Roman numerals: chart, rules, and why there is no zero

The seven letters

I is 1, V is 5, X is 10, L is 50, C is 100, D is 500 and M is 1,000. That is the entire alphabet of the system.

C and M come from centum and mille. The others are older and their origins are disputed, which is a good sign that nobody designed this on purpose.

The four rules

Add when they descend. Symbols are written largest first and their values add up. VIII is 5 + 1 + 1 + 1 = 8. MMXXVI is 1000 + 1000 + 10 + 10 + 5 + 1.

Subtract when a smaller one comes first. IV is 5 − 1 = 4; IX is 9; XL is 40; CM is 900. Only one symbol may be placed in front, so 8 is VIII and never IIX.

Only six subtractive pairs exist: IV, IX, XL, XC, CD and CM. I goes before V and X only, X before L and C, C before D and M. So 99 is XCIX — ninety plus nine — and not IC, however sensible IC looks.

Never repeat more than three times. III is 3, and 4 is IV rather than IIII. V, L and D are never repeated at all, since two of any of them makes a symbol that already exists.

The charts

Generated from the rules above rather than typed out, which is worth saying because a hand-typed chart of a hundred numerals contains mistakes nobody ever notices.

Why there is no zero

Our number system is positional: the 0 in 407 is doing real work, holding the tens column empty so the 4 stays in the hundreds. Without it, 407 and 47 would be written the same way.

Roman numerals are not positional. Each symbol carries its own value wherever it stands, so there are no columns and nothing to hold open. Nothing to write simply means nothing written, and a placeholder for it would have no job.

Which is why the system is fine for recording a quantity and unusable for arithmetic. Try multiplying XLVII by XIX and the reason Europe eventually adopted Arabic numerals becomes obvious quickly.

Where the system stops

The largest standard number is 3,999 — MMMCMXCIX. There is no symbol above M, and repeating M more than three times breaks the third rule.

The historical extension is a vinculum: a bar over a letter multiplies it by a thousand, so a barred V is 5,000 and a barred X is 10,000. It appears on monuments and rarely anywhere else, and most converters stop at 3,999 rather than attempt it.

Where you still meet them

Years, in film copyright lines, on cornerstones and in book front matter. This is the most common modern use by a wide margin, and it is why the tail of a film's credits is worth reading if you like a puzzle.

Clock faces, where IIII is the convention rather than IV. It balances the VIII opposite it, and it lets a clockmaker cast a full set from a mould holding twenty I, four V and four X. Big Ben is the famous exception.

Monarchs, popes and sequels. Elizabeth II, Rocky IV.

Outlines and front matter, where lower-case i, ii, iii number the pages before page 1.

The errors that recur

IC for 99. Logical, and not a legal pair. It is XCIX.

IIII for 4 outside a clock face.

MIM for 1999. It is MCMXCIX — a thousand, nine hundred, ninety, nine.

Reading a subtractive pair as an addition. XIX is 19, not 21; the I belongs to the X after it, not the X before it.

Converting

A converter goes both directions and applies the rules exactly, which matters most in the reading direction — a numeral copied from a monument or a film credit is where a subtractive pair is easiest to misread.

The short version

IVXLCDM. Largest first and add; one smaller symbol in front subtracts; six subtractive pairs only; nothing repeats more than three times. Nothing above 3,999 without a bar, and no zero, because there is no column for it to hold.