Good Math, Bad Math : Roman Numerals and Arithmetic
Good Math, Bad Math : Roman Numerals and Arithmetic provides a discussion of doing arithmetic with Roman Numerals.
Back in 1995, David Paredes and I wrote a chapter that discussed this in part,
The Chu-Carroll discussion is generally accurate, although it overlooks an important feature of Roman Numerals that was used in teaching arithmetic. That is the incorporation of addition and subtraction into number representations.
If you want to add IX + XI = XX (9 + 11 = 20), you can see that that IX consists of 10-1 and XI consists of 10+1, so that +1 -1 = 0.
The construction of Roman numerals directly include tens-complements and five-complements. In much of East Asia, tens-complements are taught explicitly in early grades and used in addition above 10.
Thus, if you want to add 9 + 4, you take advantage of knowing that the tens complement of 9 is 1 to turn it into
10 – 1 + 4 = 10 + 3 = 13 (of course, the fact that the name for 13 is “10 3” in Chinese-related systems helps a bit).
It didn’t make it into the final paper, but I remember reading about an approach to mathematics education in England that involved using Roman Numerals for addition and subtraction (because they were seen as conceptually simpler) and then introducing Arabic numerals for multiplication and division (because there aren’t compact algorithms for those operations with Roman Numerals).
The trade-offs between ease of acquisition and ease of use are important ones for any cognitive system. Arabic numerals are very opaque conceptually, but once mastered are a very efficient system for computation.