Day 10: Heart of Mathematics

Last time: Fibonacci Nim
Today:
- Return homework: biggest problem was lack of explanation!
- Project ideas:
  Nice to see that some of you are thinking about it:
  - Nell: Fibonacci music
  - Josh: Mathematics of Chess
  - Here's a free one: Shapes of Nature. "How do we recognize and describe shapes and structures in natural objects?" This is a sort of generalization of Fibonacci numbers....
- Prime Numbers:
  - Start from the idea of decomposing (or factoring) natural numbers into products.
    Last time (in Fibonacci Nim) we decomposed natural numbers using Fibonacci numbers and sums.
  - We can find them using the Sieve of Eratosthenes
  - The first 100 Primes (primes come around a lot less often, the higher up you go in the natural numbers)
  - How many primes are there? There are infinitely many -- they just don't stop! (But how do we know?)
  - Even though primes appear to appear irregularly amongst the natural numbers, the relative number of primes in the first n natural numbers appears to approach a known value:
    
    Number of primes in first n natural numbers = n/Ln(n)
    Again, we see that as we head off to infinity (wherever that is!) a quantity is tending toward some nice limiting value.
  - Exercises: I.2, 7, 9, 10, 19; II.7.
Next time: More on Prime Numbers, and Fermat's Last Theorem

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