Table Of ContentCO U N T L I K E A N EG Y P T I A N
Princeton University Press
Princeton and Oxford
Copyright © 2014 by Princeton University Press
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Library of Congress Cataloging-in-Publication Data
Reimer, David, 1962- author.
Count like an Egyptian / David Reimer.
pages cm
Includes index.
ISBN 978-0-691-16012-2 (hardcover : alk. paper) 1. Mathematics, Babylonian.
2. Mathematics--Egypt. I. Title.
QA22.R38 2014
513.2 11--dc23
2013036208
British Library Cataloging-in-Publication Data is available
This book has been composed in Times Ten
Printed on acid-free paper. ∞
Printed in the United States of America
1 3 5 7 9 10 8 6 4 2
CO N T E N TS
Preface vii
Introduction ix
Computation Tables xi
1
Numbers 1
2
Fractions 13
3
Operations 22
4
Simplification 55
5
Techniques and Strategies 80
6
Miscellany 121
7
Base-Based Mathematics 144
8
Judgment Day 182
Practice Solutions 209
Index 235
P R E FA C E
A number of years ago a colleague of mine looked over it does. I spent a fair amount of time dissecting tables and
my class notes for a course in the history of mathemat- algorithms in an attempt to understand this aspect of their
ics. She pointed out that I had misinterpreted the ancient system. I used pedagogy to attempt to get into the mind
Egyptian method. At that time, Egyptian computation of the ancient mathematicians. Every teacher asks ques-
was only a small part of the class, but I wanted to make tions for a specific reason. If you look at adjacent ques-
sure that everything was done properly. So I pulled all tions in a math text, you will often see that although they
my history of mathematics texts off the shelf and reread are similar, the second one often requires one additional
the Egyptian sections. What I found hardly helped me at thought or method to solve. Noticing this, you can realize
all. There were only a few trivial examples followed by what the writer is trying to teach you with this question.
abstract discussions filled with equations completely out By examining the difficulty of the different parts of the
of context. computation, you can begin to grasp what they want you
I decided that I needed to learn Egyptian mathemat- to focus on and what is a trivial detail to be glossed over.
ics the old-fashioned way. I obtained a translation of the After doing this, I went back to my history of math
Rhind Mathematical Papyrus, the only known complete texts thinking that my newfound knowledge would give
Egyptian math text, and worked my way through it as I me greater insights into the authors’ discussions. How-
would any modern textbook. I could always get correct ever, upon rereading, I couldn’t help but feel that they
answers to the exercises, but more often than not, the just didn’t get it. They continually focused on insignificant
solutions in the papyrus were solved in a different way. details or challenging problems rather than on the themes
Even worse, their solutions took far less work than mine. I central to the Egyptian system. They expressed their ideas
would stare endlessly at the point where our solutions di- in algebra, which often overly complicates simple ideas.
verged and ask myself, what did they see that I could not? Since algebra in its current form would not exist for more
Eventually I developed an instinct for the subject. By than three thousand years after the creation of the Rhind
the second pass through the book, I could often match papyrus, using modern algebra was a bit off the point.
their solutions. By the third pass, I could occasionally bet- At about the same time, I had been working on my
ter them, but I would always understand why they made notes for the Greek section of my course. At first I alter-
the choices they did. I would also be aware of what knowl- nated sections on math and historical background. How-
edge I had and they didn’t that enabled me to beat them. ever, as I delved deeper into the development of Greek
Surprisingly enough, at my peak, I found that I could per- philosophy, I noticed that the two were inextricably linked.
form most computations faster in Egyptian than I could The ideas of a philosopher would never explain how to
with modern methods. solve a particular problem, but they would give great in-
I attacked the subject both academically and pedagog- sight into why they considered the problem important.
ically. Egyptian methods produce answers efficiently, but Further study revealed connections between Greek social
it’s not clear on casual observation why it works as well as structure, historical events, cosmology, and the changes in
viii preface
their mathematics. Slowly I learned to merge the cultural While I’m fairly satisfied with my attempts in this regard,
history and the math into single documents. The positive there are a few points late in the book where it was all but
feedback from students and colleagues reading my notes impossible to find meaningful context. Minor nuances in
convinced me that my efforts were well worth it. computational methods don’t lend themselves to interest-
This book is the result of these two influences. I’ve be- ing anecdotes. In these cases, I’ve included some histori-
come convinced that you can’t appreciate Egyptian math- cal background in order to maintain the tone of the book.
ematics without doing it. Just as you can’t understand Often the connection to the material is metaphorical at
baseball by examining lists of dry statistics, you can’t learn best, but hopefully it will appeal to readers who appreci-
Egyptian math with a sampling of abstract discussions. ate the historical aspects of Egypt.
