Day 05, MAT115

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Question of the Day: How might primitive people have counted?
We're going to start by reviewing the "rules of the road" for fractions, and a curious application.
- Adding fractions is really the bane of students. For this, we need a common denominator. The rule is actually pretty simple:
  
  ${\frac{m}{n}+\frac{p}{q}=\frac{mq+np}{nq}}$
  So if you eat ${\frac{1}{4}}$ of the pie, and then ${\frac{1}{3}}$ of the pie, you've eaten a total of
  
  ${\frac{1}{4}+\frac{1}{3}=\frac{3+4}{12}=\frac{7}{12}}$
  of the pie. It's that easy!
  These types of fractions (with numerators of 1) are especially important in Egyptian division (which we'll study next week).
  Let's do that Egyptian fraction example on p. 15. It's a "do it again" situation:
  
  ${\frac{1}{3}+\frac{1}{4}+\frac{1}{8}+\frac{1}{168}}$
- Now: we want to consider an example of fractions that is ancient, and very very curious: Zeno of Elea, (ca. 490 - 430 BC)
  Is it possible to reach one wall from the opposing wall? Zeno argued that no, it's not possible.
  In the process of discussing this we'll add together an infinite number of fractions (there's a nightmare for you!):
  
  ${\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^n}+\cdots=1}$
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)
  - 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.
  - Let's begin by counting some students their way....
  - 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 1,0,1,1,0
    5. Can you go backwards? How many sheep is meant by 1,1,0,1,1,0,0?
  - The easiest way to illustrate the counting method is via a tree -- which we've seen before, when doing probabilities. Finding the sample space of four coin tosses was best done using trees.
    Let's see how we might use a tree to represent the solution to the 22 counting problem: in the linked example, we would get 10110 by writing the remainders from left-to-right starting from the bottom of the tree. (The result should always start with a 1 if done correctly, since we always end with one sheep!)
  - Can you go backwards?
    1. How many sheep is meant by 1,1,0,1,1,0,0?
    2. How many sheep is meant by 1,0,0,1,0,1,1,0,1?
    Try making a tree with these remainders.
  - and now let's count some coins:
    1. We'll be counting pennies - in small groups.
    2. Divide your penny rolls up (no need to do it evenly!)
    3. Get your answer, then swap with another nearby group.
    4. Compare (and record) your answers

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