What do you do when your child fails to understand multiplication? Do you give up? Do you repeat the process as though you were speaking louder to a hearing-impaired person?
Try teaching your child to multiply like the Egyptians, instead! It’s fun and will explain more about Mathematics than regular multiplication methods do. Sometimes all it takes is a different method for understanding to happen.
Students of Mathematical History are frequently surprised at the ingenuity of ancient people. We generally assume that they had virtually no skills or talent compared to our modern procedures.
The ability of ancient Egyptians to multiply is just one example of ancient ingenuity that can help us today. Keep in mind, the Egyptians did not have the number 0, so they could not use it as a placeholder as we do in Base 10. In fact, all they could really do was count.
Knowing this alone should show just how remarkable it is the Egyptians could multiply – but they did, and here’s how it works.
Egyptian Multiplication Example
Before explaining why this works, let’s try a couple of examples to become proficient with the method. For example let’s multiply 26 and 47. Because Multiplication is commutative with our number system we may choose either 26 or 47 to start. Let’s choose 26.
Set up a rudimentary table where 1 is on the left and 26 is on the right like this. Don’t worry about the 47 just yet.
Now double each side and keep your table all together.
Since 2 is not greater than 47 keep doubling your table until the left hand column is greater than or equal to the number not chosen, in this case 47.
Now that the left hand number is greater than or equal to the number not chosen we can figure out our answer. To do this find the unique combination of numbers in the left hand column when added together equal 47. It helps to start with the largest number less than 47 and work backwards. In our example start with 32.
32 is less than 47. So, let’s add the next number less than 32 with it to see what we get.
32 + 16 = 48. 48 is larger than 47. We do not want to use 16. Try the next lower number to see what you get.
32 + 8 = 40. 40 is less than 47 so we want to keep the 8. Add the next lower number to these two and see what you get.
32 + 8 + 4 = 44. 44 is less than but not equal to 47. Keep going. Add the next smaller number to these three and evaluate their sum.
32 + 8 + 4 + 2 = 46. Again 46 is less than and yet not equal to 47. Try the last number.
32 + 8 + 4 + 2 + 1 = 47. Success! We now have the summed up combination that is exactly equal to 47.
To find out what 26 x 47 equals we just sum up the corresponding entries in the right hand column. This would be:
832 + 208 + 104 + 52 + 26 = 1222
This is precisely what we would get through modern regular multiplication.
One More Egyptian Multiplication Example
It is helpful to choose the smaller of the two numbers to be the left hand column choice. This normally means fewer doublings. The Egyptian procedure works both ways, no matter which of the two numbers are chosen. It simply requires fewer doublings if we choose to double up to the larger number in the right hand column. We can be explain this with one more example. Let’s multiply 17 and 57. Here is the completed table and solution.
This example is easy because 16 + 1 = 17. That means 17 x 57 = 912 + 57 = 969.
If we had chosen to double 17 instead here is the larger table:
While not saving much time, effort and computation you can see we saved 1 row in our table. And when we solve we saved when we chose to double to the smaller number: here 57 = 32 + 16 + 8 + 1 to get 544 + 272 + 136 + 17 = 969. We got to the same answer. It just took more steps.
That’s all there is to multiplying like ancient Egyptians. You’re welcome to find out why this works or you can stop here and make up a work sheet for your star students.
Why Multiplying Like an Egyptian Works
First, and perhaps foremost, this doubling process is not dependent on the number system. Remember that all Egyptians knew how to do was count. Our Base 10 number system has many inherent properties that allow for easy number manipulation. The Egyptian doubling process works no matter what Base number system is being used. Try this in Base 4 or Base 7 to challenge higher level students.
Secondly, since this is a “counting” method, students who have trouble memorizing the values of particular products do not have this stumbling block. Sure this method is slower. It is another option in the event a little extra help is required.
Most importantly there is the opportunity for the gifted teacher to demonstrate factoring, and the distributive property of Algebra. Both of these are extremely helpful and important for higher order Mathematics. In our above example the “hidden” number not doubled is first factored in the appropriate way and then distributed. In our example 47 = 32 + 8 + 4 + 2 + 1.
47 x 26 = (32 + 8 + 4 + 2 + 1) 26
This when distributed becomes:
(32 x 26) + (8 x 26) + (4 x 26) + (2 x 26) + (1 x 26)
Multiplying inside these parentheses gives us:
832 + 208 + 104 + 52 + 26
That is how the distributive property and factoring are used in the Egyptian method for multiplication.
Lastly the gifted older student will soon wonder if any integer can be constructed through this doubling process. Eventually they will wonder if there are numbers that can’t be produced by choosing the right combination in the left hand column. While it is beyond this small article there is proof that by choosing a specific combination of certain powers of 2 any integer can be constructed. Leave this for the serious scholar.
Added interest is achieved through the Egyptian method of multiplication. This should provide encouragement. Who isn’t interested in ancient Egypt? Perhaps this will help encourage your student when current methods of Multiplication prove to be difficult.