Thank you!

]]>Glad you are enjoying the series. Scholars have figured out pretty well what the medieval algebraists were doing. It is hard for us to fully understand their methods because we are so familiar with numbers systems in a way they were not back then. For example, we are comfortable with using negative quantities in manipulating equations, but that was not the case back then. I consulted from time to time with Prof Jeffrey Oaks when writing about early algebra. This paper by him provides a good explanation of what they were doing.

]]>There and here, you use the terms “restoration” and “confrontation” in describing the methods of both Diophantus and Al-Khwarizmi. This seems to suggest they were doing basically the same things, but you don’t say so explicitly. How different or similar would you say their approaches were? Can we really know at this far remove?

]]>I hope which you will proceed sharing your wisdom with us.

]]>Thank you for the post. Egyptian algebra has been labeled rhetorical algebra by math historians. Unknown values were rhetorically discussed before hard-to-read scribal shorthand calculations were written down.

RMP 32 is a case in point. The algebra is trivial:

x + (1/3 + 1/4)x =2

is solved today by

(19/12)x = 2

x = 24/19 = 1 + 5/19

Ahmes solved the problem the same way, adding three level numeration aspects (proto-number theory) that otherwise well-informed scholars failed to parse as Ahmes recorded.

One aspect of Ahmes’ work scaled 5/18 by 12/12 such that

.

60/228 = (38 + 19 + 2 + 1)/228 concluded

x = 1 + (1/6 + 1/12 + 1/114 + 1/228

A second aspect scaled the entire equation by 114 scaled

http://planetmath.org/encyclopedia/SCALEDEQUATIONSRMP32.html

A third aspect included a proof that scaled the remainders to 912.

Greek algebra followed the same scaled arithmetic logic that.recorded rational numbers in exact unit fraction series. .

By the time of Diophantus, indeterminate equation algebra flowered in ways that are explained by the arrival of the Chinese Remainder Theorem (CRT) on the Silk Road. Fibonacci included a medieval version of the CRT in the “Liber Abaci” that Diophantus would have recognized..

Fibonacci’s arithmetic included exact unit fraction series the used of an algorithm. Medieval number theory scaled rational numbers n/p such that (n/p – 1/m) = (mn = p)/mp set (mn -p) = 1 as often as possible, a notation introduced by Arab algebra around 800 AD.

Conclusion: Egyptian, Greek, Arab and medieval multiplication included a dual definition that scaled rational numbers and repeated addition, as Maria is discussing.

Best Regards,

Milo Gardner

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