# Why is largest number in Roman Numerals not represented as "MMMIM"?

according to wikipedia, the largest number Roman Numeral system can represent is represented like following: (answer below has much bigger number represented)

``````MMMCMXCIX
``````

why can't it be represented like following?

``````MMMIM
``````

As a part-time computer scientist, I've tried to come up with different ways of analyzing the Roman numeral system so that the rules can be formulated in as general a way as possible, with no special cases per glyph. The one that satisifes me the most, for the time being, is one in which we analyze the seven glyphs as four sets that each have a multiplier — similar to how each column has a place-value multiplier in the Arabic numeral system.

These sets are `I V X` ; `X L C` ; `C D M` ; `M` (the Romans did not leave us perfectly symmetrical sets). You can only subtract the first glyph in each set from the second or third in the same set. This precludes `IM`.

To see this analysis in action, read this simple handout. It includes exercises to practice with and an online tool to verify your answers. For the sake of perpetual Stack Exchange availability, I'll copy a screenshot of the first page here:

• wonderful handout, I didn't find rule for how many same symbol can one place consecutively though. is it 3? Commented Aug 10, 2019 at 8:28
• Additionally, could you provide source/reference for this handout? is this grammar authoritative in any regard? Commented Aug 10, 2019 at 8:49
• @Sajuuk The source is the Wikipedia article on Roman numerals. It lists a few variants but I just described the most widely accepted one. Interestingly, I reread the article for the first time in about 8 years and found that it uses a similar analysis about place value. I just added a note about the source to the handout. Commented Aug 10, 2019 at 14:41
• As for how many of the same symbol: you can only do that for the `x1` symbols. The limit is usually 3, but a common variant of the system allows 4. (I mentioned that in one of the exercises because it greatly increases the upper bound of numbers you can write! The checker tool linked in the handout also lets you write 4 of the same symbol.) Commented Aug 10, 2019 at 14:42
• Then there's this inscription, over a fireplace at Trerice. Yes, that is a numeral '3' on the end. Commented Aug 10, 2019 at 23:05

It's because of the grammar of the Roman numerals (It's not the same as natural language grammar, but it's still grammar).

`I` does not combine with all characters. `I` combines on the left of `V` or `X` to mean `-1`. (Just like `X` combines on the left of `L` or `C` to mean `-10`, but combinations like `XM` does not work).

So the problem with `MMMIM` is that `IM` is invalid and cannot mean `999`. To subtract anything from `M`, you can use `C`.

• Can you provide source or reference for the formal grammar?
– Sajuuk
Commented Aug 9, 2019 at 12:12
• Here's something. compilers.iecc.com/comparch/article/07-03-118 Though I would have preferred a higher-order approach. Commented Aug 9, 2019 at 12:16

Maybe because you cannot skip so many slots, mind that 49 is XLIX, not IL, and 99 is XCIX, not IC. Anyhow, you can write much bigger numbers than that MMMCMXCIX (3999). By putting a horizontal line, called vinculum or overline above a Roman numeral, you multiply it by 1,000, so M with an overline is 1,000,000 that is one million. Using the overline, the biggest number that can be written is 3,999,999 (see the image).

Note, that the overline is a medieval feature.

• wikipedia used the word "another system is 'Vinculum'", is this some sort of extension to original Roman Numeral system? and by medieval feature, do you mean this is only used in europe since middle age?
– Sajuuk
Commented Aug 9, 2019 at 12:18
• @Sajuuk - Yes, you are right, since the Middle Ages. Commented Aug 9, 2019 at 12:29