Ok so if the Romans did not use things like IX and IV and XC etcetera then addition and subtraction would be almost as instant as it is in our number system.

However with the new system it seems to me like it isn't that quick, the only benefit that I can find is that it makes numbers shorter.

Does anyone have any insights as to why they introduced this concept (I think that it must have not been in use originally as I believe the system originally came from some sort of tally system).

Also if anyone has some reasoning as to why it doesn't really make addition harder that would be great.

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    $\begingroup$ Interestingly I've never considered this, but now that you've brought it up, I suspect that the reverse order is used for decreasing. Thus, VI is "V increased by I" whereas IV is "V decreased by I". And why do this after III and not after II? Perhaps because three identical objects is easier to immediately visually intuit than four identical objects, and also (or maybe instead) perhaps because something like XX is potentially ambiguous --- is this "X increased by X" or is this "X decreased by X"? $\endgroup$ Jun 15 at 19:59
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    $\begingroup$ You are backward. IV was an improvement over IIII . (It is easy to confuse III and IIII right?) And IX was an improvement over VIIII . And so on. For calculations they used something like an abacus ... small stones moved in grooves. Only after the answer was found would the result be copied onto expensive parchment. (calculus is Latin for stone) $\endgroup$ Jun 15 at 22:11
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    $\begingroup$ In my limited knowledge, echoing @GeraldEdgar's comment, I infer that people in those days did not do calculations on paper/papyrus/parchment, but did the calculations with some version of an abacus, and only recorded the outcomes in writing. In particular, the transformative rules for the notation didn't matter. $\endgroup$ Jun 16 at 20:24
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    $\begingroup$ @Jordi, it's my semi-ignorant impression that whatever inputs they had, computations were done by abacus... $\endgroup$ Jun 16 at 21:31
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    $\begingroup$ An aside ... hsm.stackexchange.com/a/776/229 ... a Roman calendar, where 28 was written XXIIX $\endgroup$ Jun 16 at 22:35

I'd like to add my comment as an answer to have memory of a side comment.

As I was saying the subtractive notation was a way of sparing characters in carving and this is the reason behind it becoming popular.

@paul garret made a very interesting comment on the fact that with Latin numerals computations were not reported on paper/parchment and computations with the abacus were effective since they did not require keeping track of intermediate steps.

This was, in fact, one of the reasons of a quarrel after Fibonacci introduced Indian digits in his Liber Abbaci (Book of Calculations).

At first computations with what we nowadays call Arabic digits were considered to be cumbersome because they required keeping track of intermediate steps. But, at the time, paper was not widespread in Europe (paper is another invention that arrived in Europe from China through Arabs) and was very expensive.

Abachists kept preferring abacus computations for years since it was cheaper and still effective when compared to computations made by "algoritmists" - such were termed those in favour of arab numerals.

Only after some years, when starting from Fabriano, Italian production of paper became quite standard and costs dropped, algoritmists won their war.

So at least twice in this story the material required for writing numbers played a role.


It seems like the "subtractive" convention was not as commonly used as I initially thought.

As is said here the additive convention was in many times utilized when the number could be further manipulated.

I believe the "subtractive" convention was mostly used for shortening.

I would like to add that I am not completely sure, but I think I have more chances of someone correcting me if I write out this answer.


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