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THE ORIGINS OF NUMERATION AND ARITHMETIC Pamela Leutwyler

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Once upon a time, in ancient Greece, there was a USED CAMEL DEALER.

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Each day, he counted his camels. This was difficult. They would not stand still. They all looked alike. And sometimes…

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It was cold and rainy.

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So one day, the used camel dealer got a good idea. Having just counted, he knew exactly How many camels were on the lot. He went into his showroom

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Where he built a shallow box on the floor In which he placed ONE pebble to represent each camel on his lot.

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If he bought 2 camels If he sold 3 camels If 2 camels are born If 1 camel dies

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This system was an improvement over running around in all kinds Of weather to chase and count the actual camels. But as he Prospered, the huge pile of pebbles became awkward to count.

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This system was an improvement over running around in all kinds Of weather to chase and count the actual camels. But as he Prospered, the huge pile of pebbles became awkward to count. Then he got a really good idea!

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The pebbles are temporarily removed Then partitions are built The first pile will function just as the original pebble board One pebblerepresents one camel

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Let’s view the pebble board from the top In this pile 1 pebble represents 1 camel he sells 1 camel 3 camels are born 1 camel dies he buys 5 camels

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In this pile 1 pebble represents 1 camel Instead of allowing the pebbles to pile up, the dealer makes this rule: When 10 pebbles accumulate in a pile, they are replaced with 1 pebble in the next pile

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In this pile 1 pebble represents 10 camels Instead of allowing the pebbles to pile up, the dealer makes this rule: When 10 pebbles accumulate in a pile, they are replaced with 1 pebble in the next pile

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Instead of allowing the pebbles to pile up, the dealer makes this rule: When 10 pebbles accumulate in a pile, they are replaced with 1 pebble in the next pile Ten tens become 1 hundred

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How many camels?

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This is beginning to look familiar. Representing a camel with a Pebble was the first level of abstraction. Next, piles of pebbles will Be replaced with symbols. Then, we do not need pebbles and boards. 325

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The laws of arithmetic arise from this pebble board

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65 48+

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65 48+ 3 Put down 3

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65 48+ 3 Carry 1 1

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65 48+ 3 Put down 3 Carry 1 1 1 Put down 1 Carry 1 1 1

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25 8-

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25 8- Borrow 1 1 1

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25 8- 1 1 1 7

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Why did the used camel dealer choose the number 10 ? Probably it was because he had 10 fingers. Our numeration system is called base ten. If we choose another number – lets say 7 - when we Make the rule about when to move pebbles, we generate a BASE SEVEN NUMERATION SYSTEM

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Module 3.3 Reading and Representing Three-Digit Numbers.

Module 3.3 Reading and Representing Three-Digit Numbers.

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