Introduction to Ethiopian Multiplication

Most people today use a calculator to multiple numbers. In school, we also learn ways of multiplying large numbers by aligning them one over the other and multiply each digit of one number by each digit in the other number and then adding all the sums. For example, if we multiple 13 x 21, the traditional way to multiple is to write the numbers like below:

            13
x          21
            13         (1 x 13 = 13)
           260         (2 x 13 = 26; spaced over one place to the left; it was really 20x13=260)    
           273        (sum of 13 and 260 is 273)

The larger the number, the bigger the problem becomes. This method requires the student to know their multiplication tables. There is another method which helps someone multiply any two numbers of any size without knowing their multiplication tables at all. The method only requires one to be able to double a number and to be able to divide a number by two.

Pick any two numbers. For example:

13 x 21

Divide the first number in half and disregard any fraction. Write this number under the first.

13 x 21

6 (13/2 is 6.5 but we are disregarding any fractions – so just write 6)

Divide in half again. Write this number underneath.

13 x 21

6

3

Continue to divide in half until you reach 1.

13 x 21

6

3

1

Now go back to the top and double the second number and write the answer just below it.

13 x 21

6      42

3

1

Double again and repeat until you get to the last line – even with 1 on the left.

13 x 21

6      42

3       84

1      168

Cross out any row in which the first number is even.

13 x 21

6      42

3       84

1      168

Add the numbers in the second column (that are not crossed out) to get the answer:

13 x 21

6      42

3       84

1      168

        273           This is the answer to the question 13 x 21.

Another example (14 x 15):

14    x 15

7          30

3          60

1           120

14x15= 210

This method works for any two numbers. Of course, with larger the numbers more doubling, halving and lines will be required but it still works.

Historical notes:

This method of multiplication goes back thousands of years and has been used by many cultures. It is commonly known as Ethiopian Multiplication or Russian Peasant Multiplication.

This lesson is inspired by Leonardo Fibonacci. He is known for bringing the Hindu-Arabic numeral system to Europe and for many other contributions to mathematics. Before Fibonacci, Europe used Roman numerals, which are very difficult to use for math. Roman accountants and mathematicians used the method described above to perform multiplication.

Advanced notes (the link of the ancient method to modern computers):

In the first example problem place a “1” in front of each row not crossed out and a “0” in front of the crossed-out row:

1           13    X     21

0          6             42

1           3             84

1           1           168

Reading the new first column from the bottom is 1101. This number is 13 in base-2 which is how modern computers and calculators complete math problems only using zeros and ones. We now see how the ancient Romans, Ethiopians and Russian peasants were many years ahead of our modern computers.

Our common numbers are base-10. Each digit represents ones, tens, hundreds and higher powers of ten. Each digit in a base-2 number represents one, two, four, eight and so on. Therefore, the number 1101 in base-2 is 1x8 plus 1x4 plus 0x2 plus 1 or 8+4+1=13.