The representation for numbers

1. 2012
Long Count calendar
Thursday, April 4, 13

2. Knowledge representation
systems from the beginning
Part one: Numbers
Mingli Yuan
Thursday, April 4, 13

3. numeral
representation
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4. Ishango bone
the Upper Paleolithic era, about 18,000 to 20,000 BC
Tally stick
Unary numeral system
The most natural way to express numbers
but how about the large numbers?
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5. First known positional numeral system
Babylonian numerals
base-60
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6. Maya numerals
maybe the first positional system with zero
base-20
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7. 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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8. Notes 2
• 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?
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9. We borrow the symbol from Babylonian numeral to
express a base-10 system
• Soundness can be established
• Clustering keep the total number of sticks unchanged
• Applying Euclidean division recursively
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10. • 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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11. Multiplication
of
Numbers
an example of reasoning on a representation
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12. 13 × 238 = ?
✓ 1 238
2 476
✓ 4 952
✓ 8 1904
13 3094
Ancient Egyptian multiplication Ahmes Papyrus
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13. Chinese multiplication table
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14. With invention of multiplication,
people found many hard problems
.........
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