Last semester, I began my math history class with some Babylonian arithmetic. The mathematics we were doing was easy—multiplying and adding numbers, solving quadratic equations by completing the square—but the base 60 system and the lack of a true zero made those basic operations challenging for my students. I was glad that the different system shook them up a little, and got them thinking about things we take for granted, but some students seemed to draw the conclusion that Babylonian mathematics was awkward and silly. The class spent a lot of time thinking about the differences between the two systems but not as much thinking about the Babylonian system on its own terms. As children, we spend several years learning how to do arithmetic; it’s not really fair to judge an unfamiliar number system based on a few days of working with it.
Count like an Egyptian by David Reimer, published in 2014 by Princeton University Press, thoughtfully avoids that pitfall. The Egyptian number system, which has some profound differences from our own, is not presented as a sideshow or tourist attraction. In addition to explaining how the numbers were written and the basic arithmetic operations carried out, Reimer analyzes the logic behind the operations that seem unusual to us. He compares learning Egyptian math to learning a new language. "Spanish is stupid," he told a junior high Spanish teacher after a run-in with an irregular verb. Irregular verbs can make a language seem arbitrary to an outsider. But of course English has more than its fair share of linguistic idiosyncrasies. Native speakers just don't notice them until they're pointed out. Reimer writes,
Egyptian mathematics has an alien feel to it. Most math historians refer to it as primitive or awkward. Even worse, many simply ignore it except for a passing reference. They look at this system and feel uncomfortable because it’s so different. They perceive apparent “flaws” and move on. They don’t understand Egyptian mathematics simply because they don’t do it enough to truly appreciate it. To someone who’s mastered it, Egyptian mathematics is beautiful. It scorns memorization and rote algorithms while it favors insight and creativity. Each problem is a puzzle that can be solved in many ways. Frequently, solutions will be surprising, something that never happens in the step-by-step drudgery that is modern computation.
Consider fractions. One of the first things you learn if you read a little bit about Egyptian mathematics is that with the exception of 2/3 and occasionally 3/4, Egyptians only used fractions with 1 in the numerator: 1/2, 1/3, 1/4, and so on. They would write other fractions as sums of unit fractions. For example, 7/24 could be written 1/4+1/24 or 1/6+1/8. (The Egyptians didn't actually write their fractions with numerators and denominators; if your only numerator is one, it's redundant to write it down every time. Instead, an Egyptian fraction would consist of a "mouth" symbol on top of the symbol of an integer. So 1/7 would be a mouth over a 7. To write a sum of two fractions, they would just write the second one after the first one.)
When I first read about Egyptian fractions, I dismissed them as awkward and inefficient. But Reimer points out that they aren't so different from our decimal system. When we write the number 0.572, we're just writing 5/10+7/100+2/1000 in a slightly different way. The denominators of those fractions follow a predictable pattern, unlike the Egyptian one, but we are still writing numbers as the sum of fractions with increasing denominators. One nice thing about our decimal system is that if we cut the number off after a few places, we have a pretty good idea of how big it is. The number 0.572 is pretty close to 0.5 and 0.57. Likewise in the Egyptian fraction system, 7/24 is pretty close to 1/4, the first term in one of the possible ways to write 7/24.
Egyptian fractions also have some advantages over decimals. For one, they will always terminate. We can't even write 1/3 as a terminating decimal. Sticking rigidly to powers of ten in our denominators limits the numbers we can represent easily. Like our fractions, Egyptian fractions are exact but use only a finite number of terms. Reimer sees Egyptian fractions as a compromise between the best qualities of our fraction and decimal systems. They are just as precise as our fractions, but like our decimals, they also make approximation easy.
But Egyptian fractions can throw some curveballs. There isn't necessarily just one way to write a number as an Egyptian fraction. For example, above I wrote 7/24=1/4+1/24 or 1/6+1/8. In that case, it is pretty clear that 1/4+1/24 is a better way to write it: 1/4 is a good approximation, while 1/6 is not. But in other cases, it isn't so clear. In the book, Reimer gives the example of 4/15, which can be written as either 1/6+1/10 or 1/5+1/15. 1/5 is a better approximation than 1/6, but there may be other reasons to choose 1/6+1/10. Egyptians used doubling a lot when multiplying, and it's easier to double fractions with even denominators than odd ones. So depending on the specific circumstance, 1/6+1/10 might be a better choice for computation. This is what Reimer is talking about when he says the system "scorns memorization and rote algorithms while it favors insight and creativity." It seems strange to have so much leeway in how to represent numbers, but it shows that creativity can have a place even in simple arithmetic.
With insights like this, Reimer not only explains the logic of the Egyptian system but also encourage us to think about the whats, hows, and whys of our own mathematics techniques. “This book is a thinly disguised critique of modern mathematics," Reimer writes. The last chapter, Judgment Day, is a "battle" between Egyptian and modern methods, but it's not just about a winner and a loser. “We will consider which system is better and what exactly ‘better’ means.” As you might guess, it's complicated.
Reimer sprinkles vignettes about Egyptian mythology and society throughout the book, and he also includes some historical information about the few Egyptian mathematical artifacts that still survive. (Papyrus generally doesn't hold up very well for 3,000 years.) He also makes it clear when his mathematical commentary is backed by evidence from Egyptian papyri and when it is his own conjecture based on his mathematical intuition. Because I'm interested in the book from the perspective of a math history teacher, I do wish there had been a little bit more about exactly what mathematics is in what papyrus, but the book is not a scholarly history of Egyptian mathematics, and that information may have distracted from the mission of getting people to try Egyptian mathematics for themselves.
Count Like an Egyptian would make an excellent addition to math classrooms at many different levels. Reimer includes problems in the text and solutions in the back of the book, so the reader can practice techniques and get a feel for exactly how the system works as they go through the book. The mathematics is basic enough to be helpful for children learning fractions or multiplication for the first time, but it's also different enough from the methods most of us know that adults will get a lot out of it as well. I used Egyptian multiplication and fractions on the first day of this semester's math history class as a way to push students out of their comfort zone and get them thinking about some of the most basic building blocks of math in a new way. With more background on the rationale behind the system, I think it was an effective way to open the class up with some interesting discussion about what numbers should do for us.