The Glade 4.0

It is often said that an important advantage of the decimal notation over the Roman one is that makes multiplication of numbers much easier. Adding CLXXVII to XXIII may be relatively straightforward--but how about multiplying the two?

It is indeed easier to multiply these as decimal numbers 177 by 23, but the Romans also had a multiplication method of their own. It was probably discovered by trial and error, and it always worked, though the Romans did not know why.

Here the method is described, and its secret explained. 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 --- 46
44 --- 92
22 --- 184
11 --- 368
5 --- 736
2 --- 1472
1 --- 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.