Day 08: Heart of Mathematics

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Announcements:
- New Assignments:
  - Reading Assignment: for Monday, read sections 3.1 (very short) and 3.2 (infinity!).
  - To hand in (due in a week: 2/18): Section 2.2, problems 16-24.
- Reminder: Assignment 3 (due Monday, 2/16): note -- I made a typo on the first problem.
Today: More on Number Systems:
- Decomposing numbers, part III: we can write natural numbers using Fibonacci numbers.
  - 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, -- what's next?
  - II.6: bunnies and Leonardo Pisano Fibonacci
    We can model the Fibonacci numbers using a coin toss.
  - Fibonacci Numbers and Nature
  - Fibonacci number decomposition and Fibonacci Nim
    - Rules:
      - Two players, n sticks/coins/bottlecaps/etc.
      - First player must take anywhere from 1 to (n-1) sticks
      - From then on, the next player may take from 1 up to twice what the previous player took.
      - Winner is the player who picks up the last stick
    - Is there a winning strategy?
    - Is there a perfect defensive strategy?
    - Let's look at the early examples: If you looked at some special cases, what can we conclude?
      
      Number of sticks 1 2 3 4 5 6 7 8
      
      Winner X P? P? P? P? P? P? P?
  - We "decompose" natural numbers using Fibonacci numbers and sums:
    every natural number is either
    1. Fibonacci, or
    2. can be written as a sum of non-consecutive Fibonacci's in a unique way.
    Examples?
    Can we justify this?
    - If a number is Fibonacci, then we're done.
    - Assume a number is not Fibonacci:
      - then there is a largest Fibonacci that "fits inside it" -- e.g. 24 = 21 + 3.
      - If the remainder (3, in the example above), after subtraction, is Fibonacci, how do we know that it is not consecutive (that is, that it is not 13)? That is, how do we know that the two Fibonacci numbers are not successive Fibonacci numbers?
      - If it is not Fibonacci, can we not simply repeat the current process of looking for a sum for a non-Fibonacci number, but using the remainder instead of the original number? (i.e. can we not simply "recurse" -- that is, do it again, and so construct a chain of numbers leading down towards 1.
        Example: 33=21+12=21+8+4=21+8+3+1
- Section 3.1: one-to-one correspondences
  - 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.
    
    Here's another one, by Robert Graves: Warning to Children
  - As we approach infinity, we start with the very idea of number: 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

Website maintained by Andy Long. Comments appreciated.

Number of sticks	1	2	3	4	5	6	7	8
Winner	X	P?	P?	P?	P?	P?	P?	P?