Egyptian Multiplication

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 multiply 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 multiplications, 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.
 

Multiplication is a more complicated idea than addition and subtraction. (See Egyptian Addition and Egyptian Subtraction .) Primitive people probably didn't need it, but when people started to do the more complicated things that civilizations do like trading and buying and selling things then it started to become a useful operation to have. If you know what the price of one item is and you want to know the price of several of them, then you have a use for a repeated addition operation. But even in ancient Egypt multiplication wasn't one of the fundamental operations. 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 .)

Before we can learn how they multiplied, first we have to learn their symbols and it would probably be a good idea for you to read my article Egyptian Addition as well.

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 multiplying, suppose you were an official in ancient Egypt and the allowance of grain for each worker is 12 hekats. (Hekat was a measure of grain that they used, maybe how much grain would fit into a certain sized basket.) You have an overseer who is in charge of a gang of 7 men. You want to know how many hekats of grain to give him to distribute to his men. Here is how you would do it.

Here are the Egyptian symbols representing the 12 hekats that one worker would get.

Here is how the scribe would figure out how many hekats the 7 workers would get. The first line represents the hekats that one worker gets. In the next line this is doubled by simply writing all of the symbols twice to represent how many two workers get. In the next line it is doubled again to represent how many 4 workers would get. Then since the numbers of workers that the first three lines represent totals up to 7, the amounts of grain are totaled up to get how much grain the 7 workers will get. This total is given in the 4th line and then it is neatened up by replacing ten ones with a ten in the fifth line to give the final answer. (See Egyptian Addition.) This represents 8 tens and 4 ones, which in our notation is 84. (And you don't even have to memorize the times table to do it.)

This method of doubling and adding up will always work. What you do is you take the number you are multiplying by, which I will call the multiplier, and you double 1 until you get up to a number that is more than half its size. If that is the number then you are done. If not you subtract it from the number and repeat the process with that number. Each time you do this you will reduce the number that you have left, so it has to eventually either come out even or reduce down to 1.

Example: 14

Doubling 1 successively you get 1,2,4,8. 8 is more than half of 14, so we stop. Subtract that from 14 and we have 6 left over. 4 is more than half of 6. Subtract that from 6 and we get 2 left over, so 14=8+6+2.

Here is an example of multiplying bigger numbers ancient Egyptian style.

34 times 56
The first line represents 56 in Egyptian symbols and in each of the next few lines after that, it gets doubled. The number in the left column represents how many times 56 this is in Egyptian symbols. The last doubling line represents 32 times 56 and since 32 is more than half of 34 that is as far as we need to go. Then we tick of the ones we are going to sum up with a slanted line. Then we add them up, both the left and the right columns showing that the left column then represents the multiplier 34 and the right column represents the answer. (See Egyptian Addition.) The final answer is represented at the bottom and corresponds to the correct answer of 1904 written in Hindu-Arabic notation.

The process works, but you can see that it might get long for large multipliers, so they used some short cuts. One short cut that they used for large multipliers was to multiply by tens, hundreds, etc.. as well as doubling. This can be done without much difficulty because all you have to do is promote your symbols one up, so for example to multiply by ten you would just replace all of the ones with tens and the tens with hundreds, etc.. And this way the most you ever have to double up to is 8. Then you can do the above problem with fewer steps like this.

The fourth line represents 56 multiplied by ten by turning the ones into tens and the tens into hundreds. The fifth line represents 56 multiplied by 20. This can be done either by doubling ten times it or multiplying its double by ten. I chose to do it by the later method because the double had already been worked out, so I could avoid regrouping that way. Then to get 34 times 56 we add up 20 times 56 and 10 times 56 and 4 times 56. (See Egyptian Addition.) After neatening things up we get the same answer.