Day 11: Heart of Mathematics

Last Time

Next Time

Announcements:
- Remember, you have an exam this Friday. It will cover through the material on prime numbers that we're doing today.
- Your problems on Bases and Fibonacci are due today. All those who've brought in something from nature involving Fibonacci numbers, please bring those up and put them on the desk, with a postit note with your name on it next to your object.
Section 2.3: Prime Cuts of Numbers
- Decomposing numbers, part III:
  - We can "decompose" (or factor) natural numbers using prime numbers and products (that is, multiplication).
  - We decompose numbers using powers (bases), and with sums (non-consecutive Fibonacci numbers). Prime numbers and products is a third way.
  - 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: therefore 1 is not a prime number.
    - 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).
    - Example: #2, p. 77
  - The distribution of primes
    - The first 100 Primes (primes come around a lot less often, the higher up you go in the natural numbers.
      Here are the the primes under 10000 -- is 1729 prime? Check out 9551 and its neighbor....)
    - Example: #35, p. 80
    - Conclusion: gaps without primes of any size exist in the natural 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
Website maintained by Andy Long. Comments appreciated.