Date: Nov 9, 2012 2:01 PM
Author: Milo Gardner
Subject: Re: Why study Egyptian fraction math?

Arithmetic and geometric proportions offer twoi of the highest forms of Egyptian fraction math. Arithmetic proportions are included in"

http://planetmath.org/encyclopedia/RMP69AndTheBerlinPaprusProportionMethod.html

MP 69 and the Berlin Papyrus: A Proportion Method

This entry outlines scribal shorthand data by translating raw unit fraction data. The shorthand data has been made readable by adding back missing data that creates accurate scribal longhand data recorded in modern base 10 arithmetic.

There are three reasons for modern students of Egyptian mathematics to study the 1650 BCE Rhind Mathematical Papyrus(RMP) problem 69. All three reasons solve two Berlin Papyrus second degree equations in scribal algebra.

The RMP problem converted 3 1/2 hekats of grain that made 80 loaves of bread into a pesu unit. Scholars Schack-Schackenburg, early on, commented on a proportion method that a Berlin Papyrus scribe, and Ahmes, the RMP scribe, utilized as a common mathematical tool.

To outline the math shorthand used by Ahmes, the RMP scribe, a three phase proof will be analyzed.

1. Initially 3 1/2 hekats of grain meal, that produced 80 loaves of bread, were combined into pesu units for distribution purposes. Ahmes' first phase calculated the pesu as a rational rational number by applying a proportion.

Ahmes conversion of 7/2 hekat, making 80 loaves of bread, to 22 18/21 pesu was achieved by inverting 7/2 to 2/7 and multiplying by 80, reporting:

80 times 2/7 equaled (160/7) pesu and (22 + 2/3 + 1/7 + 1/21) pesu

by applying the Old Kingdom duplation multiplication method.

Modern conversions of hekat, loaf, and pesu date to modern rational numbers discussed the same proportion method used in the Berlin Papyrus. In modern fractions the Berlin scribe found two squared areas, one 10 cubit by 10 cubits, and the second 20 cubits by 20 cubits, by considering the proportions: 1: 3/4, and 2: 1/3.The two Berlin Papyrus problems solved second degree equations

2. Ahmes proved the answer by returning its pesu unit fractions to 80 loaves of bread. This was done by:

(22 + 18/21) pesu times 7/2 equals 80 loaves of bread,

by applying the Old Kingdom duplation method.