Roman numerals

Roman numerals on stern of the ship Cutty Sark showing draught in feet. The numbers range from 13 to 22, from bottom to top.

Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Modern usage employs seven symbols, each with a fixed integer value:^[1]

The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced by Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some applications to this day.

One place they are often seen is on clock faces. For instance, on the clock of Big Ben (designed in 1852), the hours from 1 to 12 are written as:

The notations IV and IX can be read as "one less than five" (4) and "one less than ten" (9), although there is a tradition favouring representation of "4" as "IIII" on Roman numeral clocks.^[2]

Other common uses include year numbers on monuments and buildings and copyright dates on the title screens of movies and television programs. MCM, signifying "a thousand, and a hundred less than another thousand", means 1900, so 1912 is written MCMXII. For the years of this century, MM indicates 2000. The current year is MMXXI (2021).

Description

Roman numerals are essentially a decimal or "base ten" number system, but instead of place value notation (in which place-keeping zeros enable a digit to represent different powers of ten) the system uses a set of symbols with fixed values, including "built in" powers of ten. Tally-like combinations of these fixed symbols correspond to the (placed) digits of Arabic numerals. This structure allows for significant flexibility in notation, and many variant forms are attested.

In fact, there has never been an officially binding, or universally accepted standard for Roman numerals. Usage in ancient Rome varied greatly and became thoroughly chaotic in medieval times. Even the post-renaissance restoration of a largely "classical" notation has failed to produce total consistency: variant forms are even defended by some modern writers as offering improved "flexibility".^[3] On the other hand, especially where a Roman numeral is considered a legally binding expression of a number, as in U.S. Copyright law (where an "incorrect" or ambiguous numeral may invalidate a copyright claim, or affect the termination date of the copyright period)^[4] it is desirable to strictly follow the usual style described below.

Standard form

The following table displays how Roman Numerals are usually written:^[5]

The numerals for 4 (IV) and 9 (IX) are written using "subtractive notation",^[6] where the first symbol (I) is subtracted from the larger one (V, or X), thus avoiding the clumsier (IIII, and VIIII).^[a] Subtractive notation is also used for 40 (XL) and 90 (XC), as well as 400 (CD) and 900 (CM).^[7] These are the only subtractive forms in standard use.

A number containing several 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^[8][9]

Roman numerals for large numbers are seen in the form of year numbers, as in these examples:

• 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)
• 1954 = M + CM + L + IV = MCMLIV (as in the trailer for the movie The Last Time I Saw Paris)^[4]
• 2014 = MM + X + IV = MMXIV (the year of the games of the XXII (22nd) Olympic Winter Games (in Sochi, Russia))

The largest number that can be represented in this notation is 3,999 (MMMCMXCIX), but since the largest Roman numeral likely to be required today is MMXXI (the current year) there is no practical need for larger Roman numerals. Prior to the introduction of Arabic numerals in the West, ancient and medieval users of the system used various means to write larger numbers; see Large numbers below.

Variant forms

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

Use of additive notation

A typical clock face with Roman numerals in Bad Salzdetfurth, Germany

While subtractive notation for 4, 40 and 400 (IV, XL and CD) has been the usual form since Roman times, additive notation (IIII, XXXX and CCCC)^[10] continued to be used, including in compound numbers like XXIIII,^[11] LXXIIII,^[12] and CCCCLXXXX.^[13] The additive forms for 9, 90, and 900 (VIIII,^[10] LXXXX,^[14] and DCCCC^[15]) have also been used, although less frequently.

The two conventions could be mixed in the same document or inscription, even in the same numeral. On the numbered gates to the Colosseum, for instance, IIII is systematically used instead of IV, but subtractive notation is used for other digits; so that gate 44 is labelled XLIIII.^[16]

Modern clock faces that use Roman numerals still very often employ 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.^[17][18][19] However, this is far from universal: for example, the clock on the Palace of Westminster tower, Big Ben, uses a subtractive IV for 4 o'clock.^[18]

Isaac Asimov once mentioned an "interesting theory" that Romans avoided using IV because it was the initial letters of IVPITER, the Latin spelling of Jupiter, and might have seemed impious.^[20] He did not say whose theory it was.

