roman numerals

Roman numerals are commonly used today in numbered lists (in outline format), clockfaces, pages preceding the main body of a book, chord triads in music analysis, the numbering of movie sequels, book publication dates, successive political leaders or children with identical names, and the numbering of some sport events, such as the Olympic Games or the Super Bowls.

For arithmetics involving Roman numerals, see Roman arithmetic and Roman abacus.

Symbols

Roman numerals use a basic set of seven symbols:

Multiple symbols may be combined to produce numbers in between these values, subject to certain rules on repetition. In cases where it may be shorter, it is sometimes allowable to place a smaller symbol before a larger value, so that, for example, one may write IV or iv for four. Again, for the numbers not assigned a specific symbol, the above given symbols are combined:

• II or ii for two,
• III or iii for three,
• IV, iv, IIII or iiii for four,
• VI or vi for six,
• VII or vii for seven,
• VIII or viii for eight,
• IX or ix for nine.

For large numbers (five thousand and above), a bar is placed above a base numeral to indicate multiplication by 1000:

• V for five thousand
• X for ten thousand
• L for fifty thousand
• C for one hundred thousand
• D for five hundred thousand
• M for one million

For very large numbers (five million and above), there is no standard format, although sometimes a double bar or underline is used to indicate multiplication by 1,000,000. That means an underline X (X) is ten million.

Origins

Although the Roman numerals are now written with letters of the Roman alphabet, they were originally separate symbols. The Etruscans, for example, used I Λ X ⋔ 8 ⊕ for I V X L C M.

They appear to derive from notches on tally sticks, such as those used by Italian and Dalmatian shepherds into the 19^th century. Thus, the I descends from a notch scored across the stick. Every fifth notch was double cut (⋀, ⋁, ⋋, ⋌, etc.), and every tenth was cross cut (X), much like European tally marks today. This produced a positional system: Eight on a counting stick was eight tallies, IIIIΛIII, but this could be abbreviated ΛIII (or VIII), as the existence of a Λ implies four prior notches. Likewise, number four on the stick was the I-notch that could be felt just before the cut of the V, so it could be written as either IIII or IV. Thus the system was neither additive nor subtractive in its conception, but ordinal. When the tallies were later transferred to writing, the marks were easily identified with the existing Roman letters I, V, X.

(A folk etymology has it that the V represented a hand, and that the X was made by placing two Vs on top of each other, one inverted.)

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, was written variously as >I< or ƆIC, was then shortened to Ɔ or C, with C 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 superposed on a ⋌ or ⊢ (that is, like a Þ with a cross bar), becoming a struck-through D or a Ð by the time of Augustus, under the graphic influence of the letter D. It was later identified as the letter D, perhaps as an abbreviation of the phrase demi-mille 'half-thousand'. Meanwhile, 1000 was a circled X: Ⓧ, ⊗, ⊕, and by Augustinian times was partially identified with the Greek letter Φ. It then evolved along several independent routes. Some variants, such as Ψ and CD (more accurately a reversed D adjacent to a regular D), were historical dead ends (although one folk etymology later identified D for 500 as half of Φ for 1000 because of this CD variant), while two variants of ↀ survive to this day. One, CIƆ, led to the convention of using parentheses to indicate multiplication by 1000 (later extended to double parentheses as in ↁ, ↂ, etc.); in the other, ↀ became ∞ and ⋈, eventually changing to M under the influence of the word mille ('thousand').

Zero

In general, the number zero did not have its own Roman numeral, but the concept of zero as a number was well known by all medieval computists (responsible for calculating the date of Easter). They included zero (via the Latin word nulla meaning nothing) as one of nineteen epacts, or the age of the moon on March 22. The first three epacts were nullae, xi, and xxii (written in minuscule or lower case). The first known computist to use zero was Dionysius Exiguus in 525, but the concept of zero was no doubt well known earlier. Only one instance of a Roman numeral for zero is known. About 725, Bede or one of his colleagues used the letter N, the initial of nullae, in a table of epacts, all written in Roman numerals.

A notation for the value zero is quite distinct from the role of the digit zero in a positional notation system. The lack of a zero digit prevented Roman numerals from developing into a positional notation, and led to their gradual replacement by Hindu-Arabic numerals in the early second millennium. On the other hand, the lack of positional notation may have prevented the Romans from developing a "zero". Which affected which is not certain.

