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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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    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"? – Dave L Renfro Jun 15 '21 at 19:59
  • I wonder how the romans would have added 999+999. –  Jun 15 '21 at 20:04
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    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) – Gerald Edgar Jun 15 '21 at 22:11
  • In what ways is it an improvement aside from making things shorter? I guess that whole calculus thing makes sense, kidney stones are called "cálculos renales" in spanish. Also, thank you for the abacus information. Do you have a reference for how the abacus things work? I guess it would be harder to pass the number into an abacus with the whole $IV$ shenanigans though, or am I wrong? –  Jun 15 '21 at 22:13
  • They added... simply the procedure was different. See Abacus – Mauro ALLEGRANZA Jun 16 '21 at 07:21
  • @GeraldEdgar wandering well off-topic, "IIII" is often used on clock faces instead of "IV" because it visually balances the "VIII" on the opposite side. – Carl Witthoft Jun 16 '21 at 12:48
  • @MauroALLEGRANZA thanks, but I still think the new system complicates things. Parsing a number with the "old" roman system for abacus use is faster than the new system. Or did the romans just very seldom use written numbers? I also don't know how spoken roman numbers work. EDIT: I've been looking at the latin numeral wikipedia site and it seems spoken latin numerals are even harder. –  Jun 16 '21 at 14:27
  • Oh, I found something related here: https://www.quora.com/How-did-the-Romans-do-multiplication-with-Roman-numerals/answer/B%C3%A9a-Tremblay . It seems they many times did not use the whole IV stuff to avoid the complications I'm talking about ! –  Jun 16 '21 at 14:42
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    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. – paul garrett Jun 16 '21 at 20:24
  • so they almost never had written input ? –  Jun 16 '21 at 20:26
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    @Jordi, it's my semi-ignorant impression that whatever inputs they had, computations were done by abacus... – paul garrett Jun 16 '21 at 21:31
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    An aside ... https://hsm.stackexchange.com/a/776/229 ... a Roman calendar, where 28 was written XXIIX – Gerald Edgar Jun 16 '21 at 22:35
  • I can agree with that, but at the same time I think that parsing the operands is a non-negligible component of the computation process. Sorry if I come off as confrontational and thanks for all the input :) –  Jun 16 '21 at 23:05
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    Notations like IV and IX started replacing IIII and VIII since most often numerals were carved in materials like wood and stone and carving is a difficult procedure, so sparing characters made a difference. For the same reason, for example, though not frequent it is possible to find 18 represented as IIXX rather than XVIII (one character less). The same applies to 28 mentioned by Gerald Edgar above. – Nicola Ciccoli Jun 17 '21 at 14:49
  • @NicolaCiccoli oh, that makes a lot of sense to me. Perhaps in parchments they used more additive ? –  Jun 17 '21 at 14:53
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    @Jordi Not completely sure abut that: I guess that when conventions become common use then they extend over the original intentions. – Nicola Ciccoli Jun 17 '21 at 14:55
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    I was reading on some italian history paper, however, that in written documents in fact additive was more common until the last part of the empire. In Medioeval times it became the norm to use subtractive notations. – Nicola Ciccoli Jun 17 '21 at 15:02
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    Does saying "seventeen" with the 7 before the "teen" confuse you today when you go between written and oral forms? – Gerald Edgar Jun 18 '21 at 18:17
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    @GeraldEdgar Good point ! –  Jun 18 '21 at 18:21

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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.

Martin Vesely
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Nicola Ciccoli
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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.