*Count Like An Egyptian*is a hands-on explanation of how they handled fractions and multiplication and division. It is pitched at a level suitable for junior high school students, but is mostly not annoying for those who know a lot more, either about mathematics or about ancient history.

At the core of *Count Like an Egyptian* is a practical explanation
of how ancient Egyptians represented and manipulated fractions — as
"unit fractions" or sums of natural number inverses, along with two
thirds as a special case — and multiplied and divided — using repeated
doubling and halving. Ancillary to this are tables for such things as
doubling odd fractions (the equivalent of our times tables) and rules
for simplification, along with some general tricks and tools.

The core of this could have been fitted into a twenty page scholarly article (though no such article appears to exist), but has been fleshed out here with additional material to make it more useful and accessible to a broader audience. First of all, there are extensive worked examples and practice exercises, with solutions, which make it possible to learn how to do Egyptian mathematics rather than just learning how it was done.

Then there's a lot of historical and cultural background, conveying the context for Egyptian mathematical problems and something of the world in which the scribes who dealt with them lived. There are some outright digressions here, such as the story of the conflict between Horus and Seth, and Reimer writes at one point "I've decided to tell you about something that I personally find amusing but that has no link to the mathematics that follows", with a brief quote from a letter about students living the high life.

*Count Like an Egyptian* also touches on other ancient societies and the
history of mathematics more broadly: Archimedes and Fibonacci and ErdÃ¶s
get mentions. More substantially, a whole chapter describes Sumerian and
Babylonian base-based number systems and compares them with the Egyptian
approach. And a concluding chapter convincingly argues that Egyptian
methods are neither worse nor better than modern methods, just different.

"Egyptian mathematics presents us with a series of choices, and to make choices you have to think. Modern math gives us rules and steps. Performing its operations requires rote memorization, repetition, and perseverance. It may be tedious and boring, but if you follow the instructions to the letter, eventually you will be rewarded with the correct answer. This is the exact opposite of Egyptian math. To the uninitiated it often seems impossible, but that's because they don't know which choices to make and continually make the wrong ones. An Egyptian computation can take untold pages or a handful of lines depending upon the skill of the mathematician."

The full extent of ancient Egyptian mathematics remains unknown:

"We base all that we know about Egyptian mathematics on two ancient papyruses and a handful of scraps. One of the two papyri [sic] is incomplete. The idea that this is enough to judge the full mathematical capabilities of a civilization that lasted more than two and a half thousand years is a bit ridiculous."

Reimer gives us some glimpses of the uncertainties and even guesswork
involved in his own reconstruction, for example with an unknown solid
called a *nb.t* that has a diameter and resembles the word for "hill".

This is a lovely hardcover volume, with glossy paper and colour figures. The illustrations are mostly just decorative, with example calculations, for example, presented on coloured "scrolls", but this has been done so as not to sacrifice legibility. And in general the layout and presentation are clear and attractive.

*Count Like an Egyptian* assumes only simple numbers (fractions and
decimals), basic arithmetic (division and multiplication) and geometry
(areas and volumes of simple shapes and solids) and is pitched most
obviously at junior high school students, and perhaps precocious
elementary school students, and their teachers. It may inspire students
to think about what numbers are, by offering a different perspective on
how they are represented, and should give them a concrete understanding
that different methods exist for doing multiplication and division.
Every high school library should have a copy of *Count Like an Egyptian*,
but it would make a lovely present for anyone, of any age, with an
interest in arithmetic and ancient Egypt.

January 2015

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