Roman numerals

On clock faces, in movie credits, in royal names like Louis XIV: Roman numerals are still with us today. More than 2000 years ago the Romans used them to run an empire, count legions and date buildings, entirely without the digits 0 to 9. Instead of digits they used letters, and instead of place values a simple idea: write down symbols and add up their values. The system is quick to learn, and along the way it shows you why the way we write numbers today is so ingenious.

Seven symbols are enough

The Romans managed with just seven symbols: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000. You write a number by placing symbols side by side and adding up all their values. VIII is 5 + 1 + 1 + 1 = 8, and MMXXVI is 1000 + 1000 + 10 + 10 + 5 + 1 = 2026. Three rules keep everything tidy: as long as you only add, the symbols are sorted from large to small, I, X, C and M may appear at most three times in a row, and you never repeat V, L or D, because two of the same already make the next larger symbol.

Smaller before larger means subtract

So how do you write 4 if IIII is not allowed? With a trick: when a smaller symbol stands before a larger one, it gets subtracted. IV means 5 − 1 = 4 and IX means 10 − 1 = 9. Exactly six such subtraction pairs are allowed: IV, IX, XL (40), XC (90), CD (400) and CM (900). That turns 1994 into MCMXCIV: first M, then CM, then XC, then IV. By the way, many old clock faces still show IIII instead of IV, a charming exception that clockmakers have kept alive to this day.

With this toolkit you can write any number from 1 to 3999. Just break it apart by place: 2748 is 2000 + 700 + 40 + 8. Then translate each building block on its own, MM for 2000, DCC for 700, XL for 40 and VIII for 8, and join everything together: MMDCCXLVIII. So the Romans were already thinking in thousands, hundreds, tens and ones, they simply had no short way of writing them. For even larger numbers there were special marks, such as a bar above a symbol that multiplied its value by a thousand.

Why don't we calculate the Roman way anymore? The Romans were missing two ideas: zero and place value. In our system a digit's position reveals its value, which is why a short string of four digits is enough for 1888. In Roman numerals the same number takes thirteen symbols: MDCCCLXXXVIII. Written multiplication or division is barely possible like that, so the Romans calculated on an abacus and used the symbols only to record results. From 1202 onwards the Hindu-Arabic digits spread across Europe, pushed along by the arithmetic book Liber Abaci by the mathematician Fibonacci. They won out because you could calculate with them directly on paper. Roman numerals stayed on as beautiful decoration: on clocks, in chapter numbers and at almost every Super Bowl.