As a former professor of mine once said, “Math is not In order to make the book more accessible and ap-
a spectator sport; you need to play the game.” In order pealing, I’ve kept the discussion light and the rhetoric
to do this we need a complete working mathematical conversational. I’ve included diagrams, humor, and an-
system. The truth is that we can’t be sure how Egyptians ecdotes drawn from many historical periods. Like any
performed some of their mathematics, because they often good storyteller, I’ve accentuated the drama and made
skipped steps or simply listed some things in table form villains and heroes out of the characters within. I’ve also
with no discussion as to how they were derived. To rem- kept the mathematics light. You will need little beyond
edy this, I’ve filled in the gaps giving possible methods. addition, subtraction, and the multiplication of whole
Often I will give more than one interpretation. I’ve tried numbers. Each section takes the reader in small, easily
to clearly state when something is known and when it is understood steps. This is not intended to be a book over-
conjectured. stuffed with information but rather an easy read that
I’ve also done my best to give all of the sections some slowly but surely leads you to a deep understanding of
context consisting of light historical background. Giving the material.
someone a good reason to know something is as impor- This book could be used as a supplementary text for a
tant as conveying the knowledge itself. It not only helps us history of mathematics course. It could also serve as part
appreciate the information but also makes us more likely of an interdisciplinary course. However, it is intended to
to retain it. We need to set the material apart from the be a light enjoyable read accessible to anyone at the ju-
constant barrage of data overload we so readily discard. nior-high level on up.
I N T RO D U C T I O N
choices more carefully. As my teacher rightly pointed out,
English Is Stupid
anyone who speaks English can’t possibly criticize Span-
Far too many decades ago, I was sitting in a junior-high ish for its relatively few aberrations.
Spanish class. The topic of the hour was the conjugation Fortunately, all horrible things have to come to an
of verbs, but one of them didn’t follow the standard rules. end, and the bell rang signaling the conclusion of class. I
This bothered my budding mathematical sensibilities, slipped out desperately trying to avoid eye contact with
which required that all things follow well-regulated pat- the teacher. Before the next meeting of Spanish club,
terns. I asked the teacher why this word was different, and I had nothing left to do but reflect on my experience.
she simply responded “That’s just the way it is,” to which I I learned two important lessons that day. The first is to
replied, “Well, Spanish is stupid.” think before you speak. The second is that it’s very easy to
As her face turned red, I immediately realized my mis- perceive flaws in a system that’s alien to your experience.
take. She was upset for a variety of good reasons, one of At the same time, it’s very difficult to give it a fair evalua-
which was the comment that came from me. I was president tion of your own, because familiarity often leaves us blind
of the Spanish club. This “club” was her euphemism for her to our systems’ limitations.
detention hour where she sent students who didn’t do their Egyptian mathematics has an alien feel to it. Most
homework. I was president merely because I had logged in math historians refer to it as primitive or awkward. Even
the most hours that year. She was also upset because no worse, many simply ignore it except for a passing refer-
one likes some smart-aleck preteen to dismiss their life’s ence. They look at this system and feel uncomfortable be-
work in a few words. But I suspect what bothered her most cause it’s so different. They perceive apparent “flaws” and
was my comment itself, which was, in fact, stupid. move on. They don’t understand Egyptian mathematics
She then began a long rant, not directed at me, but simply because they don’t do it enough to truly appreciate
rather at the English language. She proceeded to list every it. To someone who’s mastered it, Egyptian mathematics
English inconsistency that popped in her head. Consider is beautiful. It scorns memorization and rote algorithms
the plural nouns “dogs,” “mice,” and “oxen.” Can you see while it favors insight and creativity. Each problem is a
any consistent rule for pluralization? Now consider verb puzzle that can be solved in many ways. Frequently, solu-
conjugations. The present-past tense pairs “am”/“was,” tions will be surprising, something that never happens in
“run”/“ran,” and “eat”/“ate” have no pattern whatsoever. the step-by-step drudgery that is modern computation.
The spellings of the English language are even worse. There are a number of good reasons to learn Egyptian
English is consistently inconsistent. In retrospect, I can’t mathematics. Puzzle lovers will find it fun and challeng-
even imagine how I learned the language. I suspect it ing. History lovers will gain a new insight into the Egyp-
would have been easier to invent a time machine, go back tian mind-set. However, I believe the most important
into the past, maim Daniel Webster, and leave a threaten- reason to study Egyptian mathematics is because it so
ing note to any dictionary-writer wannabes to make their different. We’re taught throughout our entire education
