Day 06, MAT115

Last Time

Next Time

Announcements:
- Hey, do you know what day this is?
- Today is Friday: you have an assignment due today!
- You also have a new assignment
- Last time:
  - Let's make a deal!
    - Probability "packing".
    - Using generalizations of a problem to get a clearer idea of how to solve the given problem.
    - Using simulations to check theoretical results.
Today:
- Any questions on weather prediction? RE my on-line description of The Birthday Problem (and use it to figure out if it's going to rain by Saturday). No rain yet....:(
- Here's our "Question of the day":
  
  How do we break natural numbers down using prime numbers?
- The history and make-up of numbers.
  - Of course we'll start with the simplest numbers, which are the natural numbers:
    - The natural numbers are the counting numbers: 1, 2, 3, ....
    - All natural numbers are interesting, and we can prove it!
      - Is 1 interesting?
      - Is 2 interesting?
      - Suppose there were a first uninteresting natural number....
    - The number 1729 is especially interesting: the story of Ramanujan and Hardy
      This story illustrates that we can break down numbers in interesting ways.
  - Now, zero didn't make it right into the mix: it just confused the heck out of everyone. Next came
    - "rational numbers" -- ratios of natural numbers
    - "irrational numbers" -- e.g. the side length of a square with sides of length 1
    - negative numbers
  - But then finally zero had its day -- what's its history?
    - Our author says that "Mathematics could not function without zero." (p. 7) How do we know that he's wrong?
    - What civilizations were the first to include zero as a number?
  - Prime Numbers: breaking numbers down by multiplication
    - 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.
        The general procedure (and one that we'll see repeated time and again in mathematics):
        
        do it once, then do it again, and again, and again.... until done (or tired).
        I'll call this out sometime as "do it again...."
      - Upshot: We can "decompose" (or factor) natural numbers using prime numbers and products.
- How might "primitive" people have counted? There are at least three good suggestions that I know of:
  - Tallies
  - One-to-one correspondence (see Idea #7, Infinity) (e.g. using body parts -- "one hand" of sheep, say -- meaning five); cairns
  - And then an unusual method of "counting by partitions" that Patricia Baggett and Andrzej Ehrenfeucht proposed at the 2011 National Math Meetings.
    - They proposed that primitive societies may have counted this way. Let's suppose you need to let the King know how many sheep you have:
      1. divide your sheep equally ("one for you, one for me") into two pens: either there is one left over, or not. You make a note of whether there is one left over or not.
      2. Send all the sheep in pen two (and any "left over") out to pasture, and then
      3. You divide the sheep in pen one into pens one and two: i.e., just do it again! And again, and again, and.... until you get down to a pen one with just sheep in it.
      4. Now let's see how we might record the results to send to the King.
        
        9 sheep
        22 sheep
        54 sheep
      So 9 is 1,0,0,1, and 22 is 0,1,1,0,1 Can you go backwards? How many sheep is meant by 0,0,1,1,0,1,1?
    - Let's count some students this way....
Links:
- The Birthday Problem (and using it to figure out if it's going to rain)
- The Monty Hall Dilemma Revisited: Understanding the Interaction of Problem Definition and Decision Making. Peter R. Mueser and Donald Granberg, May 1999
- A discussion of the Monty Hall Problem
- A simulation of the game
- Another simulator
Website maintained by Andy Long. Comments appreciated.