Post on 08-Jul-2015
description
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
.........
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