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

Modern mathematics including paper folding offers distractions from the central dual multiplication definition conflict. Multiplication defined as been repeated addition and scaling of rational numbers co-existed as main stream Western Tradition ideas 4,000 years ago, and maintained the tradition for 3,500 years.

Math historians report Egyptian fraction cultures formally used the paired multiplication definitions by 2050 BCE. Specifically, the Egyptian Middle Kingdom. Ahmes, a 1650 BCE scribe, recorded a 2/n table that scaled 2/3, 2/5, 2/7, …, to 2/101 to concise unit fraction series that followed a dual multiplication method.

Modern scholars scratched their collective heads during the 20th century when only reporting the additive side of the paired dual set of multiplication definitions. Ahmes 2/n table introduced 87 arithmetic, algebraic, geometric and weights and measures problems that required a dual understanding of the multiplication definitions.

Both sides of the multiplication definitions were needed by Ahmes, and Egyptian scribes, as scribes as late as Fibonacci in 1202 AD used to record the Liber Abaci, Latin speaking/writing Europe’s arithmetic, algebra, geometry and weights and measures instruction book for 250 years.

Of course, with the death of Egyptian fractions, and the birth of modern base 10 decimal arithmetic in 1600 AD, the ancient dual definition of multiplication conflict seemed to disappear. But has it?

I think not. Modern mathematical physics reports the same dual conflict in ways that would have made ancient Egyptian fraction scribes shake their heads.