Day 02, MAT115

• Announcements:
  • Reminder: turn off your rectangles!
  • You have your first assignment, to turn in next Thursday.
  • The book will be in the bookstore by the end of the week.

• Today we're going to talk about primitive counting:
  How would primitive, numerically illiterate people have counted?

• If you want to talk primitive, how about muppets?! Where do kids learn to count, but Sesame Street!

  I asked you to do the reading "From Fish to Infinity" for today, in which we learn the answer to one of life's persistent questions:

  What is "six"?

  We learn something of that, anyway -- at least about six fish. But

  "Six is more ethereal than six fish.... It applies to six of anything..., the ineffable thing they all have in common." (Steven Strogatz, author of From Fish to Infinity)

  We learn that

  • "...numbers are shortcuts for counting by ones...."
  • Furthermore, "...Humphrey might realize he can keep counting forever." -- something that will be very important when we come to talk about infinity.

  Did you notice Humphrey using any other cues to help him keep track of the number of fish?

  We encounter three important things in this example:

  1. Counting is one-to-one correspondence (it won't always work for infinity)
  2. "Blocking" is important -- fish, fish, fish
  3. Prime decomposition

• Let's dive into today's new topic: How might "primitive" people have counted? (We'll use a mathematical concept called a "tree" (a type of graph) to help us. More on those later....)

  There are at least three good suggestions that I know of:

  • Tallies

  • One-to-one correspondence
    • using body parts -- "one hand" of sheep, say -- meaning five sheep.
    • cairns

  • And then an unusual method of "counting by partitions" that Patricia Baggett and Andrzej Ehrenfeucht described at the 2011 National Math Meetings (it's from a 1827 book by John Leslie -- The Philosophy of Arithmetic -- in which Prof. Leslie describes how "savages" may have developed arithmetic.)
    • 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.

         The easiest way to illustrate the counting method is via a tree, which is a mathematical object in its own right.

         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

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

         So how do we write

         • 9 sheep
         • 31 sheep
         • 54 sheep
         Card trick interlude....

      5. Now: can you go backwards? How many sheep is meant by
         1. 1,1,1,1,0,1
         2. 1,1,0,1,1,0,0
         3. 1,0,0,1,0,1,1,0,1

         Try making a tree with these remainders. Notice that each way of writing has a leading 1 on the left. These representations always do.

      6. and now let's count some coins:
         1. We'll be counting pennies - in pairs.
         2. Split your penny rolls into two parts (no need to do it evenly!), and each of you use this method.
         3. Then try to deduce the number of pennies your partner counted.
         4. Compare (and record) your answers.

fish fish fish
fish fish fish