Egyptian Subtraction

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 subtract 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 subtractions, 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.

First of all, before I talk about how to do subtraction I need to talk a little about what it would have meant when it was first done. Subtraction represented taking away. You have a certain amount of grain, and you give out a certain amount, and you want to know how much you have left. Probably the earliest method to do subtraction was to sort of make a model of the situation and literally take away such things as pebbles in a pile. But this gets difficult when you have to deal with large numbers, so you have to use some kind of grouping system where certain objects or symbols represent more than one thing.

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.

When the Egyptians wanted to subtract or take away from they would do it like this.

The idea is that you have to take away all the symbols in the first number from the second number. But there is a problem here, because there are enough of some of them. The second number is indeed a larger number than the first, because it has more thousands, so the subtraction is possible, but some regrouping has to be done. Normally you are not supposed to use more symbols than necessary to represent a number, but when doing subtraction an exception is made, so what we have to do is break up one of the tens into ones and break up one of the hundreds into tens and break up one of the thousands into hundreds in order to do the taking away. By replacing a ten with ten ones and a hundred with ten tens and a thousand with ten hundreds, the second number can be written like this.

Now we simply take away from this to get