# Roman numerals

Entrance to section LII (52) of the Colosseum, with numerals still visible

Roman numerals, the numeric system in ancient Rome, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as follows:

I, II, III, IV, V, VI, VII, VIII, IX, and X.

The Roman numeral system is a cousin of Etruscan numerals. Use of Roman numerals continued after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced in most contexts by more convenient Hindu-Arabic numerals; however this process was gradual, and the use of Roman numerals in some minor applications continues to this day.

## Reading Roman numerals

 MMXIII "2013" as a Roman numeral

Roman Numerals, as used today, are based on seven symbols:[1]

Symbol Value
I 1
V 5
X 10
L 50
C 100
D 500
M 1,000

Numbers are formed by combining symbols together and adding the values. So II is two ones, i.e. 2, and XIII is a ten and three ones, i.e. 13. There is no zero in this system, so 207, for example, is CCVII, using the symbols for two hundreds, a five and two ones. 1066 is MLXVI, one thousand, fifty and ten, a five and a one.

Symbols are placed from left to right in order of value, starting with the largest. However, in a few specific cases,[2] to avoid four characters being repeated in succession (such as IIII or XXXX) these can be reduced using subtractive notation as follows:[3][4]

• the numeral I can be placed before V and X to make 4 units (IV) and 9 units (IX) respectively
• X can be placed before L and C to make 40 (XL) and 90 (XC) respectively
• C can be placed before D and M to make 400 and 900 according to the same pattern[5]

An example using the above rules would be 1904: this is composed of 1 (one thousand), 9 (nine hundreds), 0 (zero tens), and 4 (four units). To write the Roman numeral, each of the non-zero digits should be treated separately. Thus 1,000 = M, 900 = CM, and 4 = IV. Therefore, 1904 is MCMIV. This reflects typical modern usage rather than a universally accepted convention: historically Roman numerals were often written less consistently.[6]

An inscription on Admiralty Arch, London. The numeral is 1910, although MCMX would be more usual

A common exception to the practice of placing a smaller value before a larger in order to reduce the number of characters, is the use of IIII instead of IV for 4, especially, although by no means exclusively, on clock faces; see below. Another example of the same kind of thing is the representation of 1910 in Roman Numerals on Admiralty Arch, London - where DCCCC is used instead of CM for 900 (see illustration). In general, the "rules" about subtractively applied symbols are the most frequently "broken".

Below are some examples of the modern use of Roman Numerals.

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## History

### Pre-Roman times and Ancient Rome

Although Roman numerals are now written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used I, Λ, X, ⋔, 8, ⊕, for I, V, X, L, C, and M, of which only I and X happened to be letters in their alphabet.

There are a number of hypotheses about the origin of the system:

1. Tally sticks

One hypothesis is that the Etrusco-Roman numerals actually derive from notches on tally sticks, which continued to be used by Italian and Dalmatian shepherds into the 19th century.[8]

Thus, 'I' descends not from the letter 'I' but from a notch scored across the stick. Every fifth notch was double cut (i.e. , , , , etc.), and every tenth was cross cut (X), IIIIΛIIIIXIIIIΛIIIIXII..., much like European tally marks today. This produced a positional system: Eight on a counting stick was eight tallies, IIIIΛIII, or the eighth of a longer series of tallies; either way, it could be abbreviated ΛIII (or VIII), as the existence of a Λ implies four prior notches. By extension, eighteen was the eighth tally after the first ten, which could be abbreviated X, and so was XΛIII. Likewise, number four on the stick was the I-notch that could be felt just before the cut of the Λ (V), so it could be written as either IIII or IΛ (IV). Thus the system was neither additive nor subtractive in its conception, but ordinal. When the tallies were transferred to writing, the marks were easily identified with the existing Roman letters I, V and X. The tenth V or X along the stick received an extra stroke. Thus 50 was written variously as N, И, K, Ψ, , etc., but perhaps most often as a chicken-track shape like a superimposed V and I: . This had flattened to (an inverted T) by the time of Augustus, and soon thereafter became identified with the graphically similar letter L. Likewise, 100 was variously Ж, , , H, or as any of the symbols for 50 above plus an extra stroke. The form Ж (that is, a superimposed X and I) came to predominate. It was written variously as >I< or ƆIC, was then abbreviated to Ɔ or C, with C variant finally winning out because, as a letter, it stood for centum, Latin for "hundred".