Fractions

Even though the Romans used a decimal system for whole numbers, presumably because of the number of human fingers, they commonly used duodecimal for rational numbers, because the divisibility of twelve (12 = 2×2×3), as opposed to that of ten (10 = 2×5), makes it easier to handle such common fractions as 1/3 and 1/4. On coins, many of which had values that were duodecimal fractions of the unit as, they notated these fractional quantities with a similar system to that of whole numbers, but based on one twelfth and six twelfths instead of one unit and five units. They used a dot • to notate an uncia (one twelfth) and more dots were added up to five twelfths. Then one half (six twelfths) was notated using the letter S for semis. Dots were juxtaposed to S to notate the fractions from seven to eleven twelfths, just like bars were juxtaposed to V for whole numbers from six to nine. Each of these fractions had its own name, which was also the name used for the corresponding coin:

IIII or IV?

The notation of Roman numerals has varied through the centuries. Originally, it was common to use IIII to represent "four", because IV represented the god Jove (and later YHWH). The subtractive notation (which uses IV instead of IIII) has become universally used only in modern times. For example, Forme of Cury, a manuscript from 1390, uses IX for "nine", but IIII for "four". Another document in the same manuscript, from 1381, uses IV and IX. A third document in the same manuscript uses both IIII and IV, and IX. Constructions such as IIIII for "five", IIX for "eight" or VV for "ten" have also been discovered. Subtractive notation arose from regular Latin usage: the number "18" was duodeviginti or “two from twenty”; the number "19" was undeviginti or “one from twenty”. The use of subtractive notation increased the complexity of performing Roman arithmetic, without conveying the benefits of a full positional notation system.

Likewise, on some buildings it is possible to see MDCCCCX, for example, representing 1910 instead of MCMX - notably Admiralty Arch in London. Another notable example is on Harvard Medical School's Library which reads MDCCCCIIII for 1904.

Calendars and clocks

Clock faces that are labelled using Roman numerals conventionally show IIII for 4 o'clock and IX for 9 o'clock, using the subtractive principle in one case and not in the other. There are several suggested explanations for this, several of which may be true:

• The four-character form IIII creates a visual symmetry with the VIII on the other side, which IV would not.
• The number of symbols on the clock totals twenty 'I's, four 'V's, and four 'X's, so clock makers need only a single mold with five I's, a V, and an X in order to make the correct number of numerals for the clocks, cast four times for each clock:
  • V IIII IX
  • VI II IIX
  • VII III X
  • VIII I IX

IIX and one of the IX's can be rearranged or inverted to form XI and XII. The alternative uses seventeen 'I's, five 'V's, and four 'X's, possibly requiring the clock maker to have several different molds.

• IIII was the preferred way for the ancient Romans to write 4, since they to a large extent avoided subtraction.
• It has been suggested that since IV is the first two letters of IVPITER, the main god of the Romans, it was not appropriate to use.
• The I symbol would be the only symbol in the first 4 hours of the clock, the V symbol would only appear in the next 4 hours, and the X symbol only in the last 4 hours. This would add to the clock's radial symmetry.
• IV is difficult to read upside down and on an angle, particularly at that location on the clock.
• Louis XIV, king of France, preferred IIII over IV, ordered his clockmakers to produce clocks with IIII and not IV, and thus it has remained. ^{citation needed]}

XCIX or IC?

Rules regarding Roman numerals often state that a symbol representing 10^x may not precede any symbol larger than 10^x+1. For example, C cannot be preceded by I or V, only by X (or, of course, by a symbol representing a value equal to or larger than C). Thus, one should represent the number "ninety-nine" as XCIX, not as the "shortcut" IC. However, these rules are not universally followed.

This 'problem' manifested in questions as to why 1990 was not written as MXM instead of the universal usage MCMXC, or why 1999 was not written simply IMM or MIM as opposed to the virtually universal MCMXCIX.

The possibility of using IC for 99 provides one explanation why Cadbury's smaller, unwrapped Flakes (for inserting into ice creams) were designated 99 Flakes.

It is also worth noting that writing 99 as "IC" or "ic", in Old English texts would appear identical to the word "Ic" meaning "I".

Year in Roman numerals

In seventeenth century Europe, using Roman numerals for the year of publication for books was standard; there were many other places it was used as well. Publishers attempted to make the number easier to read by those more accustomed to Arabic positional numerals. On British title pages, there were often spaces between the groups of digits: M DCC LXI is one example. This may have come from the French, who separated the groups of digits with periods, as: M.DCC.LXI. or M. DCC. LXI. Notice the period at the end of the sequence; many countries did this for Roman numerals in general, but not necessarily Britain. (Periods were also common on each side of numerals in running text, as in "commonet .iij. viros illos".)

These practices faded from general use before the start of the twentieth century, though the cornerstones of major buildings still occasionally use them. Roman numerals are today still used on building faces for dates: 2006 can be represented as MMVI.

