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 are written with combinations of letters from the Latin alphabet, each letter with a fixed integer value. Modern style uses only these seven:

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. 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 and  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 "" on Roman numeral clocks.

Other common uses include year numbers on monuments and buildings and copyright dates on the title screens of movies and television programs. , signifying "a thousand, and a hundred less than another thousand", means 1900, so 1912 is written. For the years of the current (21st) century, indicates 2000; this year is  (2024).

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 (where an "incorrect" or ambiguous numeral may 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:

The numerals for 4 and 9  are written using subtractive notation, where the smaller symbol  is subtracted from the larger one (, or ), thus avoiding the clumsier  and. Subtractive notation is also used for 40, 90 , 400 and 900. 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:
 * &numsp;&numsp;39 = +  = .
 * &numsp;246 = +  +  = .
 * &numsp;789 = +  +  = .
 * 2,421 = +  +  +  = .

Any missing place (represented by a zero in the place-value equivalent) is omitted, as in Latin (and English) speech:
 * &numsp;160 = +  = 
 * &numsp;207 = +  = 
 * 1,009 = +  = 
 * 1,066 = +  +  = 

The largest number that can be represented in this manner is 3,999 (), but this is sufficient for the values for which Roman numerals are commonly used today, such as year numbers:
 * 1776 = +  +  +  =  (the date written on the book held by the Statue of Liberty).
 * 1918 = +  +  +  =  (the first year of the Spanish flu pandemic)
 * 1944 = +  +  +  =  (erroneous copyright notice of the 1954 movie The Last Time I Saw Paris)
 * =  (this year)

Prior to the introduction of Arabic numerals in the West, ancient and medieval users of Roman numerals used various means to write larger numbers; see large numbers below.

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 (, and ) has been the usual form since Roman times, additive notation to represent these numbers (,  and ) continued to be used, including in compound numbers like 24, 74 , and 490. The additive forms for 9, 90, and 900 (,, and ) 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, is systematically used instead of, but subtractive notation is used for ; consequently, gate 44 is labelled.

Modern clock faces that use Roman numerals still very often use for four o'clock but  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 on the Palace of Westminster tower (commonly known as Big Ben) uses a subtractive for 4 o'clock.



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

Especially on tombstones and other funerary inscriptions 5 and 50 have been occasionally written and  instead of  and, and there are instances such as  and  rather than  or.



Other subtractive forms
There are numerous historical examples of being used for 8; for example,  was used by officers of the XVIII 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, is used for the 18 days to the next Kalends, and  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: for 17,  for 18,  for 97,  for 98,  and  for 99. 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). However, the explanation does not seem to apply to and, 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 "" setting. For example, the number "499" (usually ) can be rendered as, , or. 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.
 * 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 of "22nd Legion" stood for 18, and "corrected" it to.
 * There are some examples of year numbers after 1000 written as two Roman numerals 1–99, e.g. 1613 as, corresponding to the common reading "sixteen thirteen" of such year numbers in English, or 1519 as 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 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 "" (which could be read "three-score and seventeen").
 * A medieval accounting text from 1301 renders numbers like 13,573 as "", that is, "13×1000 + 5×100 + 3×20 + 13".
 * Other numerals that do not fit the usual patterns – such as for 45, instead of the usual  — 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
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 (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, the initial of nulla or of (the Latin word for "nothing") for 0, in a table of epacts, all written in Roman numerals.

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

Fractions


The base "Roman fraction" is, indicating $\overline{CCLXXXXVII}$. The use of (as in  to indicate 7$\overline{CCCCL}$) is attested in some ancient inscriptions and also in the now rare apothecaries' system (usually in the form ): but while Roman numerals for whole numbers are essentially decimal,  does not correspond to $C V$, as one might expect, but $C V$.

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 $4/12$ 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 $6/12$ 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 for semis "half". Uncia dots were added to for fractions from seven to eleven twelfths, just as tallies were added to  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 $1/2$ to $1/2$ had a name in Roman times; these corresponded to the names of the related coins:

Other Roman fractional notations included the following:

Large numbers
The Romans developed two main ways of writing large numbers, the apostrophus and the vinculum, further extended in various ways in later times.

Apostrophus


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

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


 * ' = 500              ' = 1,000
 * ' = 5,000        ' = 10,000
 * ' = 50,000   ' = 100,000

Sometimes (1000) is reduced to,  (5,000) to ;  (10,000) to ;  (50,000) to ; and  (100,000) to. It is likely (500) reduced to  and  (1000) influenced  the later.

John Wallis is often credited with introducing the symbol for infinity $5/10$, and one conjecture is that he based it on, 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: 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.
 *  = 4,000
 *  = 25,000

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:
 *  p. = 1,332,000 paces (1,332 Roman miles).

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.

Origin
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.

Etruscan numerals
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: $6/12$, $1/3$, $1/4$, $1/2$, and $1/2$ 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.)

The symbols $1/12$ and $12/12$ resembled letters of the Etruscan alphabet, but $1/12$, $2/12$, and $1/6$ 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 (nor 𐌢𐌣 for 40), and wrote 𐌡𐌠𐌠, 𐌡𐌠𐌠𐌠 and 𐌡𐌠𐌠𐌠𐌠 for 7, 8, and 9, respectively.

Early Roman numerals
The early Roman numerals for 1, 10, and 100 were the Etruscan ones: $3/12$, $1/4$, and $4/12$. The symbols for 5 and 50 changed from $1/3$ and $5/12$ to $6/12$ and $1/2$ at some point. The latter had flattened to $7/12$ (an inverted T) by the time of Augustus, and soon afterwards became identified with the graphically similar letter $8/12$.