The hundredth V or X was marked with a box or circle. Thus 500 was like a Ɔ superimposed on a or — that is, like a Þ with a cross bar,— becoming 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 (I) or CIƆ, and half of a thousand or "five hundred" is the right half of the symbol, I) or IƆ, and this may have been converted into D.[9] This at least was the false etymology given to it later on.

Meanwhile, 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".

2. Hand signals

Alfred Hooper has an alternative hypothesis for the origin of the Roman numeral system, for small numbers.[10] Hooper contends that the digits are related to hand signals. For example, the numbers I, II, III, IIII correspond to the number of fingers held up for another to see. V, then represents that hand upright with fingers together and thumb apart. Numbers 6–10, are represented with two hands as follows (left hand, right hand) 6=(V,I), 7=(V,II), 8=(V,III), 9=(V,IIII), 10=(V,V) and X results from either crossing of the thumbs, or holding both hands up in a cross.

3. Intermediate symbols deriving from few original symbols

A third hypothesis about the origins states that the basic ciphers were I, X, C and Φ (or ⊕) and that the halfthrough ones derived from taking half of those (half a X is V, half a C is L and half a Φ/⊕ is D).[11]

### Middle Ages and Renaissance

Minuscule (lower case) letters were developed in the Middle Ages, well after the demise of the Western Roman Empire, and lower-case versions of Roman numbers are now also commonly used: i, ii, iii, iv, etc. In the Middle Ages, a j was sometimes substituted for the final i of a number, such as iij for 3 or vij for 7. This j was considered a swash variant of i. The use of a final j is still used in medical prescriptions to prevent tampering with or misinterpretation of a number after it is written.[12][13]

A unique, more comprehensive shorthand for writing Roman numerals was developed during the Middle Ages, which today are called "medieval Roman numerals." This system used almost every other letter of the Roman alphabet to stand as abbreviations for more longhand numbers (usually those that consisted of repetitions of the same symbol). They are still listed today in most dictionaries, although through disfavor are primarily out of use.[14]

Modern
number
Medieval
abbreviation
Notes
5 A Resembles an upside-down V. Also said to equal 500.
6 Ϛ Either a ligature of VI, or the Greek letter stigma (Ϛ), having the same numerical value.[15]
7 S, Z Presumed abbreviation of septem, Latin for 7.
11 O Presumed abbreviation of (e.g.) onze, French for 11.
40 F Presumed abbreviation of English forty.
70 S Also could stand for 7, and has same etymology.
80 R
90 N Presumed abbreviation of nonaginta, Latin for 90.
150 Y Possibly derived from the lowercase y's shape.
151 K This unusual abbreviation's origin is unknown; it has also been said to stand for 250.[16]
160 T Possibly derived from Greek tetra, as 4 x 40 = 160.
200 H
250 E
300 B
400 P, G
500 Q Redundant with D, abbreviation for quingenti, Latin for 500.
2000 Z

Chronograms, messages with a numbers 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.

### Modern usage

Roman numbers on stern of Cutty Sark, Greenwich, showing draft in feet.

Roman numerals remained in common use until about the 14th century, when they were outmoded by Hindu–Arabic numerals (thought to have been introduced to Europe from al-Andalus, by way of Arab traders and arithmetic treatises, around the 11th century) in practically all mathematical and economical applications. Roman numerals are still used today in several niche 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 monarchical ordinals; e.g. "II" is pronounced "the second". This tradition began in Europe sporadically in the Middle Ages, gaining widespread use in England only during the reign of Henry VIII. Previously, the monarch was not known by numeral but by an epithet such as Edward the Confessor.
• Generational suffixes for people who share the same name across generations (especially males).
• The year of production of films, television shows and other works of art within the work itself, which according to BBC News was originally done "in an attempt to disguise the age of films or television programmes."[17] Outside reference to the work will use regular Hindu–Arabic numerals.
• Hour marks on timepieces. In this context 4 is usually written IIII.
• The year of construction on building faces and cornerstones.
• Page numbering of prefaces and introductions of books.
• Book volume and chapter numbers.
• Sequels of movies, video games, and other works.
• Outlines.
• Occurrences of a recurring grand event, for instance:

In astronomy, the natural satellites or "moons" of the planets are traditionally designated by capital Roman numerals.

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

In music theory, the diatonic functions are identified using roman numerals. See: Roman numeral analysis.

In performance practice, individual strings of stringed instruments, such as the violin, are often denoted by Roman numerals, with higher numbers denoting lower strings.

