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Re: Who discovered irrational numbers?
Posted:
Mar 19, 2009 8:46 AM
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Franz,
Thank you for your series of posts today. Generally, I disagree with your methodology. However, when you mention specific RMP or other available ancient problems, as you did today with RMP 34, the scribal details of the story can be easily placed aside your often confusing work.
For example. RMP 34 and RMP 31 are matched problems. As usual, Ahmes starts with the most complicated, RMP 31, stating and solving
RMP 31: x + (2/3 + 1/2 +1/7)x = 33
and then going on to,
RMP 34: x + (2/3 + 1/2 + 1/7)x = 37
To solve both RMP 31 and 34 as a matched pair:
collect: x + (28 + 21 + 6)/42 = 33 and 37
(97/42)x = 33 and (97/42)x = 37
applying Ahmes' cross multiplication, as we do today
x = (42*33)/97 and x = (42*37)/97
solving to the vulgar fraction level,
x = 14 + 28/97, and x = 16 + 2/97
and, converting to Egyptian fractions
x = 14 + 26/97 + 2/97, and x = 16 + 2/97
solving: 2607 and 2/97
26/97 *(4/4) = (97 + 4 + 2 + 1)/(4*97) = 4' 97' 194' 388'
and,
2/97 *(56/56) = (56 + 8 + 7)/(56*97) = 56' 679' 776'
thereby, exactly and easily finding Ahmes' unit fraction answers -- as Franz did not cite in historical attested detail following Ahmes' easy to read algebra and arithmetic operations.
Best Regards,
Milo Gardner
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