One of the great advantages of our modern decimal notation is that it makes multiplication easier. How did the Romans perform multiplication anyway?
The Romans started by writing the numbers next to each other. Of course, they used Roman numerals — but to make it clear in what follows, here decimal notation will be used throughout. Suppose we need to derive 177 x 23.
After writing down the numbers (here the multiplication sign was added), halve the first one and double the second, writing the new numbers below the preceding ones. If the number being halved is odd, just ignore the remainder. Repeat this operation as long as you can:
177 x 23 88 46 44 92 22 184 11 368 5 736 2 1472 1 2944 Now cross out in the second column all numbers where the corresponding number in the first column is even:
177 x 23 88 4644 9222 18411 368 5 736 2 14721 2944 Then add up the remaining numbers:
23 + 368 + 736 + 2944 = 4071
You can verify that indeed 177 x 23 = 4071. The Romans did all this using their own cumbersome notation, but people used to handling numbers were experienced in doubling and halving, and could carry it out fairly quickly. Doubling can be relatively simple: XXIII doubled is XXXXVI, doubled again DXXXXII (Romans wrote 4 = IIII,40 = XXXX, and notations like 4 = IV were only introduced in the middle ages). Like the method we use, it reduced the multiplication of two numbers to addition, which Roman numerals could handle.
Wait, why does that work?
Let us rewrite the earlier table, but add binary representations:
177 (10110001) x 23 88 (1011000) 4644 (101100) 9222 (10110) 18411 (1011) 368 5 (101) 736 2 (10) 14721 (1) 2944
The Roman algorithm mimics binary multiplication:
The right hand column of our table, where 23 is doubled and redoubled, contains the number 23 times all powers of 2 — 1, 2, 4, 8…. By adding there the numbers corresponding to the “1″ [bits] of 177 = 10110001, we sum up the product of 23 by only those powers of 2 contained in 177. The result therefore equals 177 x 23.