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

#### Modern non-English-speaking usage

Capital Roman numerals are used to denote centuries (e.g., XVIII refers to the eighteenth century) in Bulgarian, Croatian, French, Hungarian, Italian, Polish, Portuguese, Romanian, Russian, Serbian, Georgian, and Spanish languages. This use has largely been replaced by Hindu-Arabic numerals (e.g. 18.) in Czech and Slovak languages.

Boris Yeltsin's autograph, 10 November 1988. The month is specified by Roman numerals.

In Central Europe, Italy, Russia, and in Bulgarian, Croatian, Portuguese, Romanian, and Serbian languages, mixed Roman and Hindu-Arabic numerals are used to record dates (usually on tombstones, but also elsewhere, such as in formal letters and official documents). The month is written in Roman numerals while the day is in Hindu-Arabic numerals: 14.  VI. 1789 is 14 June 1789. This use has largely been replaced by Hindu-Arabic numerals (e.g. 14.06.1789) in Czech, Slovene, Slovak, Polish, Portuguese and Russian languages.

In the Baltic and Eastern Europe nations, Roman numerals are used to represent the days of the week in hours-of-operation signs displayed in windows or on doors of businesses, and also sometimes in railway and bus timetables. Monday is represented by I, which is the initial day of the week. Sunday is represented by VII, which is the final day of the week. 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. The following example hours-of-operation table would be for a business whose hours of operation are 9:30 AM to 5:30 PM on Mondays, Wednesdays, and Thursdays; 9:30 AM to 7:00 PM on Tuesdays and Fridays; and 9:30 AM to 1:00 PM on Saturdays; and which is closed on Sundays.

 I 9:30–17:30 II 9:30–19:00 III 9:30–17:30 IV 9:30–17:30 V 9:30–19:00 VI 9:30–13:00 VII —
Sign at km. 17·9 on route SS4 Salaria north of Rome

In Rome, Greece, Romania, and other European countries to a lesser extent, Roman numerals are used for floor numbering. Likewise, apartments in central Amsterdam are indicated as 138-III, with both an Hindu-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 kilometer 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 km. 17·900.

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## Special values

### Zero

The number zero does not have its own Roman numeral, but the word nulla (the Latin word meaning "none") was used by medieval computists in lieu of 0. Dionysius Exiguus was known to use nulla alongside Roman numerals in 525.[18][19] About 725, Bede or one of his colleagues used the letter N, the initial of nulla, in a table of epacts, all written in Roman numerals.[20]

### 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.

Though the Romans used a decimal system for whole numbers, reflecting how they counted in Latin, they used a duodecimal system for fractions, because the divisibility of twelve (12 = 3 × 2 × 2) makes it easier to handle the common fractions of 1/3 and 1/4 than does a system based on ten (10 = 2 × 5). On coins, many of which had values that were duodecimal fractions of the unit as, they used a tally-like notational system based on twelfths and halves. A dot (•) indicated an uncia "twelfth", the source of the English words inch and ounce; dots were repeated for fractions up to five twelfths. Six twelfths (one half) was abbreviated as the letter 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.

Each of these fractions had a name, which was also the name of the corresponding coin:

Fraction Roman Numeral Name (nominative and genitive) Meaning
1/12 uncia, unciae "ounce"
2/12 = 1/6 •• or : sextans, sextantis "sixth"
3/12 = 1/4 ••• or quadrans, quadrantis "quarter"
4/12 = 1/3 •••• or :: triens, trientis "third"
5/12 ••••• or :·: quincunx, quincuncis "five-ounce" (quinque unciaequincunx)
6/12 = 1/2 S semis, semissis "half"
7/12 S• septunx, septuncis "seven-ounce" (septem unciaeseptunx)
8/12 = 2/3 S•• or S: bes, bessis "twice" (as in "twice a third")
9/12 = 3/4 S••• or S dodrans, dodrantis
or nonuncium, nonuncii
"less a quarter" (de-quadransdodrans)
or "ninth ounce" (nona uncianonuncium)
10/12 = 5/6 S•••• or S:: dextans, dextantis
or decunx, decuncis
"less a sixth" (de-sextansdextans)
or "ten ounces" (decem unciaedecunx)
11/12 S••••• or S:·: deunx, deuncis "less an ounce" (de-unciadeunx)
12/12 = 1 I as, assis "unit"

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.

