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### Rhind Papyrus Problems

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Date: 03/23/99 at 10:37:41
From: Byron
Subject: The Rhind Papyrus equations

We would like to know the equations on the Rhind Papyrus.
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Date: 03/23/99 at 13:35:53
From: Doctor Rick
Subject: Re: The Rhind Papyrus equations

Hi, Byron, welcome to Ask Dr. Math!

I have been reading about Egyptian and other ancient math lately - I
find it fascinating!

The Rhind papyrus, named for the man who discovered it in 1858, is
sometimes called the Ahmes papyrus for the scribe who wrote it around
1650 BC. It is a roll 1 foot high and 18 feet long. It starts, as I
understand it, with some numerical tables - tables of "Egyptian
fractions." After these come 84 math problems with solutions. A lot of
the problems would look pretty simple to you - for instance, if you
have 1, 2, 6, 7, 8, or 9 loaves and you want to divide them equally
among 10 men, what fraction of a loaf does each man get?

This kind of problem was harder for the Egyptians because of the way
they wrote fractions. An Egyptian fraction is a sum of fractions that
all have 1 in the numerator (as we write it). Actually they would also
use 2/3; they had a special symbol for it.

Other problems are multiplication and division problems. Problem 13,
for instance, is to multiply 1/16 + 1/112 by 1 + 1/2 + 1/4. Problem 70
is to find the quotient of 100 divided by (7 + 1/2 + 1/4 + 1/8).

Still other problems deal with proportions. Problem 63: How do you
divide 700 loaves among 4 people in the ratio 2/3 : 1/2 : 1/3 : 1/4.

There are a few problems that might be called algebra, although I
think some math historians would not want to call it this. These were
still pretty simple given our algebraic methods, for instance, problem
24 (in algebraic terminology): what is x if x + x/7 = 19? They didn't
use the distributive property as we would. Instead they guessed a
value for x; say, 7. With this value, x + x/7 = 8. If you multiply x
by 19/8 (2 + 1/4 + 1/8 as an Egyptian fraction), you get the correct
value of x, which is 133/8 or 16 + 1/2 + 1/8. But my source says that
there is one such problem that is solved by the distributive property:
Problem 30, x + 2/3 x + 1/2 x + 1/7 x = 37, is solved by dividing 37
by (1 + 2/3 + 1/2 + 1/7).

That's about it, but remember, this is from 1650 BC, and it is
supposed to be a copy of a document 200 or more years older! The math
you learn in school has been built over many generations on
foundations like what we find in the Rhind papyrus. If you were left
to figure out math on your own, you might never come up with the
things the Egyptians had figured out.

My source is a book, _A History of Mathematics_, 2nd ed., by C. Boyer
and U. Merzbach, 1988.

You can find some on-line material on Egyptian mathematics here:

History Topics Index, MacTutor Math History Archives (look for
Babylonian and Egyptian Mathematics)
http://www-groups.dcs.st-and.ac.uk/~history/HistoryTopics.html

http://www.interlochen.k12.mi.us/Math/EgyptianMath/MathInEgypt.html

This last site includes this page on the Rhind papyrus (differing in
some minor details from what my source said):

http://www.interlochen.k12.mi.us/Math/EgyptianMath/Rhind.html

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
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