Day 3, MAT115

• Reminder: turn off your rectangles!
• Your first assignment is due next time, this Thursday. Please: show work! Don't just write down answers.

• You have a new assignment, due next Tuesday. Remember to justify your work!

We talked about some possible counting strategies:

• Tallies:
  

• One-to-one correspondence
  • using body parts -- "one hand" of sheep, say -- meaning five sheep.
  • cairns: 'An old Scottish Gaelic blessing is "Cuiridh mi clach air do charn, "I'll put a stone on your cairn." In Highland folklore it is believed that the Highland Clans, before they fought in a battle, each man would place a stone in a pile. Those who survived the battle returned and removed a stone from the pile. The stones that remained were built into a cairn to honour the dead.

• 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. Record the results to send to the King. (So how do we do that? Maybe string and knots?)

    The easiest way to illustrate the counting method is via a tree.

    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!)

    

    The answer will be written as 1, 0, 1, 1, 0, starting from the left with the final 1, which I always put in a triangle to distinguish it from the remainders.

    That is, from the bottom up, left to right. This is important! We have to have a consistent scheme for writing.

    Do you have any questions about this strategy?

    We also learned a few lessons from Sesame Street last time:

    • One-to-one correspondence with the "counting numbers" (we'll call them the natural numbers -- 1, 2, 3, 4, 5, ....).
    • Prime Factorization -- 6=2*3=3*2
    • "Blocking" numbers helps us to get the count: fish, fish, fish
      fish, fish, fish

• Now let me introduce the Babylonians (or, rather, their mathematics).
  • Babylon, then (source)
    

  • Babylon, today

  • I want to introduce you to their mathematics via a "detective story": I hope that you all like detective stories.... The story is called Detective in the times of Babylon. Let's see if we can solve this mystery before the end of class.

    Feel free to discuss with your neighbor!

    Prof. Buck guesses that the document comes from the city of Nippur, in what is now Iraq:

    

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• Time out for a card trick.... By the way, every card trick we do will have some mathematical relevance, or we wouldn't do it!

  As usual, on Thursday I'm hoping that some of you will explain how it works, for extra credit (an extra GOHF card). You need to write up your solution for extra credit.

  I'll be tougher this time, however: your "solution" must have some mathematics in it. It must, indeed, be a correct solution. You must be able to perform the trick yourself.

  • I shuffle the deck, then lay out eight cards, turned down.
  • I tell my assistant to choose three consecutive cards, examine them, remember them, and put them back in place (and I'm not looking -- I swear!).
  • I ask my assistant to sum them up (explaining about face cards, etc.), then I magically "read their mind", and tell them exactly what three cards they chose. How does the trick work?

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• What can we deduce about
  • the Babylonians and place value?
  • the purpose of the table?
  • the Babylonian's missing number(s)?
  • the way Babylonians looked at numbers and their world?
  • the Babylonian influence in our world today?

  How would the Babylonians write these numbers:

  1. 47
  2. 573
  3. 3600
  4. 14001

• Zero, My Hero