Dr. David P. Stern explains A Different Kind of Multiplication:
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.
The algorithm is far from clear:
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.Now cross out in the second column all numbers where the corresponding number in the first column is even.
Then add up the remaining numbers.
You’ll have to read the article to learn why that odd process would yield the product of two numbers.