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Roman numerals

Numbers in the Roman numeral system

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Roman Numbers on Cutty Sark , before the ship was burnt in May 2007. The metal sheathing is not copper but Muntz alloy
Alex1011 · CC BY-SA 3.0

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The use of Roman numerals has continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced by the positional Hindu–Arabic numeral system; however, this process was gradual, and the use of Roman numerals has persisted in some contexts, such as on clock faces. For instance, on the clock of Big Ben (designed in 1852), the hours from 1 to 12 are written as:

I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII

Other common uses include year numbers on monuments and buildings and copyright dates on the title screens of films and television programmes. MCM signifies 1000 + 1000 − 100 = 1900, so 1912 is written MCMXII. For the years of the 21st century, MM indicates 2000, so 2026 (the current year) is written MMXXVI.

Description

Roman numerals use different symbols for each power of ten, and there is no zero symbol, in contrast with the place value notation of Arabic numerals (in which place-keeping zeros enable the same digit to represent different powers of ten).

This allows some flexibility in notation, and there has never been an official or universally accepted standard for Roman numerals. Usage varied greatly in ancient Rome and became thoroughly chaotic in medieval times. The more recent restoration of a largely "classical" notation has gained popularity among some, while variant forms are used by some modern writers as seeking more "flexibility". Roman numerals may be considered legally binding expressions of a number, as in U.S. copyright law before the Berne Convention Implementation Act of 1988 (where an "incorrect" or ambiguous numeral in a copyright notice could invalidate a copyright claim or affect the termination date of the copyright period).

Standard form

The following table displays how Roman numerals are usually written in modern times:

The numerals for 4 (IV) and 9 (IX) are written using subtractive notation, where the smaller symbol (I) is subtracted from the larger one (V or X), instead of IIII and VIIII. Subtractive notation is also used for 40 (XL), 90 (XC), 400 (CD) and 900 (CM). These are the only subtractive forms in standard use.

A number containing two or more decimal digits is built by appending the Roman numeral equivalent for each, from highest to lowest, as in the following examples:

  • 39 = XXX + IX = XXXIX.
  • 246 = CC + XL + VI = CCXLVI.
  • 789 = DCC + LXXX + IX = DCCLXXXIX.
  • 2,421 = MM + CD + XX + I = MMCDXXI.

Any missing place (represented by a zero in the place-value equivalent) is omitted, as in Latin (and English) speech:

  • 160 = C + LX = CLX
  • 207 = CC + VII = CCVII
  • 1,009 = M + IX = MIX
  • 1,066 = M + LX + VI = MLXVI

The largest number that can be represented in this manner is 3,999 (MMMCMXCIX), but this is sufficient for the values for which Roman numerals are commonly used today, such as year numbers:

  • 1776 = M + DCC + LXX + VI = MDCCLXXVI (the date written on the book held by the Statue of Liberty).
  • 1918 = M + CM + X + VIII = MCMXVIII (the first year of the Spanish flu pandemic)
  • 1944 = M + CM + XL + IV = MCMXLIV (erroneous copyright notice of the 1954 movie The Last Time I Saw Paris)
  • 2026 = MMXXVI (this year)

For larger numbers (4,000 and larger): Both before and after the introduction of Arabic numerals in the West, from ancient times through medieval and modern, users of Roman numerals have used various means to write larger numbers .

Other forms

Forms exist that vary in one way or another from the general standard represented above.

Other additive forms

While subtractive notation for 4, 40, and 400 (IV, XL, and CD) is the modern standard, additive notation to represent these numbers (IIII, XXXX, and CCCC) was often used, including in compound numbers like 24 (XXIIII), 74 (LXXIIII), and 490 (CCCCLXXXX). The additive forms for 9, 90, and 900 (VIIII, LXXXX, and DCCCC) have also been used, although less often.

The two conventions could be mixed in the same document or inscription, even in the same numeral. For example, on the numbered gates to the Colosseum, IIII is systematically used instead of IV, but subtractive notation is used for XL; consequently, gate 44 is labelled XLIIII.

Especially on tombstones and other funerary inscriptions, 5 and 50 have been occasionally written IIIII and XXXXX instead of V and L, and there are instances such as IIIIII and XXXXXX rather than VI or LX.

