If we don’t have a calculator with us, the fastest way we know to do the multiplications is in column. But there is another method dating back to the ancient egyptians, reached us through a papyrus from 1650 BC, which is based solely on doubling and adding.
Although it may seem counterintuitive, the doubling method actually hides several mathematical properties, including the binary writing of a number and the distributive property of multiplication.
To multiply, just double.
Egyptian multiplication, also called the doubling method, has come down to us today thanks to the discovery of the Rhind Papyrusa writing dating back to 1650 BCwhen the scribe Ahmes transcribed an earlier papyrus from 2000 BC containing tables of fractions and arithmetic, algebraic and geometric problems.
The method is very simple, although not immediate, and consists in the execution of two different repeated doublings:
- the first, under the first factor of multiplication, consists in start from 1 and double – then we continue with 2, then 4, 8, 16, … – until we get to the greatest power of 2 less than the first factor (in the case of the example below, since the first factor is equal to 36, the greatest power of 2 less than 36 is 25 = 32)
- the second, under the second factor of multiplication, consists in start from the value of the factor (in the case of the example below, 17) and double – so in our case 17, 34, 68, … – as many times as there are doubles in the first column.
At this point, we choose under the first factor i numbers That, added together, they result in exactly the first factor (in our case, we choose 32 and 4, which add up to 36). We are sure that this operation is feasible because it is precisely the writing the first factor in binary and every integer can be written as a sum of powers of 2.
Now we choose the corresponding numbers which we find in the column below the second factor and them let’s add up. Their sum is precisely the result of the multiplication (in our case, they are 68 and 544, which added together give 612, which is precisely the result of the operation 36 x 17 – try it with a calculator to believe it!).
The method uses binary decomposition and the distributive property
Looking at it this way, it seems incredible that this apparently random method gives us the result of the multiplication. In truth, what lies behind it is something very sophisticated, namely the binary system, which is the basis of modern computing.
When we write the powers of 2 under the first factor and then choose those that, added together, give the first factor as a result, we are actually breaking down the number in his binary components. In our case, we wrote 36 like 4+32, that is 22+25. Without knowing it, in the second column we have written the very powers of 2 that we see in the first column, multiplied by the second factor, so in our case 17.
So when we add them up, we are essentially using the distributive property: we broke the first factor down into powers of 2, and then instead of multiplying the first factor by the second, we multiplied the various powers of two by the second factor, then added the result.
