See the previous blog entry for the problem. I don't want to repeat it here. The question is: "How and why does the curious system of "Ancient Roman Multiplication" work?
The answer lies in the hint "the ancient Roman need only know how to double and half." Let's look for a binary solution. Translating 41 to binary representation gives 101001 which just happens to represent the column of numbers being halved. The 1's represent the odd numbers and the o's represent the even numbers, starting at the bottom with 1 and going to the top - 41.
Let's see what a binary representation of the problem would look like:
23-----------> 010111
41-----------> 101001
---------------______
--------------> 010111
-----------> 010111
---------->010111
-----------__________
--------->11110101111
Which translates back to 3AF hexadecimal or 943 decimal.
And, by the way, the three intermediate numbers in the multiplication (after inserting the proper number of placeholders) are 23 (17 hex), 184 (B8 hex), and 736 (2E0 hex) which shouldn't surprise anyone who has been following along.
OK, now you're just making my head hurt worse.
ReplyDeleteYou posted too fast. I had no time to work on it until late last night, but I got it then. Very cute. I doubt the Romans understood why it worked. (At least not in the way we understand it.)
ReplyDeleteLee