Day 10: Heart of Mathematics

Last time: Fibonacci Nim
Today:
- Reminder:
  - Assignment 1 will be due on Friday (the Thirteenth).
  - Project Assignment 1: by Friday, 9/20, pick your partner(s) (you can go alone!) for the poster project; it would also be nice if you have a topic, but that's not required!
- Project idea: Shapes of Nature. "How do we recognize and describe shapes and structures in natural objects?" This is a sort of generalization of Fibonacci numbers....
- Aaron asked: "What will test questions look like?"
  - There will be some factual information: e.g., "Write the first 10 Fibonacci numbers, and explain how they are generated," or "Describe the Pidgeonhole Principle."
  - There will be some questions like this, relative to your reading and what we've done in class: "Playing Fibonacci Nim from a starting position of 50, what is your next move if you began play with a removal of 1 counter, and your opponent then removed 2?"
  - And then there will be some problems that look more like the Mindscapes, for which you will need to show some insight.
- Prime Numbers:
  - Start from the idea of decomposing (or factoring) natural numbers using division. (Last time we decomposed natural numbers using Fibonacci numbers and addition!)
  - 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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