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]

SymbolIVXLCDM
Value1510501005001,000

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 in most contexts by the more convenient Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some minor 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:

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

The notations IV and IX can be read as "one less than five" (4) and "one less than ten" (9), although there is a strong 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 this century, MM indicates 2000. Thus the current year is MMXIX (2019).

Description

There is not, and never has been, an "official", "binding", or universally accepted standard for Roman numerals. Usage in ancient Rome varied greatly and remained somewhat inconsistent in medieval times and later.[3] The "rules" of the system as it is now applied have been established only by general usage over the centuries.

A "base 10" system

Roman numerals are essentially a decimal or "base 10" number system. Powers of ten – thousands, hundreds, tens and units – are written separately, from left to right, in that order. In the absence of "place keeping" zeros, different symbols are used for each power of ten, but a common pattern is used for each of them.

The underlying form of this pattern employs the symbols I and V (representing 1 and 5) as simple tally marks, to build the numbers from 1 to 9. Each marker for 1 (I) adds a unit value up to 5 (V), and is then added to (V) to make the numbers from 6 to 9. Finally the unit symbol for the next power completes a "finger count" sequence:

I, II, III, IIII, V, VI, VII, VIII, VIIII, X.

At some early time the Romans started to use the abbreviated forms IV ("one less than 5") for IIII and IX ("one less than 10") for VIIII - a convention that has been widely, although not universally, used ever since.[a] This convention is called "subtractive" notation,[4] as opposed to the purely "additive" notation of IIII and VIIII.[5] Thus the numbers from 1 to 10 are generally written as

I, II, III, IV, V, VI, VII, VIII, IX, X.[6]

The multiples of 10, from 10 to 100, are written according to the same pattern, with X, L, and C taking the place of I, V, and X

X, XX, XXX, XL, L, LX, LXX, LXXX, XC, C.

Note that 40 is usually written XL ("10 less than 50") rather than XXXX, and 90 as XC ("10 less than 100") rather than LXXXX.

Similarly, the multiples of 100, 100 to 1000, are written as

C, CC, CCC, CD, D, DC, DCC, DCCC, CM, M.

where CD is to be read as "100 less than 500" (that is, 400), and CM as "100 less than 1000" (that is, 900).

Since the system has no standard symbols for 5,000 and 10,000, the full pattern cannot be extended to the multiples of 1000 – restricting the "thousands" range of "normal" Roman numerals to 1,000, 2,000 and 3,000:

M, MM, MMM.

A number containing several decimal places is represented, as in the Arabic system, by writing its power-of-ten parts — thousands, hundreds, tens and units — in sequence, from left to right, in descending order of value. For example:

  • 39 = 30 + 9 = XXX + IX = XXXIX.
  • 246 = 200 + 40 + 6 = CC + XL + VI = CCXLVI.
  • 789 = 700 + 80 + 9 = DCC + LXXX + IX = DCCLXXXIX.
  • 2,421 = 2000 + 400 + 20 + 1 = MM + CD + XX + I = MMCDXXI.

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

  • 160 = 100 + 60 = C + LX = CLX
  • 207 = 200 + 7 = CC + VII = CCVII
  • 1,009 = 1,000 + 9 = M + IX = MIX
  • 1,066 = 1,000 + 60 + 6 = M + LX + VI = MLXVI[7][8]

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

The largest number that can be represented in this notation is 3,999 (3,000 + 900 + 90 + 9 = MMM + CM + XC + IX = MMMCMXCIX).[b]

Use of additive notation

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

While subtractive notation for multiples of 4 (IV, XL and CD) has been the usual form since Roman times, additive notation (IIII, XXXX,[11] and CCCC[11]) continued to be used, including in compound numbers like XXIIII,[12] LXXIIII,[13] and CCCCLXXXX.[14] The additive forms for 9, 90, and 900 (VIIII,[11]LXXXX,[15] and DCCCC[16]) 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.[17] Isaac Asimov speculates that the use of IV, as the initial letters of "IVPITER" (a classical Latin spelling of the name of the Roman god Jupiter) may have been felt to have been impious in this context.[18]

Modern clock faces that use Roman numerals still usually 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.[19][20][21] 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.[20]

The year number on Admiralty Arch, London (1910). 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 - a classical use of additive notation for MCMX (1910), as seen on Admiralty Arch, London, to the more unusual, if not unique MDCDIII for MCMIII (1903), on the north entrance to the Saint Louis Art Museum.[22]

Sometimes 5 and 50 have been written IIIII and XXXXX instead of V and L, and there are instances such as IIIIII and XXXXXX rather than VI or LX.[23][24]

Irregular subtractive notation

Epitaph of centurion Marcus Caelius, showing "XIIX"

The irregular use of subtractive notation, such as IIIXX for 17,[25] IIXX for 18,[26] IIIC for 97,[27] IIC for 98,[28][29] and IC for 99[30] were occasionally used in more modern times. A possible explanation is that the word for 18 in Latin was duodeviginti, literally "two from twenty". Similarly, the words for 98 and 99 were duodecentum (two from hundred) and undecentum (one from hundred), respectively.[31] These ways of saying 18, 98 and 99 have been attributed to influence from the Etruscans, who would say ciem zaθrum (three from twenty) for 17, eslem zaθrum (two from twenty) for 18 and θunem zaθrum (one from twenty) for 19.[32] Apparently, at least one ancient stonecutter mistakenly thought that the IIXX of "22nd Legion" stood for 18, and "corrected" it to XVIII.[33] 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.

Another example of irregular subtractive notation is the use of XIIX for 18. It was used by officers of the XVIII Roman Legion to write their number.[34][35] The notation appears prominently on the cenotaph of their senior centurion Marcus Caelius (c. 45 BC – AD 9). There does not seem to be a linguistic explanation for this use, although it is one stroke shorter than XVIII.

Rare variants

While the subtractive and additive notations seem to have been used interchangeably through history, some other Roman numerals have been occasionally observed 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 (twentieth second).[33]
Excerpt from Bibliothèque nationale de France.[36] The Roman numeral for 500 is VC, 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 XVCXIX as in French quinze-cent-dix-neuf (fifteen-hundred-ten-nine), and similar readings in other languages.[37]
  • 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).[37] Similarly, in some English documents one finds, for example, 77 written as "iiixxxvii" (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

Not all combinations of symbols used in Roman numerals are intended to be taken numerically. For example, "XXX" and "XL" have other connotations in addition to their values as Roman numerals, while "IXL" more often than not means "I excel", and is in any case not an unambiguous Roman numeral.

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