2 6) 
(3 1 3 1 3 1 3 1 3 1 3) 
congruent to 5, mod 10
check digit: 5 
```
- Questions/Comments on old stuff?
- Infinity!
  - Infinity - so mysterious....
  - Q: Are mathematicians trying to take all the fun out of it by studying it?! A: Of course not! We relish the mind-blowing mystery. And there's plenty of fun to be had....
    
    William Blake (from Auguries of Innocence):
    To see a World in a Grain of Sand
    And a Heaven in a Wild Flower,
    Hold Infinity in the palm of your hand
    And Eternity in an hour.
- We started last time with the idea of number (e.g. What does "2" mean?).
- What is infinity?
  - Is it a number? We agreed that it's not.
  - Is it just one thing? Not sure yet....
  - What's the difference between infinity and eternity? Eternity is an infinite amount of time. But if infinity isn't a number, how much is an infinite amount?
- What is a one-to-one correspondence, and how is it useful?
- Let's look at some problems (pp. 141-144):
  - III.16
  - IV.22
- The natural numbers are infinite in number (the size of the set is larger than any natural number).
- What is Cardinality?
  Two sets have the same cardinality if there is a one-to-one correspondence between them.
  Intuitively: they're the same size.
- A subset is never bigger in size than the set itself.
  And if the sets are finite, the proper subset is always smaller....
  but if the set is infinite, we may actually be able to throw away elements of a set and not change the size of the set!
- Let's look at these curious problems (pp. 157):
  - 14 - counting cubes
  - 16 - Hotel Cardinality
  - 17 - Hotel Cardinality (cont.)
  - 18 - Hotel Cardinality and the Infinite Life Insurance Co.
  - Hotel Cardinality and the Limitless Bus Barn Co.
- The ping-pong ball conundrum
- The ping-pong ball conundrum is related to other paradoxical ideas, such as Zeno's paradoxes

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