The document discusses the history of numerical representation systems from ancient times to their modern positional form. It describes early counting systems using tools like tally sticks and the Ishango bone that used markings to represent small quantities. Later, the Babylonians developed a positional numeral system in base-60 around 3000 BC and the Maya also used a positional system in base-20 around AD 400. The document then examines how positional systems establish soundness and completeness to represent all numbers and discusses early methods for multiplication problems using representations like in the Ahmes Papyrus.

1. 2012
Long Count calendar
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2. Knowledge representation
systems from the beginning
Part one: Numbers
Mingli Yuan
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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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