Primitive Counting

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How might "primitive" people have counted? There are at least three good suggestions that I have heard of:

  • Tallies, starting perhaps with the Ishango bone: here's a video with its history.

  • One-to-one correspondence, and related methods (e.g. using body parts -- "one hand" of sheep, say -- meaning five); cairns are an example of a one-to-one correspondence, and a way of counting: 'An old Scottish Gaelic blessing is Cuiridh mi clach air do chàrn, "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.' (source)

    (Paul Scott has lots of other good material, including stuff on calculus!)

  • And then an unusual method of "counting by partitions" that Patricia Baggett and Andrzej Ehrenfeucht proposed at the 2011 National Math Meetings. I have modified their algorithm slightly (to conform more to the original document, from 1820, from which they took the idea).
    • 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, writing from right-to-left 9 is 1,0,0,1, and 22 is 1,0,1,1,0

    • 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:
      CountingTree.jpgWe write 1,0,1,1,0

      (or just 10110),

      recording the numbers as we find them from right to left.

      Notice that there will always be a 1 on the far left

      (representing the 1 in the triangle) --

      representing the last sheep.

      In the previous example, get 10110 by writing right-to-left.

    • 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?

    • Let's count some students this way....

Link to Binary Arithmetic

We certainly have seen that ancients practiced binary arithmetic (especially the Egyptians). My slight modification to the "rules" proposed by the authors of this method links the method directly to binary arithmetic. If you look at the representation of each number above, you will see that it is the binary expansion of the number backwards.

If we followed the method proposed by the authors, we would have stopped when we got to three or two items, and we wouldn't have reached the binary representation.