 HubPages»
 Education and Science»
 Math
Multiplication: Do It Like the Egyptians
I just came across a new hub by Cooldudemaths and it reminded me I have a few (interpret several) really fun Math projects I have come across as an adjunct instructor for a local community college. I found over the years that my students relate to topics that can engage them from unexpected sources. This particular lesson can be used in a high school classroom and adjusted to be included in a beginning analysis class. This example can be used to present topics from the Associative Property to Factoring and the Distributive Property. For higher levels one can use this to talk about binary operations and other philosophical topics.
It is always best to start at the beginning. For simplicity let's only talk about multiplying positive whole numbers. The Egyptians did not know about negative numbers or what 0 meant. In your best Egyptian style headdress explain to your class that they multiplied by a series of doubling steps. Then they would sum up the relevant parts to give the correct answer.
The following table will show how to set up a table of values for multiplying 23 by 42.
Find 23 * 42 = ???
23
 42
 

start with a single 42
 1
 42

double the above row
 2
 84

double the above row
 4
 168

double the above row
 8
 336

double the above row
 16
 672

notice how the fist column is larger than 23
 32
 stop

Finding the Solution
Now the Egyptians didn't add they way we do. They didn't know what base 10 was. Their number symbols were different than ours too. They did know how to double amounts though. They simply matched up the amount and then portion the total in known quantity amounts. So their method is to double upwards from a unit amount until the multiple factor is exceeded.
To solve start in the column headed up by 23. Starting with the largest number less than 23 or 16. Now add numbers backwards until you total exactly 23. I can't add 16 + 8 because this is > 23. The first two lines to add together then are 16 + 4 = 20. Keep going. 16 + 4 + 2 = 22 OK so the last addition will be the right combination. 16 + 4 + 2 + 1 = 23
To find the answer add together the numbers in the right hand column under the 42 that are associated with the numbers found in the left hand column. This turns out to be: 672 + 168 + 84 + 42 = 966
As long as we may be talking about properties of numbers let's demonstrate that the Commutative Property holds even for the Egyptian scribes. Here is the table for 42 * 23 = ???
Find 42 * 23 = ???
start with a single 23
 1
 23


double the above row
 2
 46

double the above row
 4
 92

double the above row
 8
 184

double the above row
 16
 368

double the above row
 32
 736

this double makes 64 > 42
 64
 stop

Solving 42 * 23 = ???
In this example we start with 32. But 32 + 16 = 48 > 42. This means we need to use this combination 32 + 8 + 2 = 42 That means if we add the numbers in the right hand column in the same rows you get 736 + 184 + 26 = 966 SURPRISE! You got the same result.
Multiplying this ancient way works every time. All the student need to do is set up a table. It is easy to see that the left hand column is the base 2 or exponential series with 2 as the base and the exponent in whole numbers. This means the left hand column will always follow the series 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 and so on. They will still want to make tables because it is the right hand column that will not be obvious.
Concepts
Now you can work in the ties between how the ancient Egyptians method and our method are really compatible. This will include at least the Distributive Property and Factoring. You can link in other properties if you really have their attention. These two are the obvious ones to show.
The trick is to use the left hand binary series. This is the Factoring skill. For example let's look at 55 * 105. Notice that 55 becomes 32 + 16 +4 + 2 + 1 = 55.
55 * 105 becomes (32 + 16 + 4 + 2 + 1 ) * 105
The Distributive Property turns this in to:
32*105 + 16*105 + 4*105 + 2*105 + 1*105 =
3360 + 1680 + 420 + 210 + 105=
5775
Math History
There are countless other historical approaches to Math that we have replaced with modern techniques. We have had thousands of years to design our number system not to mention the style formatting the numerals. There are ancient examples of how preceding cultures developed various types of problem solving solutions that are just as interesting.
Sometimes it is not even the remotely ancient you should consider. It is interesting how people solved problems before Algebraic symbols and concepts existed. That was just a little over 500 years ago. It was cumbersome before more modern systems were introduced. Presenting Math in a non traditional way can present a different face on understanding. These historical discussions let those who have a stumbling block with normal teaching processes to see how the many facets fit together. Often it is just the alternative method that is enough for them to realize what is happening and why.