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# Knowledge Representation Systems From The Beginning 01- Numbers

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The representation for numbers

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### Knowledge Representation Systems From The Beginning 01- Numbers

1. 1. 2012 Long Count calendarThursday, April 4, 13
2. 2. Knowledge representation systems from the beginning Part one: Numbers Mingli YuanThursday, April 4, 13
3. 3. numeral representationThursday, April 4, 13
4. 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?Thursday, April 4, 13
5. 5. First known positional numeral system Babylonian numerals base-60Thursday, April 4, 13
6. 6. Maya numerals maybe the ﬁrst positional system with zero base-20Thursday, April 4, 13
7. 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... → numberThursday, April 4, 13
8. 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?Thursday, April 4, 13
9. 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 recursivelyThursday, April 4, 13
10. 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 orderThursday, April 4, 13
11. 11. Multiplication of Numbers an example of reasoning on a representationThursday, April 4, 13
12. 12. 13 × 238 = ? ✓ 1 238 2 476 ✓ 4 952 ✓ 8 1904 13 3094 Ancient Egyptian multiplication Ahmes PapyrusThursday, April 4, 13
13. 13. Chinese multiplication tableThursday, April 4, 13
14. 14. With invention of multiplication, people found many hard problems .........Thursday, April 4, 13