Long Count calendar

2012

Thursday, April 4, 13

Knowledge representation systems from the beginning

Part one: Numbers

Mingli Yuan

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representation

numeral

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Unary numeral system

The most natural way to express numbersbut how about the large numbers?

Tally stick

Ishango bonethe Upper Paleolithic era, about 18,000 to 20,000 BC

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First known positional numeral systemBabylonian numerals

base-60

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Maya numeralsmaybe the first positional system with zero

base-20

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Note 1

• Symbol systems expand the land of our mind

• Counting

• Stick symbol: object → stick

• Small numbers

• Base symbol: stick → base

• Big numbers

• symbol string: base, base... → number

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• Counting is the origin of numbers, so sticks (unary positional system) is obviously good enough.

• Questions are not so obviously trivial:

• Is the (non-unary) positional system faithful or soundness?

• Is the (non-unary) positional system complete?

Notes 2

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• Soundness can be established

• Clustering keep the total number of sticks unchanged

• Applying Euclidean division recursively

We borrow the symbol from Babylonian numeral to express a base-10 system

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• Completeness can also be established

• Positional symbols can be arranged into a natural order by the order of base symbols

• This natural order is the same with the counting order

↔↔↔↔↔↔↔↔↔↔↔↔

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Multiplicationof

Numbers

an example of reasoning on a representation

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Ancient Egyptian multiplication Ahmes Papyrus

✓ 1 238

2 476

✓ 4 952

✓ 8 1904

13 3094

13 × 238 = ?

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Chinese multiplication table

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With invention of multiplication, people found many hard problems

.........

Thursday, April 4, 13