# Rules of Roman Numerals

I've been working on a problem that involves discovering valid methods of expressing natural numbers as Roman Numerals, and I came across a few oddities in the numbering system.

For example, the number 5 could be most succinctly expressed as $\texttt{V}$, but as per the rules I've seen online, could also be expressed as $\texttt{IVI}$.

Are there any rules that bar the second expression from being valid? Or are the rules for roman numerals such that multiple valid expressions express the same number.

# Edit

A sample set of rules I've seen online:

1. Only one I, X, and C can be used as the leading numeral in part of a subtractive pair.
2. I can only be placed before V or X in a subtractive pair.
3. X can only be placed before L or C in a subtractive pair.
4. C can only be placed before D or M in a subtractive pair.
5. Other than subtractive pairs, numerals must be in descending order (meaning that if you drop the first term of each subtractive pair, then the numerals will be in descending order).
6. M, C, and X cannot be equalled or exceeded by smaller denominations.
7. D, L, and V can each only appear once.
8. Only M can be repeated 4 or more times.
• On old clocks you will find both IIII and IV. – Yves Daoust Aug 10 '18 at 14:36
• Oh yes I know, but is IVI valid? – Don Thousand Aug 10 '18 at 14:50
• By virtue of which rule would it be ? – Yves Daoust Aug 10 '18 at 14:57
• I mean, what rule makes it invalid, no? – Don Thousand Aug 10 '18 at 15:01
• Please exhibit "the rules I've seen online" that might allow this. – Yves Daoust Aug 10 '18 at 15:02

Please keep in mind that strict rules for Roman numerals were established in a recent time. Besides having different signs for the digits, subtractive system was not fixed. In an inscription found in Forum Popilii and dated between 172 BC and 158 BC, 84 is written as XXCIIII (quoted from Georges Ifrah, The Universal History of Numbers: I have the Italian edition, so I cannot provide the page number for the English edition).

As per your question, subtractive and addictive rule cannot be applied to the same number (V, in your case). IIII and XXXX are widely used too: subtractive rule was not mandatory, but rather used as a shortcut.

• Quite interesting! That is actually exactly the rule I was looking for. Thank you for both the answer and additional insight! – Don Thousand Aug 11 '18 at 21:09

"Roman numerals" presumably means how the actual Romans actually wrote down numbers.

They would never have written five as IVI, full stop.

If you're following a particular set of formal rules that are not "do things as the actual Romans would have done them", then follow those rules. Be prepared to get nonsense results if the rules you follow are sloppily phrased.

• I'm more looking for a rule that shows this is incorrect. Examples like these make Roman Numerals hard to quantitfy. – Don Thousand Aug 10 '18 at 14:25
• @RushabhMehta: If you're looking for rules, make up your own! It will be just as authoritative as anything else you can find, in that there are no universal authoritative written-down rules for what "Roman numerals" means or doesn't mean. – Henning Makholm Aug 10 '18 at 14:30
• Fair enough I guess. – Don Thousand Aug 10 '18 at 14:32

Wikipedia explicitly enumerates the patterns for the units, the tens and the hundredths.

https://en.wikipedia.org/wiki/Roman_numerals#Basic_decimal_pattern

This doesn't leave room for extravagant expressions, though some variants are described. Subtractive-additive forms were of course not used.

• I'm talking about subtractive forms in this question, so while this reference is interesting, I don't think it is particularly relevant. – Don Thousand Aug 10 '18 at 14:50
• @RushabhMehta: did you read the material at all ??? This is absolutely relevant. – Yves Daoust Aug 10 '18 at 14:55