Other Roman fractions include the following:

• 1/8 sescuncia, sescunciae (from sesqui- + uncia, i.e. 1½ uncias), represented by a sequence of the symbols for the semuncia and the uncia.
• 1/24 semuncia, semunciae (from semi- + uncia, i.e. ½ uncia), represented by several variant glyphs deriving from the shape of the Greek letter Sigma (Σ), one variant resembling the pound sign (£) without the horizontal line(s) and another resembling the Cyrillic letter (Є).
• 1/36 binae sextulae, binarum sextularum ("two sextulas") or duella, duellae, represented by (ƧƧ), a sequence of two reversed Ss.
• 1/48 sicilicus, sicilici, represented by (Ɔ), a reversed C.
• 1/72 sextula, sextulae (1/6 of an uncia), represented by (Ƨ), a reversed S.
• 1/144 = 12−2dimidia sextula, dimidiae sextulae ("half a sextula"), represented by (ƻ), a reversed S crossed by a horizontal line.
• 1/288 scripulum, scripuli (a scruple), represented by the symbol ().
• 1/1728 = 12−3siliqua, siliquae, represented by a symbol resembling closing guillemets (»).

### Large numbers

1630 on the Westerkerk in Amsterdam
Roman numerals, 16th century

In the Middle Ages, a horizontal line was used above a particular numeral to represent one thousand times that numeral, and additional vertical lines on both sides of the numeral to denote one hundred times the number, as in these examples:

The same overline was also used with a different meaning, to clarify that the characters were numerals. Sometimes both underline and overline were used, e. g. MCMLXVII, and in certain (serif) typefaces, particularly Times New Roman, the capital letters when used without spaces simulates the appearance of the under/over bar, e.g. MCMLXVII.

Sometimes 500, usually D, was written as original |Ɔ, while 1,000, usually M, was written as original C|Ɔ. This is a system of encasing numbers to denote thousands (imagine the Cs and Ɔs as parentheses), which has its origins in Etruscan numeral usage. The D and M used to represent 500 and 1,000 were most likely derived from and CIƆ, respectively, and subsequently influenced by assumed abbreviations.

An extra Ɔ denoted 500, and multiple extra Ɔs are used to denote 5,000, 50,000, etc. For example:

Base number 1 extra Ɔ 2 extra Ɔs 3 extra Ɔs CIƆ = 1,000 CCIƆƆ = 10,000 CCCIƆƆƆ = 100,000 IƆ = 500 CIƆƆ = 1,500 CCIƆƆƆ = 10,500 CCCIƆƆƆƆ = 100,500 IƆƆ = 5,000 CCIƆƆƆƆ = 15,000 CCCIƆƆƆƆƆ = 105,000 IƆƆƆ = 50,000 CCCIƆƆƆƆƆƆ = 150,000

(Sometimes (CIƆ) 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, (IƆƆ) for 5,000) was reduced to (); (CCIƆƆ) for 10,000 to (); and similarly () and ().)

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## "IIII" on clocks

A typical clock face with Roman numerals in Bad Salzdetfurth, Germany
The Shepherd gate clock with Roman numbers up to XXIII (and 0), in Greenwich

Clock faces that are labeled using Roman numerals conventionally (Big Ben is an exception) show IIII for four o'clock and IX for nine o'clock, using the subtractive principle in one case and not the other. There are many suggested explanations for this:

• Many clocks use IIII because that was the tradition established by the earliest surviving clock, which is the Wells Cathedral clock built between 1386 and 1392. It used IIII because that was the typical method used to denote 4 in contemporary manuscripts (as iiij or iiii). That clock had an asymmetrical 24-hour dial and used Hindu-Arabic numerals for a minute dial and a moon dial, so theories depending on a symmetrical 12-hour clock face do not apply.[21]
• Perhaps IV was avoided because IV represented the Roman god Jupiter, whose Latin name, IVPPITER, begins with IV. This suggestion has been attributed to Isaac Asimov.[22]
• Louis XIV, king of France, who preferred IIII over IV, ordered his clockmakers to produce clocks with IIII and not IV, and thus it has remained.[23]
• Using standard numerals, two sets of figures would be similar and therefore confusable by children and others unused to reading clockfaces: IV and VI are similar, as are IX and XI. As the first pair are upside down on the face, an additional level of confusion would be introduced—a confusion avoided by using IIII to provide a clear distinction from VI.
• The four-character form IIII creates a visual symmetry with the VIII on the other side, which the two-character IV would not.
• With IIII, the number of symbols on the clock totals twenty Is, four Vs, and four Xs,[24] so clock makers need only a single mould with a V, five Is, and an X in order to make the correct number of numerals for their clocks: VIIIIIX. This is cast four times for each clock and the twelve required numerals are separated:
• V IIII IX
• VI II IIX
• VII III X
• VIII I IX
The IIX and one of the IXs are rotated 180° to form XI and XII. The alternative with IV uses seventeen Is, five Vs, and four Xs, requiring the clock maker to have several different patterns.
• Only the I symbol would be seen in the first four hours of the clock, the V symbol would only appear in the next four hours, and the X symbol only in the last four hours. This would add to the clock's radial symmetry.
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## References

