Roman numerals

The seven letters are younger than the counting they record. Rome's numerals descend from the Etruscan signs for 1, 5, 10, 50 and 100, and those are usually traced in turn to notched tally sticks: a cut for each unit, a doubled cut for every fifth, a cross cut for every tenth. It is a tidy reconstruction, and widely repeated, but no ancient tally stick survives because wood rots, so it remains conjecture; tally sticks of that design were reportedly still being cut by Italian and Dalmatian shepherds in the nineteenth century. The mnemonic link between C, M and centum, mille came later. A thousand was first written Φ, or CIↃ, and D for five hundred is generally explained as half of that sign.

The system has no zero and never needed one: values are added rather than placed, so no column sits empty waiting for a marker. When medieval computists did need a zero in their Easter tables they wrote a word instead — Dionysius Exiguus used nulla in 525, and around 725 Bede used the single letter N in a table of epacts.

Subtractive notation took more than a millennium to settle. Roman inscriptions mix the conventions freely: gate 44 of the Colosseum is cut as XLIIII, and officers of Legio XVIII wrote eighteen as XIIX — the order of the Latin duodeviginti, "two from twenty" — a form that appears on the cenotaph of their senior centurion Marcus Caelius, who died in AD 9. Masons kept the older habit long afterwards: Admiralty Arch in London carries MDCCCCX for 1910 rather than MCMX. The rule taught today, with exactly six subtractive pairs and no alternatives, hardened only in the later Middle Ages and the centuries of print that followed.

Key properties

• Standard notation uses exactly seven letters: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1,000.
• Only six subtractive pairs are allowed — IV, IX, XL, XC, CD and CM. A letter is never subtracted from more than ten times its own value, so IC and IM are not numerals.
• Standard notation spells every integer from 1 to 3,999 exactly one way, and 3,999 = MMMCMXCIX is as far as it reaches.
• The longest standard numeral is MMMDCCCLXXXVIII for 3,888, at fifteen letters.
• There is no symbol for zero, and none for a negative value or a decimal fraction. Fractions were handled by a separate duodecimal set, with S for one half and a dot for each twelfth.
• 1666 = MDCLXVI uses each of the seven letters exactly once, in descending order.
• An overline — the vinculum — multiplies a letter by 1,000, so V̅ = 5,000 and the largest value with a single bar is 3,999,999.
• In the apostrophus system CIↃ = 1,000, IↃↃ = 5,000, CCIↃↃ = 10,000, IↃↃↃ = 50,000 and CCCIↃↃↃ = 100,000; these forms were combined additively rather than subtractively.

Where they turn up

• Clock and watch dials: the Wells Cathedral clock, whose mechanism dates to between 1386 and 1392, is the example usually cited for the IIII convention, while the Elizabeth Tower dial at Westminster uses IV. The explanations offered for IIII — balance against the VIII opposite, economy in casting moulds, a king's complaint — are folklore rather than documented history.
• Film and television copyright notices carry the year as a numeral, which is how 1984 appears as MCMLXXXIV. The familiar claim that this was meant to disguise a production's age is a widely repeated explanation, not a documented policy.
• Regnal and papal names, ship names and sequels: Elizabeth II, Benedict XVI, Star Wars Episode IV.
• Chemistry: Stock nomenclature puts the oxidation state in Roman numerals, as in iron(III) chloride.
• Super Bowls have been numbered in Roman numerals since the first few games, with one deliberate exception — Super Bowl 50 in 2016 was branded in Arabic numerals before the NFL returned to LI the following year.
• Book front matter is still paginated in minuscule numerals (i, ii, iii), and outline levels, act and scene references and statute subdivisions use them too.