The symbol for 100 was written variously as $2/3$ or $9/12$, and was then abbreviated to $3/4$ or $10/12$, with $5/6$ (which matched the Latin letter C) finally winning out. It might have helped that C was the initial letter of CENTUM, Latin for "hundred".

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

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

According to Paul Kayser, the basic numerical symbols were, , and  (or ) and the intermediate ones were derived by taking half of those (half an  is , half a  is  and half a  is ). Then 𐌟 and ↆ developed as mentioned above.



Classical Roman numerals
The Colosseum was constructed in Rome in CE 72–80, and while the original perimeter wall has largely disappeared, the numbered entrances from (23) to  (54) survive, 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 is used for 40,  is avoided in favour of : in fact, gate 44 is labelled.

Use in the Middle Ages and Renaissance
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:, , , , and so on.



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

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 "" for "", or "" for ""), while others serve as abbreviations for compound numerals ("" for "", or "" for ""). Although they are still listed today in some dictionaries, they are long out of use.

A superscript "o" (sometimes written directly above the symbol) was sometimes used as an ordinal indicator.

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

Modern use
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:


 * Names of monarchs and popes, e.g. Elizabeth II of the United Kingdom, Pope Benedict XVI. These are referred to as regnal numbers and are usually read as ordinals; e.g. is pronounced "the second". This tradition began in Europe sporadically in the Middle Ages, gaining widespread use in England during the reign of Henry VIII. Previously, the monarch was not known by numeral but by an epithet such as Edward the Confessor. Some monarchs (e.g. Charles IV of Spain, Louis XIV of France and William IV of Great Britain) seem to have preferred the use of  instead of  on their coinage (see illustration).
 * Generational suffixes, particularly in the U.S., for people sharing the same name across generations, such as William Howard Taft IV. These are also usually read as ordinals.
 * In the French Republican Calendar, initiated during the French Revolution, years were numbered by Roman numerals – from the year (1792) when this calendar was introduced to the year  (1805) when it was abandoned.
 * Cambridge Main Public Library 1888.jpg, (USA) 1888, displayed in "standard" Roman numerals on its facade.]] The year of production of films, television shows and other works of art within the work itself. Outside reference to the work will use regular Arabic numerals.
 * Hour marks on timepieces. In this context, 4 is often written.
 * The year of construction on building façades and cornerstones.
 * Page numbering of prefaces and introductions of books, and sometimes of appendices and annexes, too.
 * Book volume and chapter numbers, as well as the several acts within a play (e.g. Act, Scene 2).
 * Sequels to some films, video games, and other works (as in Rocky II, Grand Theft Auto V, Myst III: Exile).
 * Outlines that use numbers to show hierarchical relationships.
 * Occurrences of a recurring grand event, for instance:
 * The Summer and Winter Olympic Games (e.g. the XXI Olympic Winter Games; the Games of the XXX Olympiad).
 * The Super Bowl, the annual championship game of the National Football League (e.g. Super Bowl XLII; Super Bowl 50 was a one-time exception ).
 * WrestleMania, the annual professional wrestling event for the WWE (e.g. WrestleMania XXX). This usage has also been inconsistent.

Specific disciplines
In astronautics, United States rocket model variants are sometimes designated by Roman numerals, e.g. Titan I, Titan II, Titan III, Saturn I, Saturn V.

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

In chemistry, Roman numerals are sometimes used to denote the groups of the periodic table, but this has officially been deprecated in favour of Arabic numerals. 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 " 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 mathematics (including trigonometry, statistics, and calculus), when a graph includes negative numbers, its quadrants are named using, , , and. 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 Nazi III Panzerkorps) with Arabic numerals being used for divisions and armies.

In music, Roman numerals are used in several contexts:
 * Movements are often numbered using Roman numerals.
 * In Roman numeral analysis, harmonic function is identified using Roman numerals.
 * Individual strings of stringed instruments, such as the violin, are often denoted by Roman numerals, with higher numbers denoting lower strings.

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

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 ", while a lower-ranking cricket or American football team might be the "3rd ".

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

In Ireland, Roman numerals were used until the late 1980s to indicate the month on postage Franking. In documents, Roman numerals are sometimes still used to indicate the month to avoid confusion over day/month/year or month/day/year formats.

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

Modern use in European languages other than English
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, e.g. the French XVIIIe siècle and the Spanish siglo (not siglo) for "18th century". Some Slavic and Turkic languages (especially in and adjacent to Russia) similarly favor Roman numerals (e.g. Russian XVIII век, Azeri XVIII əsr or Polish wiek XVIII ). On the other hand, in Turkish and some Central European Slavic languages, like most Germanic languages, one writes "18." (with a period) before the local word for "century" (e.g. Turkish 18. yüzyıl, Czech 18. století).



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 is written in Roman numerals, while the day is in Arabic numerals: "4..1789" and ".4.1789" both refer unambiguously to 4 June 1789.



Roman numerals are sometimes used to represent the in hours-of-operation signs displayed in windows or on doors of businesses, and also sometimes in railway and bus timetables. Monday, taken as the first day of the week, is represented by. Sunday is represented by. 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. For instance, apartments in central Amsterdam are indicated as 138-, 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 to  for the smaller intervals. The sign $11/12$ thus marks 17.9 km.

Certain romance-speaking countries use Roman numerals to designate assemblies of their national legislatures. For instance, the composition of the Italian Parliament from 2018 to 2022 (elected in the 2018 Italian general election) is called the XVIII Legislature of the Italian Republic (or more commonly the "XVIII 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.

Unicode
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. This range includes both upper- and lowercase numerals, as well as pre-combined characters for numbers up to 12 (Ⅻ or ). 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". The block also includes some apostrophus symbols for large numbers, an old variant of "" (50) similar to the Etruscan character, the Claudian letter "reversed C", etc.