# Roman numerals

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 style uses 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 MMXXII (2022).

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.

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), 90 (XC), 400 (CD) and 900 (CM).[7] 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:

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

In practice, Roman numerals for large numbers are currently used mainly for year numbers, as in these examples:

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 MMXXII (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.

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

While subtractive notation for 4, 40 and 400 (IV, XL and CD) has been the usual form since Roman times, additive notation to represent these numbers (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 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; so that gate 44 is labelled XLIIII.[16][17]

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.[18][19][20] 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.[19]

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.[21] He did not say whose theory it was.

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.[22]

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.[23][24]

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.[25] 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.

There is however some historic use of subtractive notation other than that described in the above "standard": in particular IIIXX for 17,[26] IIXX for 18,[27] IIIC for 97,[28] IIC for 98,[29][30] and IC for 99.[31] 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).[32] 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.[33][34] 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.[35]

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.

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.

"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.[41][42] 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.[43]

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.[44]

The base "Roman fraction" is S, indicating 12. The use of S (as in VIIS to indicate 712) is attested in some ancient inscriptions[45] and also in the now rare apothecaries' system (usually in the form SS):[44] but while Roman numerals for whole numbers are essentially decimal S does not correspond to 510, as one might expect, but 612.

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 13 and 14 than does a system based on ten (10 = 2 × 5). Notation for fractions other than 12 is mainly found on surviving Roman coins, many of which had values that were duodecimal fractions of the unit as. Fractions less than 12 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.[46] 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 112 to 1212 had a name in Roman times; these corresponded to the names of the related coins:

During the centuries that Roman numerals remained the standard way of writing numbers throughout Europe, there were various extensions to the system designed to indicate larger numbers, none of which were ever standardised.

One of these was the apostrophus,[47] in which 500 was written as IↃ, while 1,000 was written as CIↃ.[21] This is a system of encasing numbers to denote thousands (imagine the Cs and s as parentheses), which has its origins in Etruscan numeral usage. The IↃ and CIↃ used to represent 500 and 1,000 most likely preceded, and subsequently influenced, the adoption of "D" and "M" in conventional Roman numerals.

Each additional set of C and surrounding CIↃ raises the value by a power of ten: CCIↃↃ represents 10,000 and CCCIↃↃↃ represents 100,000. Similarly, each additional to the right of IↃ raises the value by a power of ten: IↃↃ represents 5,000 and IↃↃↃ represents 50,000. Numerals larger than CCCIↃↃↃ do not occur.[48]

Page from a 16th-century manual, showing a mixture of apostrophus and vinculum numbers (see in particular the ways of writing 10,000).

Sometimes CIↃ was reduced to for 1,000. John Wallis is often credited for introducing the symbol for infinity (modern ∞), and one conjecture is that he based it on this usage, since 1,000 was hyperbolically used to represent very large numbers. Similarly, IↃↃ for 5,000 was reduced to ; CCIↃↃ for 10,000 to ; IↃↃↃ for 50,000 to (); and CCCIↃↃↃ () for 100,000 to . [49]

Another system was the vinculum, in which conventional Roman numerals were multiplied by 1,000 by adding a "bar" or "overline".[49] It was a common alternative to the apostrophic ↀ during the Imperial era: both systems were in simultaneous use around the Roman world (M for '1000' was not in use until the Medieval period).[50] [51] The use of vinculum for multiples of 1,000 can be observed, for example, on the milestones erected by Roman soldiers along the Antonine Wall in the mid-2nd century AD.[52] There is some scope for confusion when an overline is meant to denote multiples of 1,000, and when not. The Greeks and Romans often overlined letters acting as numerals to highlight them from the general body of the text, without any numerical significance. This stylistic convention was, for example, also in use in the inscriptions of the Antonine Wall,[53] and the reader is required to decipher the intended meaning of the overline from the context. The vinculum for marking 1,000s continued in use in the Middle Ages, though it became known more commonly as titulus.[54]

Some modern sources describe Vinculum as if it were a part of the current "standard".[55] However, this is purely hypothetical, since no common modern usage requires numbers larger than the current year (MMXXII). Nonetheless, here are some examples, to give an idea of how it might be used:

Another inconsistent medieval usage was the addition of vertical lines (or brackets) before and after the numeral to multiply it by 10 (or 100): thus M for 10,000 as an alternative form for X. In combination with the overline the bracketed forms might be used to raise the multiplier to (say) ten (or one hundred) thousand, thus:

This use of lines is distinct from the custom, once very common, of adding both underline and overline (or very large serifs) to a Roman numeral, simply to make it clear that it is a number, e.g. for 1967.

The system is closely associated with the ancient city-state of Rome and the Empire that it created. However, due to the scarcity of surviving examples, the origins of the system are obscure and there are several competing theories, all largely conjectural.

Rome was founded sometime between 850 and 750 BC. At the time, the region was inhabited by diverse populations of which the Etruscans were the most advanced. The ancient Romans themselves admitted that the basis of much of their civilization was Etruscan. Rome itself was located next to the southern edge of the Etruscan domain, which covered a large part of north-central Italy.

