How Roman numerals work
Seven letters and two rules. Everything else about Roman numerals follows from those, including why the awkward ones are awkward.
The seven letters
| Letter | Value |
|---|---|
| I | 1 |
| V | 5 |
| X | 10 |
| L | 50 |
| C | 100 |
| D | 500 |
| M | 1,000 |
They alternate between ones and fives: I and V, X and L, C and D, then M. There is no letter for five thousand, which is why the system stops at 3,999.
Rule one: add as you go
Read left to right and add. MMXXVI is 1000 + 1000 + 10 + 10 + 5 + 1, which is 2,026. Letters are written largest first, and the same letter is never repeated more than three times in a row.
The three-in-a-row limit is what forces the second rule. Without it, four would be IIII and nine would be VIIII, and both were in fact written that way for centuries.
Rule two: a smaller letter in front subtracts
When a smaller letter stands immediately before a larger one, it is taken away instead of added. IV is one before five, so four. IX is one before ten, so nine.
Only six such pairs are allowed, and the restriction is what keeps the system readable. A letter is only ever subtracted from the next two values above it, and only I, X and C are ever subtracted at all, never V, L or D.
| Pair | Value | How it reads |
|---|---|---|
| IV | 4 | I before V takes one from five |
| IX | 9 | I before X takes one from ten |
| XL | 40 | X before L takes ten from fifty |
| XC | 90 | X before C takes ten from a hundred |
| CD | 400 | C before D takes a hundred from five hundred |
| CM | 900 | C before M takes a hundred from a thousand |
Building one from a number
Take each column separately (thousands, hundreds, tens, units), write each as its own numeral, and put them one after another. The columns never interact, which is what makes the system easy to write and hard to calculate with.
- 49 is 40 + 9, so XL + IX, giving XLIX
- 1987 is 1000 + 900 + 80 + 7, so M + CM + LXXX + VII, giving MCMLXXXVII
| Number | Numeral |
|---|---|
| 49 | XLIX |
| 1987 | MCMLXXXVII |
| 2026 | MMXXVI |
| 3999 | MMMCMXCIX |
Why there is no zero
The system has no symbol for zero and no place value, so nothing holds a column open. That is not an oversight: a numeral is a list of quantities added together, and an absent quantity is simply not written. It is also why arithmetic in Roman numerals is so difficult, and why the place-value system replaced it for calculation while the numerals stayed on for labels, clock faces and dates.
Above 3,999
The largest number with a standard numeral is 3,999, written MMMCMXCIX. Larger numbers were written with a vinculum, a bar over a letter that multiplies it by a thousand, so V̄ is five thousand. The bar is not part of the standard modern form, it does not survive being typed into most systems, and the converter does not use it.