Egyptian Division

The mathematical notation and the techniques we use to compute with it are so concise and removed from the physical world that it is easy for mathematics students to lose awareness of its meaning when using it, so I think it is a useful thing for helping students understand the meaning of arithmetic to learn some more primitive methods for doing it. The ancient Egyptian system is particularly good for this because of its simplicity and concreteness. (See also How to Add and Subtract with a Counting Board  for another interesting method.)This article is about how to divide using Egyptian symbols. It is not entirely historically accurate, because I will be using hieroglyphic symbols, when all the documentary evidence says that they would have used a more shorthand system, the hieratic symbols when they would have been doing dividing, but one might imagine that in some pre historical time they might have thought of how to do the operation in the way that I will be describing it below, and also we need to make it simpler, because we haven't been trained for years and years in a scribe school.

Like multiplication, division wasn't a fundamental operation for the ancient Egyptians. (See Egyptian Multiplication.) The two fundamental operations were addition and doubling and problems that involved multiplication were solved by a combination of these two operations. (See Egyptian Addition.) But the ancient Egyptians and other early civilizations needed to solve the kinds of problems we think of as division problems in order to share things.

Before we can learn how they divided, first we have to learn their symbols and it would be a good idea also for you to read Egyptian Addition and Egyptian Multiplication.

If you were a young scribe apprentice in ancient Egypt you would learn that each line represents 1 thing and when you get up to enough lines so that there is one for each finger on your two hands, then you replace them with a symbol like this.

Then for as many of these as you have fingers you write and this continues for a number of symbols that stand for our modern day powers of ten.

Now to dividing, if you were an Egyptian official and you wanted to distribute (1960) loaves of bread to (56) people you would need to know how many each person should get. The Egyptians seemed to have been fascinated with problems of distribution, but seemingly more with ones of uneven distribution. The Rhind Papyrus, one of the major sources for our present day knowledge about Egyptian mathematics, has several interesting problems about unequal distributions. Here is a page out of it. It was written by a scribe named Ahmes.

To keep things simpler for now, though, let's look at how they would solve the above equal distribution problem.

To do this, they would simply do the reverse of what they did when they multiplied. The instruction would be given to treat 56 so as to get 1960, and the goal is to get

in the right side and the number of the left side will be the answer. Here's how we do it. The idea is to try to use up the bread. Line (1) represents how much you would use if you gave just one loaf to each person. One line (2) we double this to see what happens if we give two loaves to each person. We neaten it up on line (3) and see that this is still nowhere near large enough, so we double it again, so line (4) represents how many loaves we would give out if we gave four to each person. That is still small, so we double it again to get on line (5) how many loaves would be given out if we gave eight to each person. We don't actually need that one, so a clever scribe might have not done it. After eight we could double again to get the result of giving out sixteen loaves, but a good shortcut is to go now to multiples of ten, so line (6) represents how many loaves we would give out if we gave ten to each person. To figure this out all we have to do is turn all the symbols for one loaf into ten loaf symbols and all of the symbols for ten loaves into ones for a hundred loaves. Then on line (7) we can do the same thing with line (3) to get the number of loaves we would give out if we gave twenty to each person.

Now at this point a good scribe could probably see that there are ways of combining these numbers to get the total, so he would stop multiplying and start looking at ways to add things together to get the desired number of loaves. (See Egyptian Addition.) The way he might see this is that there are more than half of the amount we are looking for on line (7) and also line (6) combined with line (7) looks like it is pretty close. Now a good scribe, who is well practiced in this, might do what I have done in lines (8) through (12) in his head, but I have written it out to make it clearer. There is a bit more guesswork here to do than with multiplication, because you have to figure out which of lines (1), (3), (4), (5), (6), and (7) to use. Putting them all together would clearly be too big. The strategy to use is to start with the biggest ones and add them until you find that the next one will make too much. So in line (8) first we put together lines (6) and (7) to get how many loaves get used if we give thirty to each person. That isn't quite enough, but it is getting closer and if we add line (5), representing giving out eight more loaves, it will be too much. But it looks like with have enough to add line (4) to it and give out four more loaves to each person, so line (9) represents how many loaves we would give out if we did that. In line (10) we neaten this up and see that we still have a little bit of bread left, not enough to give two loaves to each person, but enough to add line (1) to it and give out one more loaf, so line (11) represents how many we would give out if we did that and in line (12) we neaten it up and find that it is just right, so the left side of line (11) represents how many loaves each person should get, a total of thirty five.