Fraction Division Models in Ancient Egypt

by Milo Gardner

This is a description accompanying the Fraction Division page. Here is the Math 2.0 email group discussion thread where it came up.

Two of your three methods

- Rectangle model
- Common denominator
- Ratio to one

were recorded in 1650 BCE by Ahmes, and Egyptian fraction hieratic texts that date to 2050 BCE. For example, a fourth modern method "invert and multiply" method is contained in your post, as well as several of Ahmes' problems.

Several of of Ahmes' problems

http://ahmespapyrus.blogspot. com/2009/01/ahmes-papyrus-new- and-old.html

"divided and multiplied" . One problem was RMP 38.

In RMP 38 Ahmes began with a hekat, a unit of modern 4800 ccm size. The volume unit was scaled to 320/320 writing 320 ro and multiplied by 7/22 per:

1. 320 ro x 7/22 = 2240/22 = 101 + 9/11 (in unit fractions that need not be cited here)

A scribal proof of the accuracy of the 101 + 9/11 vulgar fraction answer was provided by

2. (101 + 9/11) x 22/7= 2240/22 x 22/7 = 320 ro

Ahmes shows to modern students that numbers partitioned by 1/n multiplications (statements) can be returned to initial numbers by inverting 1/n to n and multiplying (proof).

A modern version "divide and multiply" method applies properties of the ancient common denominator, unitizing, and proportion ideas contained in Egyptian fraction arithmetic.

The ancient "divide and multiple" method was also discussed in the Berlin Papyrus: http://planetmath.org/ encyclopedia/ BerlinPapyrusAndSecondDegreeEq uations.html, by solving two 2nd degree equations.

My view is that the history of mathematics offers students several ancient doors to understand modern arithmetic operations, including fractional ones that introduce algebra.