Modern clock faces that use Roman numerals still very often use IIII for four o'clock but IX for nine o'clock, a practice that goes back to very early clocks such as the Wells Cathedral clock of the late 14th century. However, this is far from universal: for example, the clock of Elizabeth Tower at the Palace of Westminster uses a subtractive IV for 4 o'clock.

Several monumental inscriptions created in the early 20th century use variant forms for "1900" (usually written MCM). These vary from MDCCCCX for 1910 as seen on Admiralty Arch, London, to the more unusual, if not unique MDCDIII for 1903, on the north entrance to the Saint Louis Art Museum.

Other subtractive forms

There are numerous historical examples of IIX being used for 8; for example, XIIX was used by officers of the Legio XVIII (18th Roman legion) to write their number. The notation appears prominently on the cenotaph of their senior centurion Marcus Caelius (c. 45 BC – 9 AD). On the publicly displayed official Roman calendars known as Fasti, XIIX is used for the 18 days to the next Kalends, and XXIIX for the 28 days in February. The latter can be seen on the sole extant pre-Julian calendar, the Fasti Antiates Maiores. There are historical examples of other subtractive forms: IIIXX for 17, IIXX for 18, IIIC for 97, IIC for 98, and IC for 99. A possible explanation is that the word for 18 in Latin is duodeviginti‍— literally "two from twenty"⁠— while 98 is duodecentum (two from hundred) and 99 is undecentum (one from hundred). However, the explanation does not seem to apply to IIIXX and IIIC, since the Latin words for 17 and 97 were septendecim (seven ten) and nonaginta septem (ninety seven), respectively.

The ROMAN() function in Microsoft Excel supports multiple subtraction modes depending on the "Form" setting. For example, the number "499" (usually CDXCIX) can be rendered as LDVLIV ((500-50)+(50-5)+(5-1)), XDIX ((500-10)+(10-1)), VDIV ((500-5)+(5-1)) or ID (500-1). The relevant Microsoft help page offers no explanation for this function other than to describe its output as "more concise".

Non-standard variants

There are also historical examples of other additive and multiplicative forms, and forms which seem to reflect spoken phrases. Some of these variants may have been regarded as errors even by contemporaries.

  • IIXX was how people associated with the XXII Roman Legion used to write their number. The practice may have been due to a common way to say "twenty-second" in Latin, namely duo et vice(n)sima (literally "two and twentieth") rather than the "regular" vice(n)sima secunda (twenty second). Apparently, at least one ancient stonecutter mistakenly thought that the IIXX of "22nd Legion" stood for 18, and "corrected" it to XVIII.
  • Other numerals that do not fit the usual patterns—such as VXL for 45, instead of the usual XLV—may be due to scribal errors, or the writer's lack of familiarity with the system, rather than being genuine variant usage.

Zero

As a non-positional numeral system, Roman numerals have no "place-keeping" zeros. Furthermore, the system as used by the Romans lacked a numeral for the number zero itself (that is, what remains after 1 is subtracted from 1). The word nulla (the Latin word meaning "none") was used to represent 0, although the earliest attested instances are medieval. For instance Dionysius Exiguus used nulla alongside Roman numerals in a manuscript from 525 AD. About 725, Bede or one of his colleagues used the letter N, the initial of nulla or of nihil (the Latin word for "nothing") for 0, in a table of epacts, all written in Roman numerals.

The use of N to indicate "none" long survived in the historic apothecaries' system of measurement: used well into the 20th century to designate quantities in pharmaceutical prescriptions.

In later times, the Arabic numeral "0" has been used as a zero to open enumerations with Roman numbers. Examples include the 24-hour Shepherd Gate Clock from 1852 and tarot packs such as the 15th-century Sola Busca and the 20th century Rider–Waite packs.

Fractions

The base "Roman fraction" is S, indicating 1⁄2. The use of S (as in VIIS to indicate 7+1⁄2) is attested in some ancient inscriptions and in the now rare apothecaries' system (usually in the form SS): but while Roman numerals for whole numbers are essentially decimal, S does not correspond to 5⁄10, as one might expect, but 6⁄12.