The Roman numerals, in particular, are directly derived from the Etruscan number symbols: "𐌠", "𐌡", "𐌢", "𐌣", and "𐌟" for 1, 5, 10, 50, and 100 (They had more symbols for larger numbers, but it is unknown which symbol represents which number). As in the basic Roman system, the Etruscans wrote the symbols that added to the desired number, from higher to lower value. Thus the number 87, for example, would be written 50 + 10 + 10 + 10 + 5 + 1 + 1 = 𐌣𐌢𐌢𐌢𐌡𐌠𐌠 (this would appear as 𐌠𐌠𐌡𐌢𐌢𐌢𐌣 since Etruscan was written from right to left.)[56]

The symbols "𐌠" and "𐌡" resembled letters of the Etruscan alphabet, but "𐌢", "𐌣", and "𐌟" did not. The Etruscans used the subtractive notation, too, but not like the Romans. They wrote 17, 18, and 19 as "𐌠𐌠𐌠𐌢𐌢", "𐌠𐌠𐌢𐌢", and 𐌠𐌢𐌢, mirroring the way they spoke those numbers ("three from twenty", etc.); and similarly for 27, 28, 29, 37, 38, etc. However they did not write "𐌠𐌡" for 4 (or "𐌢𐌣" for 40), and wrote "𐌡𐌠𐌠", "𐌡𐌠𐌠𐌠" and "𐌡𐌠𐌠𐌠𐌠" for 7, 8, and 9, respectively.[56]

The early Roman numerals for 1, 10, and 100 were the Etruscan ones: "I", "X", and "Ж". The symbols for 5 and 50 changed from Ʌ and "𐌣" to V and ↆ at some point. The latter had flattened to (an inverted T) by the time of Augustus, and soon afterwards became identified with the graphically similar letter L.[48]

The symbol for 100 was written variously as >I< or ↃIC, was then abbreviated to or C, with C (which matched a Latin letter) finally winning out. It may have helped that C is the initial of centum, Latin for "hundred".

The numbers 500 and 1000 were denoted by V or X overlaid with a box or circle. Thus 500 was like a superimposed on a Þ. It became D or Ð by the time of Augustus, under the graphic influence of the letter D. It was later identified as the letter D; an alternative symbol for "thousand" was a CIↃ, and half of a thousand or "five hundred" is the right half of the symbol, IↃ, and this may have been converted into D.[21]

The notation for 1000 was a circled or boxed X: Ⓧ, , , and by Augustinian times was partially identified with the Greek letter Φ phi. Over time, the symbol changed to Ψ and . The latter symbol further evolved into , then , and eventually changed to M under the influence of the Latin word mille "thousand".[48]

According to Paul Kayser, the basic numerical symbols were I, X, C and Φ (or ) and the intermediate ones were derived by taking half of those (half an X is V, half a C is L and half a Φ/⊕ is D).[57]

The Colosseum was constructed in Rome in CE 72–80,[58] and while the original perimeter wall has largely disappeared, the numbered entrances from XXIII (23) to LIIII (54) survive,[59] to demonstrate that in Imperial times Roman numerals had already assumed their classical form: as largely standardised in current use. The most obvious anomaly (a common one that persisted for centuries) is the inconsistent use of subtractive notation - while XL is used for 40, IV is avoided in favour of IIII: in fact gate 44 is labelled XLIIII.

Lower case, or minuscule, letters were developed in the Middle Ages, well after the demise of the Western Roman Empire, and since that time lower-case versions of Roman numbers have also been commonly used: i, ii, iii, iv, and so on.

Since the Middle Ages, a "j" has sometimes been substituted for the final "i" of a "lower-case" Roman numeral, such as "iij" for 3 or "vij" for 7. This "j" can be considered a swash variant of "i". Into the early 20th century, the use of a final "j" was still sometimes used in medical prescriptions to prevent tampering with or misinterpretation of a number after it was written.[60]

Numerals in documents and inscriptions from the Middle Ages sometimes include additional symbols, which today are called "medieval Roman numerals". Some simply substitute another letter for the standard one (such as "A" for "V", or "Q" for "D"), while others serve as abbreviations for compound numerals ("O" for "XI", or "F" for "XL"). Although they are still listed today in some dictionaries, they are long out of use.[61]

Chronograms, messages with dates encoded into them, were popular during the Renaissance era. The chronogram would be a phrase containing the letters I, V, X, L, C, D, and M. By putting these letters together, the reader would obtain a number, usually indicating a particular year.

By the 11th century, Arabic numerals had been introduced into Europe from al-Andalus, by way of Arab traders and arithmetic treatises. Roman numerals, however, proved very persistent, remaining in common use in the West well into the 14th and 15th centuries, even in accounting and other business records (where the actual calculations would have been made using an abacus). Replacement by their more convenient "Arabic" equivalents was quite gradual, and Roman numerals are still used today in certain contexts. A few examples of their current use are:

In astronomy, the natural satellites or "moons" of the planets are traditionally designated by capital Roman numerals appended to the planet's name. For example, Titan's designation is Saturn VI.

