Roman numerals
Seven signs — I, V, X, L, C, D, M — and two rules: addition and subtraction. Learn to read and write Roman numerals in the subtractive spelling, meet the limit at 3999, and find out why this system lost to positional notation.
Before you start
This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:
- AdditionAddition combines two or more numbers into one — the sum. Learn the names of the parts, the laws of addition (commutativity, associativity, identity element) and how to add in columns with carrying.
- SubtractionSubtraction is the inverse of addition — take the subtrahend away from the minuend to get the difference. Learn the names, the properties, the link to addition and how to subtract in columns with borrowing.
Where this is used
Real situations where you count exactly the way this lesson teaches:
- Clocks and old buildingsA town-hall clock face writes four as IIII or IV and twelve as XII — and the date MDCCLXXXVIII carved above the door reads 1788, so the building is over 230 years old.
- Books and documentsFront matter is paginated I–XVIII, and chapter XIV is the fourteenth chapter. In contracts, clause § XII and point IV.3 pin down a place in the document exactly as a page number does.
- History and heritage conservationA conservator describing a 13th-century church means the years 1201–1300 — century XIII is not the 1300s, and being a hundred years out changes the dating of the whole building.
- Medicine and pharmacyOn a patient chart the severity grade is written in Roman numerals and the dose in Arabic ones, precisely so the two cannot be confused: an entry reading “hypertension II, 2 tablets daily” carries two different numbers, and only the second says how much to give.
All formulas
The seven signs
the whole alphabet of the Roman system
The additive rule
signs in non-increasing order are added
Subtractive spelling
a smaller sign before a larger one is subtracted
A year on a foundation stone
CM = 900, XC = 90, IV = 4
Before positional notation reached Europe, numbers were written with Roman signs. Nobody calculates in them today, but they are still read every day: on clock faces, in chapter and century numbers, on plaques giving a date of construction, and in the page numbers of a preface.
The seven signs
The whole system rests on seven letters:
Notice how they alternate: one and five, ten and fifty, a hundred and five hundred. There is no sign for zero — and none is needed here, because the value of a sign does not depend on where it stands.
The additive rule
Signs are read left to right and added, as long as they stand in non-increasing order:
The same sign may be repeated at most three times — and only the signs I, X, C and M. V, L and D are never repeated, because two fives make a ten, and a ten has a sign of its own.
The subtractive spelling
Since IIII is not allowed, needs another way of being written. The solution: a sign of smaller value placed before a larger one is subtracted from it.
There are exactly six such pairs, and only these are valid:
| Spelling | Value | Spelling | Value |
|---|---|---|---|
| IV | 4 | XC | 90 |
| IX | 9 | CD | 400 |
| XL | 40 | CM | 900 |
So subtraction is allowed only for I before V and X, X before L and C, and C before D and M — always one sign, always one rung down. That is why is XLIX () and not IL.
Reading a Roman numeral
Work from the left, comparing each sign with the next: if the next one is larger, subtract the current one, otherwise add it.
Writing a number in Roman numerals
Going the other way, start from the largest values: take the biggest sign (or subtractive pair) that fits inside the number, subtract its value and repeat.
Always working down from the largest value gives you the subtractive spelling automatically — which is the correct one.
Where the system runs out
The classical spelling reaches , that is MMMCMXCIX. Above it you would have to write MMMM, which breaks the rule of three repetitions; in antiquity larger numbers were marked with a bar over the sign, multiplying its value by a thousand. For school and for everyday use the range up to is ample — a date, a century and a chapter number all fit inside it.
Why people stopped calculating this way
The Roman system is additive, not positional: X means ten wherever it stands. Arabic notation works the other way round — the same digit means three units in and three hundred in . It is the place of a digit that makes column addition and written multiplication possible; in Roman numerals there is nothing to line up under anything.
If you want to see the same number written in other systems — binary, octal or hexadecimal — convert it with our number-systems tool.
Practice
Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.
Common mistakes
- More than three identical signs — is IV, not IIII; is XL, not XXXX.
- An invalid subtractive pair — is XLIX, not IL; subtraction is allowed only in the six listed cases.
- Repeating V, L or D — VV, LL and DD do not exist; X, C and M take their place.
- Confusing a century with its years — century XIII means the years 1201–1300, not the 1300s.
- Subtracting two signs at once — IIX is not a way of writing ; eight is VIII.
Formula card
Topic: Roman numerals
The seven signs
the whole alphabet of the Roman system
The additive rule
signs in non-increasing order are added
Subtractive spelling
a smaller sign before a larger one is subtracted
A year on a foundation stone
CM = 900, XC = 90, IV = 4
