Roman Numeral Converter

How to use the roman numeral converter

Type a number and get its numeral, or type a numeral and get its number. The breakdown row underneath shows the addition the letters stand for, which is the part worth reading: 1994 is MCMXCIV because it is 1000 plus 900 plus 90 plus 4, and each of those last three is a subtractive pair rather than a run of repeats.

The seven letters never change value. I is 1, V is 5, X is 10, L is 50, C is 100, D is 500 and M is 1000. Everything else in the system is a rule about the order you may put them in, and there are only two rules that matter.

1. 1

   Larger before smaller means add

   Reading left to right, values normally descend and you total them up. MMXXVI is 1000 plus 1000 plus 10 plus 10 plus 5 plus 1, which is 2026.

2. 2

   Smaller before larger means subtract

   A smaller letter placed before a larger one is taken away from it. IV is 4 and IX is 9, which is how the system avoids ever writing four of the same letter in a row.

3. 3

   Only I, X and C may subtract

   And only from the next two values up. So IV, IX, XL, XC, CD and CM are the complete list of six legal pairs. V, L and D never subtract from anything.

That short list is why this converter rejects rather than guesses. IIII breaks the repeat rule, VX puts a five-letter in front of a larger one, and IC tries to subtract across two orders of magnitude at once. Every one of them has an obvious intended meaning, and none of them is how the number is written.

So why does the clock on the wall say IIII?

Because clock dials follow a horological convention rather than the grammar this tool enforces, and they have done so for centuries. Look at almost any traditional dial with Roman hours and the four is written IIII, not IV, while every other hour on the same face follows the ordinary rules.

Several explanations get repeated, and they are worth separating from each other by how well they hold up. The visual argument is the strongest of them: IIII balances the VIII sitting opposite it, and the dial divides neatly into a first group using only I, a second using V, and a third using X.

The famous counterexample is the Great Clock at Westminster, the dial usually called Big Ben, which uses IV. So the convention is a strong tendency rather than a law, and a numeral is not wrong for following either side of it. What is true is that IIII on a clock face is a real and long-standing exception to the rule this converter enforces everywhere else.

Treat it the way you would any craft convention. If you are labelling chapters, dating a film, or checking a monument inscription, use the standard subtractive form the tool produces. If you are designing a watch face, IIII is the traditional choice and nobody in horology will blink at it.

What the system genuinely cannot express

Roman numerals are additive rather than positional, and that single difference explains every limitation. There is no column that could be empty, so there is nothing for a zero to mark and no symbol was ever needed for one. The Romans handled the idea of nothing with the word nulla when they had to write it at all.

There are no negative numbers either, and no fractions in the letter system. Roman fractions existed but used a separate duodecimal notation based on twelfths, with its own symbols, which is a different system rather than an extension of this one.

The 3999 ceiling is the practical one. Roman writers did have ways past it: the vinculum drew a bar over a letter to multiply it by 1000, so a barred X meant ten thousand, and an earlier apostrophus notation used enclosing marks for the same job. Neither survives as plain typeable text.

That is why this converter stops where it does. Emitting a barred M would mean choosing between a combining overline character that renders inconsistently, an image, or a made-up convention, and none of those is something you could paste into a document and trust. A clean refusal at 3999 is more honest than a numeral that looks different everywhere it is displayed.

The formula, worked line by line

Encoding is a single greedy pass down a ladder of thirteen rungs. The ladder holds the seven letters and the six subtractive pairs together in descending value order, and the pairs sit between the letters at exactly the values they represent. Take the largest rung that fits, subtract it, repeat.

Putting the pairs on the same ladder as the letters is what makes the greedy pass correct. When the algorithm reaches 4 it finds the IV rung before it finds the I rung, so it can never emit IIII, and the same holds at 9, 40, 90, 400 and 900.

ladder = M 1000, CM 900, D 500, CD 400, C 100, XC 90, L 50, XL 40, X 10, IX 9, V 5, IV 4, I 1
encode: take the largest rung that fits, append its symbol, subtract its value, repeat
decode: sum each letter, but subtract any letter smaller than the one after it
valid = M{0,3} then (CM | CD | D?C{0,3}) then (XC | XL | L?X{0,3}) then (IX | IV | V?I{0,3})
accepted only if encode(decode(input)) is identical to the input

Decoding is the reverse scan, and on its own it is far too generous. A plain left-to-right sum with subtraction happily reports 4 for IIII and 99 for IC, because both strings do describe a coherent arithmetic even though neither is a legal spelling.

So the tool applies two gates. The grammar pattern above rejects malformed shapes outright, and then the parsed value is re-encoded and compared against what you typed. Only a string that survives both is accepted, which guarantees every input the tool takes is one it could have produced itself.

The range is small enough to verify exhaustively rather than argue about, so the test suite does exactly that. All 3999 values are encoded, decoded and compared back, producing 3999 distinct numerals with no collisions and no value that fails its own grammar.

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