The Romans used a duodecimal rather than a decimal system for fractions, as the divisibility of twelve (12 = 22 × 3) makes it easier to handle the common fractions of 1⁄3 and 1⁄4 than does a system based on ten (10 = 2 × 5). Notation for fractions other than 1⁄2 is mainly found on surviving Roman coins, many of which had values that were duodecimal fractions of the unit as. Fractions less than 1⁄2 are indicated by a dot (·) for each uncia "twelfth", the source of the English words inch and ounce; dots are repeated for fractions up to five twelfths. Six twelfths (one half), is S for semis "half". Uncia dots were added to S for fractions from seven to eleven twelfths, just as tallies were added to V for whole numbers from six to nine. The arrangement of the dots was variable and not necessarily linear. Five dots arranged like (⁙) (as on the face of a die) are known as a quincunx, from the name of the Roman fraction/coin. The Latin words sextans and quadrans are the source of the English words sextant and quadrant.

Each fraction from 1⁄12 to 12⁄12 had a name in Roman times; these corresponded to the names of the related coins:

Other Roman fractional notations included the following:

Fractions could also be indicated with a slash through the last letter in a numeral (e.g. Ɨ), which subtracted the number by an amount less than one (usually 1⁄2).

Large numbers

The modern form can only write numbers up to 3999, and without M in early Roman times only numbers up to 899 could be written. Various schemes have been used over time to write larger numbers.

Apostrophus

Using the apostrophus method, 500 is written as IↃ, while 1,000 is written as CIↃ. This system of encasing numbers to denote thousands (imagine the Cs and Ↄs as parentheses) had its origins in Etruscan numeral usage.

Each additional set of C and Ↄ surrounding CIↃ raises the value by a factor of ten: CCIↃↃ represents 10,000 and CCCIↃↃↃ represents 100,000. Similarly, each additional Ↄ to the right of IↃ raises the value by a factor of ten: IↃↃ represents 5,000 and IↃↃↃ represents 50,000. Numerals larger than CCCIↃↃↃ do not occur.

  • IↃ = 500
  • CIↃ = 1,000
  • IↃↃ = 5,000
  • CCIↃↃ = 10,000
  • IↃↃↃ = 50,000
  • CCCIↃↃↃ = 100,000

Sometimes IↃ (500) is reduced to D, CIↃ (1,000) to ↀ, IↃↃ (5,000) to ↁ; CCIↃↃ (10,000) to ↂ; IↃↃↃ (50,000) to ↇ; and CCCIↃↃↃ (100,000) to ↈ. It is likely CIↃ (1000) influenced the later M.

John Wallis is often credited with introducing the ⟨∞⟩ symbol for infinity, and one conjecture is his basing it on ↀ, since 1,000 was hyperbolically used to represent very large numbers.

Vinculum

Using the vinculum, conventional Roman numerals are multiplied by 1,000 by adding a "bar" or "overline", thus:

  • IV = 4,000
  • XXV = 25,000

The vinculum came into use in the late Republic, and it was a common alternative to the apostrophic ↀ during the Imperial era around the Roman world (M for '1000' was not in use until the Medieval period). It continued in use in the Middle Ages, though it became known more commonly as titulus, and it appears in modern editions of classical and medieval Latin texts.

In an extension of the vinculum, a three-sided box (now sometimes printed as two vertical lines and a vinculum) is used to multiply by 100,000, thus:

Vinculum notation is distinct from the custom of adding an overline to a numeral simply to indicate that it is a number. Both usages can be seen on Roman inscriptions of the same period and general location, such as on the Antonine Wall.

Other

  • There are some examples of year numbers after 1000 written as two Roman numerals 1–99, e.g. 1613 as XVIXIII, corresponding to the common reading "sixteen thirteen" of such year numbers in English, or 1519 as XXIX as in French quinze-cent-dix-neuf (fifteen-hundred and nineteen), and similar readings in other languages.
  • In some French texts from the 15th century and later, one finds constructions like IIIIXXXIX for 99, reflecting the French reading of that number as quatre-vingt-dix-neuf (four-score and nineteen). Similarly, in some English documents one finds, for example, 77 written as "iiixxxvii" (which could be read "three-score and seventeen").
  • A medieval accounting text from 1301 renders numbers like 13,573 as "XIII. M. V. C. III. XX. XIII", that is, "13×1000 + 5×100 + 3×20 + 13".

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