1. ^ Alphabetic symbols for larger numbers, such as Q for 500,000, have also been used to various degrees of standardization.Gordon, Arthur E. (1982). Illustrated Introduction to Latin Epigraphy. Berkeley: University of California Press. ISBN 0520050797.
2. ^ Reddy, Indra K. & Khan, Mansoor A. (2003). Essential Math and Calculations for Pharmacy Technicians (CRC Press) http://books.google.com/books?id=U3QY7gz0C2cC `|url=` missing title (help).
3. ^ Dela Cruz, M.L.P. & Torres, H.D. (2009). Number Smart Quest for Mastery: Teacher's Edition (Rex Bookstore, Inc.) http://books.google.com/books?id=PVK6lt2xXz4C `|url=` missing title (help).
4. ^ Martelli, Alex & Ascher, David (2002). Python Cookbook. O'Reilly Media Inc.
5. ^ Stroh, Michael. Trick question: How to spell 1999? Numerals: Maybe the Roman Empire fell because their computers couldn't handle calculations in Latin. The Baltimore Sun, December 27, 1998.
6. ^ Adams, Cecil (February 23, 1990). "The Straight Dope". The Straight Dope.
7. ^ Hayes, David P. "Guide to Roman Numerals". Copyright Registration and Renewal Information Chart and Web Site.
8. ^ Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Translated by David Bellos, E. F. Harding, Sophie Wood, Ian Monk. John Wiley & Sons.
9. ^ Asimov, Issac (1966, 1977). Asimov On Numbers. Pocket Books, a division of Simon & Schuster, Inc. p. 9.
10. ^ Alfred Hooper. The River Mathematics (New York, H. Holt, 1945).
11. ^ Keyser, Paul (1988). "The Origin of the Latin Numerals 1 to 1000". American Journal of Archaeology 92: 529–546.
12. ^ Sturmer, Julius W. Course in Pharmaceutical and Chemical Arithmetic, 3rd ed. (LaFayette, IN: Burt-Terry-Wilson, 1906). p25 Retrieved on 2010-03-15.
13. ^ Bastedo, Walter A. Materia Medica: Pharmacology, Therapeutics and Prescription Writing for Students and Practitioners, 2nd ed. (Philadelphia, PA: W.B. Saunders, 1919) p582 Retrieved on 2010-03-15.
14. ^ Capelli, A. Dictionary of Latin Abbreviations. 1912.
15. ^ Perry, David J. Proposal to Add Additional Ancient Roman Characters to UCS.
16. ^ Bang, Jørgen. Fremmedordbog, Berlingske Ordbøger, 1962 (Danish)
17. ^ Owen, Rob (2012-01-13). "TV Q&A: ABC News, 'Storage Wars' and 'The Big Bang Theory'". Pittsburgh Post-Gazette. Retrieved 2012-01-13.
18. ^ Faith Wallis, trans. Bede: The Reckoning of Time (725), Liverpool: Liverpool Univ. Pr., 2004. ISBN 0-85323-693-3.
19. ^ Byrhtferth's Enchiridion (1016). Edited by Peter S. Baker and Michael Lapidge. Early English Text Society 1995. ISBN 978-0-19-722416-8.
20. ^ C. W. Jones, ed., Opera Didascalica, vol. 123C in Corpus Christianorum, Series Latina.
21. ^ Paul Lewis, Clocking the fours: A new theory about IIII; see the clock
22. ^ http://www.voxinghistory.com/?tag=roman_numerals
23. ^ W.I. Milham, Time & Timekeepers (New York: Macmillan, 1947) p. 196
24. ^ FAQ: Roman IIII vs. IV on Clock Dials – Donn Lathrop's page on IIII vs. IV.
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Last modified on 14 June 2013, at 19:29