The author claims, apparently confidently, that the Romans used this method but had no idea why it worked. It's not clear how he knows this. I dare say most pe

The author claims, apparently confidently, that the Romans used this method but had no idea why it worked. It's not clear how he knows this. I dare say most people who used it didn't know why it works, but then probably most people today don't know why long multiplication works. Barring good evidence to the contrary, I bet that whoever thought the method up had a pretty good idea of why it worked. Of course it's possible that the Romans just stole the idea from the Egyptians. Or that someone thought up the method and then went to his grave without ever explaining it to anyone else, and that no one else tried to understand it. But I don't believe the author knows. Incidentally, you can understand the method without having heard of the binary system. For instance, here's how I might explain it to an ancient Roman who happened to speak English. I might start by putting together a rectangle with, say, 10 rows of 4 stones. If you double one number and halve the other, without any remainder, then the product of the numbers remains the same. (Rearrange the rectangle to be 5 rows of 8.) On the other hand, if the number you're halving is odd, there is a remainder and the product of the numbers has decreased -- by exactly the value of the other number. (Rearrange the rectangle again: 2 rows of 16, with a "half-row" of 8 left over.) As you keep doing this, the product stays the same every time you halve an even number (rearrange again: 1 row of 32) but when you halve an odd number you always lose some, as we did a moment ago. At the very end you end up with 1 times something, and of course that product is easy to do. Now the product of the original numbers is just your final 1-times-something product, plus the bits you lost along the way. We're done. (It's more elegant mathematically to make zero-times-something the base case, but probably harder to explain.)