Day 12: Heart of Mathematics

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Announcements:
- Remember, you have an exam this Friday -- next class period.
  - It will cover through the material on prime numbers
  - There will be questions on the test to see if you've done the assigned reading. If you've not done the reading, you'll be unlikely to get them. If you've done the reading, they should be easy. You make the call....
- Notice that you have an assignment posted for the infinity unit (it's coming in parts). It's on the assignment page.
- If anyone requested accomodation for disability on this exam, they should provide me a reminder -- just to be sure.
- Your problems on Bases and Fibonacci are returned today. Thanks to all who brought in something from nature involving Fibonacci numbers. That was worth two points on your homework.
  Let's look over one nice effort....
Section 2.3: Prime Cuts of Numbers
- How many primes are there? There are infinitely many -- they just don't stop! (But how do we know? We prove this theorem!)
  - A natural number n may or may not divide natural number m evenly, but there's always a unique way of writing the attempt (that is, exactly one way -- no more):
    
    m=qn+r
    where 0 ≤ r ≤ n-1.
    Obviously, if r=0 then n divides m.
  - We want to show that there is a prime number bigger than any number you can give.
- Unanswered questions about prime numbers:
  - Goldbach Question: Can every even natural number greater than 2 be written as the sum of two primes?
  - Twin Prime Question: are there infinitely many pairs of prime numbers that differ from one another by two? (3,5; 5,7; 11,13; etc. -- can you find another pair in the sieve?)
  Mathematicians don't know the answers yet!;)
Section 3.1: one-to-one correspondences
- Part of your homework for the infinity unit: Mindscapes pp. 141-143, II.10; III.16, 18, 20. Please type up your solutions. The Infinity unit problems will be due Friday, 10/3.
- 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 start with the idea of number (e.g. What does "2" mean?).
- What is infinity? Is it a number? Is it just one thing?
- How do you know that two sets of objects have the same number of objects in them when you're not allowed to count?
- What is a one-to-one correspondence, and how is it useful?
- Let's look at some problems:
  - II.6, II.9
  - IV.22
Section 3.2 (Read for Monday): Comparing the Infinite
- 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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