Photo by Caroline Delbert
The math wizards at Numberphile have brought back an age-old multiplying algorithm known as halves and doubles, peasant math, Egyptian math, or—as math presenter Johnny Ball describes it—Russian multiplying.
To do the method, begin by writing the two numbers you want to multiply at the top of two columns. In the left column, you progressively halve the number and take the integer floor of any “and a half” values, all the way down to 1. In the right column, you double the number as many times as there are digits in the left column.
Photo by Caroline Delbert
With your completed table, scan through and remove any rows where the left column has an even value. That includes the original term at the very top.
Photo by Caroline Delbert
Now, when you add the remaining terms in the right column, you get the solution.
Photo by Caroline Delbert
The method works for all numbers, and it works either way you arrange your original terms.
Photo by Caroline Delbert
Wait, What's Going On Here?
Johnny Ball presents the method as a fun thing he was taught long ago while hanging out in the “children’s room” of a pub. (Never change, U.K.) The person who taught it to him called it Russian multiplication, and Ball explains that the method originated not in Russia, but thousands of years before in ancient Egypt. And, critically, the method maps to a binary number system.
Binary is the common term for what’s technically the base-2 number system, where values are represented by 0 and 1 positioned in powers of 2. Here are a few integers written out as binary values so you can see the value at each position.
Photo by Caroline Delbert
If you‘re noticing that the “halving and doubling” method bears at least a superficial resemblance to how binary values double from column to column, you’re right. Making binary numbers from decimal (base-10) values is kind of like making change: you find the highest denomination that fits your number without going over, then subtract it. Then you find the highest denomination that fits your new number without going over, and subtract that.
Ancient people still had to do math, but imagine calculating even fairly simple things without any scratch paper—or if you were never taught to write. The method most children learn in school, long multiplication, involves many steps with separate products you have to note and recombine later. Halving and doubling let our ancestors use physical counters and do calculations by “making change.”
If you’re like me, you spent much of the Numberphile video wondering if the method was a trick, like that trick that relies on math facts about the number 9 to make sure you guess someone's “secret number” right every time. And maybe you also wondered about exceptions to this method. As far as I can tell, there really are none. Even a power of 2 itself will break down so that you’ve crossed out everything in both columns except the final one—the final “1,” actually—and that single value is your answer.
Photo by Caroline Delbert