Are there numerical algorithms for Roman numerals? In positional number systems there are algorithms for performing certain operations, like long division, to name one of the simplest. This works for positional systems, whatever base. I realize in number theory there are very advanced algorithms, typically for working with Very Long Numbers. (disclaimer: except for a Fourier transform I don't know any of them, I'm not a mathematician.)  
I was wondering how the Romans could do anything numerical with their odd Roman numerals. You can't divide MMDCCI by LXXIII using long division. So, question: are there numerical methods for Roman numerals, and if not, how did the Romans divide MMDCCI by LXXIII?

I think Joseph's answer is great and works with all such systems. But I happen to be familiar with the ways the Egyptians and Sumerians did their early calculations, and I think it is relevant in the sense that it is also a non-positional number system.
Egypt's number system is both non-positional and base 10 (that is, centered around powers of 10) like the Roman system, but it doesn't have the interesting feature where 4 is IV instead of IIII (using Roman Numerals to represent the Egyptian numbers as well). Egyptian scribes would divide by using a sort of mixture of guess and check, and repeated squaring.
It's easy to see through an example. Suppose we were to divide 153 by 9. Then we write down 1 and the divisor, and double both sides until the right side is bigger than 153. 
$$\begin{align}
153 / 9 \\
1 \quad &;\quad 9 \\
2 \quad &;\quad 18 \\
4 \quad &;\quad 36 \\
8 \quad &;\quad 72 \\
16 \quad &;\quad 144 \\
32 \quad &;\quad 288
\end{align}$$
Now we stop as $288 > 153$. The next step is to represent 153 by adding values on the right (there is only one way to do this). This is where I imagine they used guess and check - they would take the largest number (144 here) and keep adding the next-largest one that doesn't make it too large. 144 + 72 is too big, so we don't use 72. 36 and 18 are also too big. But 9 is just right.
So since we know that $144 + 9 = 153$, we can take their respective 2 powers to see that 153 divided by 9 is $1 + 16 = 17$. And so it is. That's pretty interesting, I think.
I also imagine that if one were calculating things all the time, they'd get really good at using 2 powers and so wouldn't even have to think about it. I also happen to know that they would sometimes guess a number early on and then just work with the difference. This is a lot like Joseph's answer: the Egyptians might guess 20, for example. Then they'd see that $20 \cdot 9 = 180$ and $180 - 153 = 27$. So then they'd just find 27 divided by 9 and subtract it from 20. But again, they would use 2 powers.