The year number on Admiralty Arch, London. The year 1910 is rendered as MDCCCCX, rather than the more usual MCMX

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.^[21]

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.^[22][23]

Irregular subtractive notation

There is a common belief that any smaller digit placed to the left of a larger digit is subtracted from the total, and that by clever choices a long Roman numeral can be "compressed". The best known example of this is the ROMAN() function in Microsoft Excel, which can turn 499 into CDXCIX, LDVLIV, XDIX, VDIV, or ID depending on the "Form" setting.^[24] There is no indication this is anything other than an invention by the programmer, and the universal-subtraction belief may be a result of modern users trying to rationalize the syntax of Roman numerals.

Epitaph of centurion Marcus Caelius, showing "XIIX"

There is however some historic use of subtractive notation other than that described in the above "standard": in particular IIIXX for 17,^[25] IIXX for 18,^[26] IIIC for 97,^[27] IIC for 98,^[28][29] and IC for 99.^[30] A possible explanation is that the word for 18 in Latin is duodeviginti, literally "two from twenty", 98 is duodecentum (two from hundred), and 99 is undecentum (one from hundred).^[31] 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.

There are multiple examples of IIX being used for 8. There does not seem to be a linguistic explanation for this use, although it is one stroke shorter than VIII. XIIX was used by officers of the XVIII Roman Legion to write their number.^[32][33] The notation appears prominently on the cenotaph of their senior centurion Marcus Caelius (c. 45 BC – AD 9). 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.^[34]

Rare variants

While irregular subtractive and additive notation has been used at least occasionally throughout history, some Roman numerals have been observed in documents and inscriptions that do not fit either system. Some of these variants do not seem to have been used outside specific contexts, and may have been regarded as errors even by contemporaries.

Padlock used on the north gate of the Irish town of Athlone. "1613" in the date is rendered XVIXIII, (literally "16, 13") instead of MDCXIII

• 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).^[35] Apparently, at least one ancient stonecutter mistakenly thought that the IIXX of "22nd Legion" stood for 18, and "corrected" it to XVIII.^[35]

Excerpt from Bibliothèque nationale de France.^[36] The Roman numeral for 500 is rendered as V^C, instead of D

• 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 XV^CXIX as in French quinze-cent-dix-neuf (fifteen-hundred and nineteen), and similar readings in other languages.^[37]
• In some French texts from the 15th century and later one finds constructions like IIII^XXXIX for 99, reflecting the French reading of that number as quatre-vingt-dix-neuf (four-score and nineteen).^[37] Similarly, in some English documents one finds, for example, 77 written as "iii^xxxvii" (which could be read "three-score and seventeen").^[38]
• Another 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".^[39]
• 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.

Non-numeric combinations

As Roman numerals are composed of ordinary alphabetic characters, there may sometimes be confusion with other uses of the same letters. For example, "XXX" and "XL" have other connotations in addition to their values as Roman numerals, while "IXL" more often than not is a gramogram of "I excel", and is in any case not an unambiguous Roman numeral.

Zero

"Place-keeping" zeros are alien to the system of Roman numerals - however the actual number zero (what remains after 1 is subtracted from 1) was also missing from the classical Roman numeral system. 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 A.D.525.^[40][41] 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.^[42]

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.^[43]

Fractions

A triens coin (1⁄3 or 4⁄12 of an as). Note the four dots (····) indicating its value.

A semis coin (1⁄2 or 6⁄12 of an as). Note the S indicating its value.

The base "Roman fraction" is S, indicating 1⁄2. The use of S (as in VIIS to indicate 71⁄2) is attested in some ancient inscriptions^[44] and also in the now rare apothecaries' system (usually in the form SS):^[43] 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.^[45] 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: