I'll give you a study guide next Tuesday. In the meantime, you should at least be aware that you should have done all the assigned reading: you will be responsible for the content, even if we have not explicitly studied it in class.
I will put a few modest questions on the exam, to check to see if you did your reading.
- What are natural numbers?
- What is a prime number?
- A natural number that can be divided evenly (that is, without remainder) by only two distinct natural numbers: 1 and itself.
- NOTE: when mathematicians say "distinct" natural numbers, they mean that they need to be different; and so the number 1 is not prime.
- We can find the primes using the Sieve of Eratosthenes (here's an animation)
- Prime Factorization Theorem:
- Every natural number greater than 1 is either prime, or it can be expressed as a product of prime numbers (in one and only one way -- order of the product aside).
- Examples:
- 42=2*21=2*3*7
- 8=2*2*2 (prime factors can repeat)
- 1729 = 7*247 = 7*13*19
- If you can't factor by any prime less than or equal to the number's square root, then the number is prime.
- 127: its square root is approximately 11.27. We don't need to go past 11 to find its factors.
- Upshot: We can "decompose" (or factor) natural numbers using prime numbers and products.
- Bunnies and Leonardo Pisano Fibonacci
- Start with a new-born pair of "immortal" bunnies (they never die).
- A new-born pair requires a month to mature.
- The following month, the pair produces a new pair, which is subject to the same rules.
- Do it again, do it again, do it again!
- The Fibonacci numbers as a recurrence relation
(do it again, do it again, do it again!). - Fibonacci
Numbers and Nature:
- Let's look at some pine cones....
- dandelions (Here's a scanned dandelion)
- Spirals in daisies:
- Similar things happen in artichokes, pineapples, etc.
- Fibonacci number decomposition and Fibonacci Nim
- Rules:
- There are two players, Player 1 and Player 2. Player 1 will go first.
- An arbitrary number of counters is thrown down (e.g. toothpicks, pennies).
- Player 1 must pick up at least one counter, but may not pick up all the counters.
- Player 2 must then pick up at least one counter, and not more than twice the number of counters picked up on Player 1's turn.
- The players continue taking turns, subject to these same conditions: pick up at least one, no more than twice what the preceding player took.
- The player who picks up the final counter wins.
- Let's play a sample game: Suppose that there are 27 counters on the table.
- You play a sample game: Suppose that there are 43 counters on the table.
- Rules:
- More
details on Fibonacci numbers and nim. It turns out that
Every natural number is either
- Fibonacci, or
- can be written as a sum of non-consecutive Fibonacci's in a unique way.
Examples?
