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# history-page6 - • By 100 AD the Chinese were handling...

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Other Numbers Positive Rationals Why are they natural objects to consider? Sharing. The problem of measurement. Around 1000 BC, the Egyptians used unit fractions (and 2 / 3). Unit fractions were written with an ellipse over the denominator. Writing proper fractions as sum of distinct unit fractions is an interesting exercise. Nonuniqueness: 1 /n = 1 /(n + 1 ) + 1 /(n(n + 1 )) Simplicity · 2 /n table in the A’hmosè Papyrus for odd n from 3 to 101. 1. Frequently uses the identity 2 n = 1 (n + 1 )/ 2 + 1 n(n + 1 )/ 2 . 2. When possible, it uses 2 3 m = 1 2 m + 1 6 m . 3. Multiplication based on doubling made this table especially handy. In ancient Babylon, the sexagesimal place-value notation extended to fractions. Boyer: “the best that any civilization a±orded until the time of the Renaissance." Ancient Greeks were ²ne with geometric concept of a ratio but didn’t really work with com- mon (as opposed to unit) fractions notationally.
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Unformatted text preview: • By 100 AD, the Chinese were handling fractions much as we do today. • The use of decimal fractions really got entrenched in the West around 1600. (This late date would help explain the old British monetary system, etc.) Irrationals • That 2 and other integers weren’t the square of any positive rational number was discovered by the Greeks around 400 BC. ◦ Arguably precipitated by their lousy number system. (Compare to epicycles.) ◦ Evidence seems to indicate that our standard algebraic proof (known to Aristotle) wasn’t used. ◦ Possibly a geometric proof based on a diagram like: and a calculation like p 2 = 2 q 2 ⇒ p/q = ( 2 q − p)/(p − q) . • This indicated that there were irrational lengths, but it didn’t explain what all lengths do look like....
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