David M. Raley
I am not a Roman Numeral "Mathematician" but I can stumble around with the process. They were still teaching the "modern" system when I was in grammar school. The major difference between modern Roman Numerals and the system that the Romans actually used is that the ancient system was entirely additive. That is, IX would be eleven rather than 9, so would XI. Nine could be IIIIV or VIIII; or even IIVII. 666 could be IIIXXXCCCCCCXXXIII or CCCIIIXXXXXXIIICCC or a great number of other arrangements, as well as DCLXVI. The reason that six was so prominent in Roman life is that Venus, the "Mother of Heaven" was in the realm of the sixth crystal sphere. To roll three sixes in dice was thought a sign of good fortune. You may recall that Claudius was passing the time before a campaign playing dice..."...then I laughed and fell at ease, for I had rolled a Venus...".
As with our system, multiplication and division are only forms of addition and subtraction, but with Romans it is more painfully obvious. Lacking the concept of numerical fractions they made extensive use of classes of measures. For example II feet and VI inches rather than 2.5 feet.
The way I learned to add was to group and reduce:
Add XXVVVIII and XXXVII = XXVVVIIIXXXVII = XXXXXVVVVIIIII = XXXXXVVVVVII = XXXXXXXV = LXXV
Step by step multiplication:
X times VII = IIIII IIIII -- VII; IIIII IIII -- VII; IIIII III -- VIIVII; IIIII II -- VIIVIIVII; etc until you have VIIVIIVIIVIIVIIVIIVIIVIIVIIVII. Group and reduce.
VVVVVVVVVVIIIIIIIIIIIIIIIIIIII = VVVVVVVVVVVVVV = LVVVV = LXX
There are no insights to be gained from the study of Roman Numerals. We are well shed of the system.