A little counting

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In the movie Clan of the Cave Bear, Ayla is adopted into a clan of Neanderthals. She is possibly a Cro-Magnan, the next rung on the evolutionary ladder. During this scene, she has been with the clan for several years and appears to be an early teen. Her adoptive father is teaching her to count. He shows her that most members of the clan can count to three. "The clans knows this much." He can count to five . "Only Mog-ur knows this much." She then shows him how to count to ten and then twenty. Ayla is at the head of this class!

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Finger counting is very old and current. Many of you learned to count and add on your fingers. Do you remember the touch system? Counting is still important. Below are just a few of the finger symbols for counting between oriental merchants.

In the next illustration you will see symbols used in Oceana and by Bombay traders. It is a base five system of counting.


Very early forms of counting: The earliest records of counting do not come from words or fingers but from physical evidence --- scratches on sticks, stones, or bones. For example, the oldest "mathematical artifact" currently known was discovered in the mountains between South Africa and Swaziland.


It is a piece of baboon fibula with 29 notches, dated 35,000 BC. No one knows what it means. The Isango bone shown above (left) is from a wolf was found in Africa and is now in the Brussels Museum. The other stone, the so-called ochre stone, above was discovered just in early 2002 at the very southern tip of South Africa. (The complete story is at www.nsf.gov.)

Tallying: The double tally stick was used by the Bank of England. If someone lent the Bank money, the amount was cut on a stick and the stick was then cut in half --- with the grain of the wood. The piece retained by the Bank was called the foil, and the other half was called the stock. It was the receipt issued by the Bank. The holder of said became a "stockholder" and

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owned "bank stock". When the holder would return, the stock was carefully checked against the foil; if they agreed, the owner would be paid the correct amount in kind or currency. A written certificate that was presented for remittance and checked against its security later became a "check ".

Multicolored knotted cords, called quipus were also used by the Incas of Peru. The conquering Spaniards noted that in each Inca village there were four quipus keepers, who maintained complex accounts and performed a function similar to today's city treasurer, historian, and secretary. This particular example is actually a quipu-book that supposedly contains the history of the world.


The comedy of ciphering

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Lou has to make seven batches of thirteen donuts each for the officers. He cooks 28 donuts, because he claims that $7\times 13=28$. His partner Bud tells him that $7\times 4=28$. Lou shows him in three different ways that what he says is true, namely that $7\times 13=28.$

What kind of math is that?

Lou makes three mistakes:

(1) In the first "proof that $7\times 4=91$," Lou simply divides all wrong. It's clever. Do you know exactly his mistakes?

Note in the picture below the different form of division symbolism. The exact form does vary from place to place, and even though we think the methods we use have been around forever, some of them are quite recent.

MATHDid you know that during the 16 $^{\text{th}}$ century

was learned only in the most prestigious colleges???

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(2) In the second example, Lou puts the $7$ in the wrong place.

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(3) In the third example, Lou counts the one's as units, but actually they represent ten's.


Now its your turn.

  1. Can you find another pair numbers to multiply and correctly make all the mistakes that Lou did? (Hint. there is at least one pair.)

  2. Make a guess as to why the notches were carved into the Isango bone.

  3. Make a tally stick. See how impossible they are to forge.

  4. Whatever could the ochre stone mean? It is for counting? for art/design? for ceremony? (At this it is completely unknown; it may never be known with certainty.)


$\pi $ given to 7,000 places. .


$\pi $ is now known to 206,000,000,000 decimal places. Typing the value at 10 digits per inch, the length of the typescript would be MATH miles, more than the distance to the moon.

This document created by Scientific WorkPlace 4.1.