In chemistry, Roman numerals are often used to denote the groups of the periodic table. They are also used in the IUPAC nomenclature of inorganic chemistry, for the oxidation number of cations which can take on several different positive charges. They are also used for naming phases of polymorphic crystals, such as ice.

In education, school grades (in the sense of year-groups rather than test scores) are sometimes referred to by a Roman numeral; for example, "grade IX" is sometimes seen for "grade 9".

In entomology, the broods of the thirteen and seventeen year periodical cicadas are identified by Roman numerals.

In graphic design stylised Roman numerals may represent numeric values.

In law, Roman numerals are commonly used to help organize legal codes as part of an alphanumeric outline.

In advanced mathematics (including trigonometry, statistics, and calculus), when a graph includes negative numbers, its quadrants are named using I, II, III, and IV. These quadrant names signify positive numbers on both axes, negative numbers on the X axis, negative numbers on both axes, and negative numbers on the Y axis, respectively. The use of Roman numerals to designate quadrants avoids confusion, since Arabic numerals are used for the actual data represented in the graph.

In military unit designation, Roman numerals are often used to distinguish between units at different levels. This reduces possible confusion, especially when viewing operational or strategic level maps. In particular, army corps are often numbered using Roman numerals (for example the American XVIII Airborne Corps or the WW2-era German III Panzerkorps) with Arabic numerals being used for divisions and armies.

In pharmacy, Roman numerals were used with the now largely obsolete apothecaries' system of measurement: including SS to denote "one half" and N to denote "zero".[44][66]

In photography, Roman numerals (with zero) are used to denote varying levels of brightness when using the Zone System.

In seismology, Roman numerals are used to designate degrees of the Mercalli intensity scale of earthquakes.

In sport the team containing the "top" players and representing a nation or province, a club or a school at the highest level in (say) rugby union is often called the "1st XV", while a lower-ranking cricket or American football team might be the "3rd XI".

In tarot, Roman numerals (with zero) are used to denote the cards of the Major Arcana.

In theology and biblical scholarship, the Septuagint is often referred to as LXX, as this translation of the Old Testament into Greek is named for the legendary number of its translators (septuaginta being Latin for "seventy").

Some uses that are rare or never seen in English speaking countries may be relatively common in parts of continental Europe and in other regions (e.g. Latin America) that use a European language other than English. For instance:

Capital or small capital Roman numerals are widely used in Romance languages to denote centuries, e.g. the French xviiie siècle[67] and the Spanish siglo XVIII mean "18th century". Slavic languages in and adjacent to Russia similarly favor Roman numerals ( xviii век). On the other hand, in Slavic languages in Central Europe, like most Germanic languages, one writes "18." (with a period) before the local word for "century".

Mixed Roman and Arabic numerals are sometimes used in numeric representations of dates (especially in formal letters and official documents, but also on tombstones). The month is written in Roman numerals, while the day is in Arabic numerals: "4.VI.1789" and "VI.4.1789" both refer unambiguously to 4 June 1789.

Roman numerals are sometimes used to represent the days of the week in hours-of-operation signs displayed in windows or on doors of businesses,[68] and also sometimes in railway and bus timetables. Monday, taken as the first day of the week, is represented by I. Sunday is represented by VII. The hours of operation signs are tables composed of two columns where the left column is the day of the week in Roman numerals and the right column is a range of hours of operation from starting time to closing time. In the example case (left), the business opens from 10 AM to 7 PM on weekdays, 10 AM to 5 PM on Saturdays and is closed on Sundays. Note that the listing uses 24-hour time.

Roman numerals may also be used for floor numbering.[69][70] For instance, apartments in central Amsterdam are indicated as 138-III, with both an Arabic numeral (number of the block or house) and a Roman numeral (floor number). The apartment on the ground floor is indicated as 138-huis.

In Italy, where roads outside built-up areas have kilometre signs, major roads and motorways also mark 100-metre subdivisionals, using Roman numerals from I to IX for the smaller intervals. The sign IX/17 thus marks 17.9 km.

Certain Spanish-speaking Latin American countries use Roman numerals to designate assemblies of their national legislatures. For instance, the composition of the Mexican Congress of the Union from 2018 to 2021 (elected in the 2018 Mexican general election) is called the LXIV Legislature of the Mexican Congress (or more commonly the "LXIV Legislature").

A notable exception to the use of Roman numerals in Europe is in Greece, where Greek numerals (based on the Greek alphabet) are generally used in contexts where Roman numerals would be used elsewhere.

The "Number Forms" block of the Unicode computer character set standard has a number of Roman numeral symbols in the range of code points from U+2160 to U+2188.[71] This range includes both upper- and lowercase numerals, as well as pre-combined characters for numbers up to 12 (Ⅻ or XII). One justification for the existence of pre-combined numbers is to facilitate the setting of multiple-letter numbers (such as VIII) on a single horizontal line in Asian vertical text. The Unicode standard, however, includes special Roman numeral code points for compatibility only, stating that "[f]or most purposes, it is preferable to compose the Roman numerals from sequences of the appropriate Latin letters".[72] The block also includes some apostrophus symbols for large numbers, an old variant of "L" (50) similar to the Etruscan character, the Claudian letter "reversed C", etc.