The two guidelines used by Reimer to write this book are that to really understand how the Egyptians in the time of the pharaohs did mathematics and in particular how they did their computations, one needs to understand some of its culture, but most of all, you have to learn to think the Egyptian way, and that means in the first place to forget how we do our calculations. As he teaches the history of mathematics, he makes his students do computations in the Egyptian system and illustrates that there are several advantages to their way of doing, and that the system is not as weird as it seems to be. So what he has compiled here is a course in Egyptian arithmetic that is pleasantly illustrated with historical and mythological stories.

Reimer's adagio is that you cannot understand Egyptian math without doing it. If you are a native English speaking person, learning a foreign language such as French is difficult because the rules are different but most of all you are very sensitive to all the exceptions to the rules, while in English there are perhaps even more exceptions that you forgot about, since you speak it without consciously thinking about the rules. The same is true for Egyptian mathematics. In our system, we have to look up (or learn by heart) all the multiplication tables. This is not needed is Egyptian arithmetic, but there were other tables to be used to sum and/or simplify numbers.

Let me give a brief idea of what it means to 'count like an Egyptian'. The Egyptians had different symbols for $10^k$ for $k=0,1,2,3,4,5,6$, the latter being almost equivalent to infinity. Repetition of these symbols denoted multiples of these quantities. However, Reimer switches to our more familiar notation for integers. Egyptians also had a notation for unit fractions of the form $1/k$. They just wrote the number $k$ under a symbol of a mouth. This is simplified by Reimer for example as $\overline{4}$ to denote 1/4 or 0.25. Placing the symbols next to each other means summation as in our positional system. So 3.5 is denoted as $3\;\overline{2}$ and an approximation for $\pi$ would be $3.14=3\;\overline{10}\;\overline{25}$. You immediately see aberrations from our system like double digits $7.4=7\;\overline{5}\;\overline{5}$ and nonunicity as in $7.25=7\;\overline{5}\;\overline{20}=7\;\overline{4}$. The latter is shorter, but the first has the advantage to see that if there are more digits that are left out, then the "rounding error" is at most $\overline{40}$. Multiplication and division were relatively simple because the only thing one needs to do is to repeatedly multiply and/or divide by 2 and add some numbers. For example to divide 115.5 by 42, one constructs the table with rows $[2^k~~2^k42]$ and select in the right column the numbers that sum up to $115\;\overline{2}$ (see check marks). Adding the corresponding powers of 2 in the first column gives the quotient. Thus $$ \def\check{\mbox{✓}} \begin{array}{rrr} & 115\;\overline{2}& \\\hline 1 & 42 & \\ 2 & 84 & \check \\ \overline{2} & 21 & \check \\ \overline{4} & 10\;\overline{2} & \check \\\hline 2\;\overline{2}\;\overline{4} & & \end{array} $$ giving $2\;\overline{2}\;\overline{4}=2.75$. To multiply $2\;\overline{2}\;\overline{4}$ by 42, the same table is used but now checking the rows whose elements in the first column add to $2\;\overline{2}\;\overline{4}$ (which are obviously the same rows) and adding the corresponding numbers in the second column (which clearly gives back $115\;\overline{2}$).

There was one exception to the use of only unit fractions namely $2/3$, denoted by Reimer as $\stackrel{=}{3}$. This exception is not so surprising in view of the powers of 2, since it is division by "one and a half". So, instead of powers of 2, one may also put $\stackrel{=}{3}$ or $\overline{3}$ in the left column as a shortcut. Moreover since a row like $[8~~~7]$ in the table actually represents an equality $8x=7$ or $8=7y$, depending on whether you multiply or divide, it can be turned around into an equivalent row $[\overline{7}~~~\overline{8}]$. Many other "tricks" like this are revealed one by one. In this way Reimer solves several arithmetical problems. Among the more complex problems are doubling of fractions and simplification after additing. For example doubling $\overline{7}$ gives $\overline{4}\;\overline{28}$. After adding many of these unit fractions the sum has to be reduced to a simpler form taking into account possible "carry digits". For example $\overline{14}\;\overline{21}\;\overline{42}\;=\overline{7}$ (in our language it would involve least common multiples of the denominators). For these one may need look-up tables. These tables can be downloaded and printed from the book's website. Along the road, the toolbox of computational rules is continuously extended. With these techniques problems of ratios, slopes, surface area, volumes, etc. are solved.

It is a textbook in the sense that it contains many examples and problems that the reader should solve on his or her own (with solutions at the end of the book), but on the other hand it has those entertaining stories, anecdotes and historical information in the background where you learn about Egyptian society, history and mythology, and it is marvelously illustrated. Tables like the one above are usually printed on a background image that represents a papyrus scroll. Since this was probably the main medium on which the scribes and the engineers of the pyramids did their computations. For less important daily use, broken pottery was the more common writing surface, and for the really important tables the much more expensive leather was used. Reimer uses the corresponding background images in the book and there are several other types of tables and many colourful illustrations as well. Also the square format of the book with two-column layout makes the book (literally) stand out.

Reimer's laudatory praise of the Egyptian system goes on for some 144 pages. But then in the next 60 pages, he briefly explains about other base-based systems like the Mayan system (base 20) and the inelegant Roman system, but most of all the Babylonian system (base 60). After some Babylonian arithmetic, he makes a fair comparison with the Egyptian system and it seems that our system (resembling the Babylonian rather than the Egyptian system) has some (slight) advantages. However Reimer cannot resist mentioning some of the disadvantages in which the Egyptian system prevails over ours like easy estimation of the error when the number represents a "rounded" approximation, or the fact that all Egyptian fractions terminate after a finite number of "unit fractions" without repetition unlike in our system for numbers like 1/3 or 4/17 etc. The bottom line of his discourse is that asking the question which system is the better one is not so simple to answer. It is not obvious that our system is so much better. Of course our system is more apt for us (or for machines) to do calculations just following recipes, which need no insight or wit, but what we loose is that the Egyptian system keeps the practitioner sharp, forcing him or her to think about the problem and the result of the calculations.

Reimer succeeds very well in transferring his enthusiasm for the Egyptian system to the reader. The reactions from his students who were used for a try-out are claimed to be positive. But even if you do not want to graduate as an Egyptian scribe, you may be charmed by the witty Egyptian system and you will be delighted by the colourful illustrations and Reimer's entertaining account of it all.

## Comments

Tag53

07 / 19 / 14

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## lookup table errors

The book is very interesting, however, there are a couple of errors in the provided lookup tables provided.

the sum table for /13 is not correct, if the first three terms are kept then "/98 /182" must be added behind the list to make it be equal to "/13", or the whole list could be changed to "/16 /104 /208" or to "/26 /39 /78".

The 2x odd table is also incorrect for the "/97" entry, the middle term "/697" should be "/679".