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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?

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I guess they rather used an abacus... See also this page. –  t.b. Jul 5 '11 at 9:20
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@stevenvh - how is doing it on pen and paper any different from having an abacus? They are both menial computational aids. Maybe you are just culturally biased because you leared "long division" first as a kid! –  Willie Wong Jul 5 '11 at 10:09
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Presumably the Roman numeral system is a system for writing numbers, not for calculation. And they used a different system (movable stones) for calculations. The Latin word for "stone" is "calculus" after all. abacuslessons.com/roman-abacus.html –  GEdgar Jul 5 '11 at 12:30
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The Romans had calculators. They were slaves, usually Greek. –  André Nicolas Jul 5 '11 at 14:48
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@Theo Buehler: Any of the larger histories of mathematics has a chapter on Hellenistic mathematics, and in particular on the work of Ptolemy. His Almagest is quite readable, particularly the first chapter that introduces the computational procedures. I would recommend the Toomer translation. –  André Nicolas Jul 5 '11 at 19:51
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up vote 3 down vote accepted

This web page has information about some possible approaches to doing arithmetic using Roman Numerals: http://turner.faculty.swau.edu/mathematics/materialslibrary/roman/

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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.

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Are there numerical algorithms for Roman numerals?

Yes there are, all the algorithms for our current numbers can work on Roman Numerals ( as far as the numbers can be represented in Roman Numeral Form). All you have to do is to represent each number in it's Roman Form, why would dividing , adding or any other operation on numbers depend on how the number is represented?

You can't divide MMDCCI by LXXIII using long division

Yes you can, Here is one way to divide MMDCCI by LXXIII : Convert them to their decimal ( arabic numeral ) equivalants, carry out the long division as usual, at the end substitute roman numerals in place of every number appearing in the division. But this does not answer the question of how Romans did it, just that it is possible to divided MMDCCI by LXXIII using long division and arabic numerals as we are used to, the algorihm is a composite of Long Division and Converting numbers between two different types of their representation.

how did the Romans divide MMDCCI by LXXIII?

How did the Romans actually divide MMDCCI by LXXIII? One simple way would have been by adding LXXIII as many times to LXXIII until the results was greater than MMDCCI, a truely long divison algorithm, it is a way that although maybe not acceptable by todays standards but at least in principal they could do it.

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downvote for an algorithm that is correct? what's the deal! –  Arjang Jul 5 '11 at 9:54
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The questioner wanted to know how the Romans did it before the invention of Arabic numerals. –  Doug Chatham Jul 5 '11 at 16:48
    
@Doug : so why not tag it with math-history nstead? It seemed as if it was asking for numerical methods to work with Roman Numerals are to be different than those with any other system. –  Arjang Jul 5 '11 at 19:16
    
@Doug : the last paragraph show the flaw in the original question to what was really intended. –  Arjang Jul 5 '11 at 19:31
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Arjang: Is this really so difficult to understand? You ripped one sentence out of context and chose to answer this one sentence, while the question clearly asks for something else. Your answer just begs for downvotes. –  t.b. Jul 5 '11 at 19:36
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