history-page6 - By 100 AD, the Chinese were handling...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 werent 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) wasnt 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 didnt explain what all lengths do look like....
View Full Document

Ask a homework question - tutors are online