The film industry has used them perhaps since its inception to denote the year a film was made, so that it could be redistributed later, either locally or to a foreign country, without making it immediately clear to viewers what the actual date was. This became more useful when films were broadcast on television to partially conceal the age of films. From this came the policy of the broadcasting industry, including the BBC, to use them to denote the year in which a television program was made. (The Australian Broadcasting Corporation has largely stopped this practice, but still occasionally lapses.)

Other modern usage by English-speaking peoples

Roman numerals remained in common use until about the 14th century, when they were replaced by Arabic numerals (thought to have been introduced to Europe from al-Andalus, by way of Arab traders and arithmetic treatises, around the 11th century). The use of Roman numerals today is mostly restricted to ordinal numbers, such as volumes or chapters in a book or the numbers identifying monarchs or popes (eg. Elizabeth II, Benedict XVI, etc.)

Sometimes the numerals are written using lower-case letters (thus: i, ii, iii, iv, etc.), particularly if numbering paragraphs or sections within chapters, or for the pagination of the front matter of a book.

Undergraduate degrees at British universities are generally graded using I, IIi, IIii, III for first, upper second (often pronounced "two one"), lower second (often pronounced "two two") and third class respectively.

Modern English usage also employs Roman numerals in many books (especially anthologies), movies (eg. Star Trek), sporting events (eg. the Olympic Games, the Super Bowl), and historic events (eg. World War I, World War II). The common unifying theme seems to be stories or events that are episodic or annual in nature, with the use of classical numbering suggesting importance or timelessness.

Sports teams can be referred to as the number of players in the squad with roman numerals. In rugby union, the 1st XV of a particular club would be the 1st and best team the club has, likewise for the XIII in rugby league, and XI for football (soccer), field hockey and cricket.

In chemistry, Roman numerals were used to denote the group in the periodic table of the elements. But there was not international agreement as to whether the group of metals which dissolve in water should be called Group IA or IB, for example, so although references may use them, the international norm has recently switched to Arabic numerals.

In astronomy, the natural satellites or "moons" of the planets are traditionally designated by capital Roman numerals, at first by order from the center of the planet, as the four Galilean satellites of Jupiter are numbered, and later by order of discovery; e.g., Callisto was "Jupiter IV" or "J IV". With recent discoveries—Jupiter currently has 63 known satellites—as well as computerization, this is somewhat disparaged for the minor worlds, at least in computerized listings. Science fiction, and not astronomy per se, has adopted the use for numbering the planets around a star; e.g., Planet Earth is called "Sol III".

In music theory, diatonic functions, scale degrees and the triads that have these degrees as their roots are often identified by Roman numerals (as in chord symbols). Upper-case Roman numerals indicate major triads while lower-case Roman numerals indicate minor triads, as the following chart illustrates. The leading tone triad is diminished.

Modern non-English speaking usage

The above uses are customary for English-speaking countries. Although many of them are also maintained in other countries, those countries have additional uses for Roman numerals which are unknown in English-speaking regions.

The French, the Portuguese, the Polish, and the Spanish use capital Roman numerals to denote centuries. For example, 'XVIII' refers to the eighteenth century, so as to avoid confusion between the '18^th century' and the '1800s'. (The Italians usually take the opposite approach, basing names of centuries on the digits of the years; quattrocento for example is the common Italian name for secolo XV, the fifteenth century.) Some scholars in English-speaking countries have adopted the French method, among them Lyon Sprague de Camp.

In Germany, Poland, and Russia, and in Spanish and Portuguese, mixed Roman numerals are used to record dates. Just as an old clock recorded the hour by Roman numerals while the minutes were measured in Arabic numerals, the month is written in Roman numerals while the day is in Arabic numerals: 14-VI-1789 is June the fourteenth, 1789. This is how dates are inscribed on the walls of the Kremlin, for example. This method has the advantage that days and months are not confused in rapid note-taking, and that any range of days or months can be expressed without confusion. For instance, V-VIII is May to August, while 1-V-31-VIII is May first to August thirty-first.

In Eastern Europe, especially the Baltic 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. 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 9:30AM to 5:30PM on Mondays, Wednesdays, and Thursdays; 9:30AM to 7:00PM on Tuesdays and Fridays; and 9:30AM to 1:00PM on Saturdays; and which is closed on Sundays.

Since the French use capital Roman numerals to refer to the quarters of the year ('III' is the third quarter), and this has become the norm in some European standards organisation, the mixed Roman-Arabic method of recording the date has switched to lowercase Roman numerals in many circles, as '4-viii-1961'. (ISO has since specified that dates should be given in all Arabic numerals, in ISO 8601 formats.)

In Romania, Roman numerals are used for floor numbering. Likewise 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'.

Roman numerals are rarely used in Asia. The motion picture rating system in Hong Kong uses categories I, IIA, IIB, and III based on Roman numerals.

Alternate forms

In the Middle Ages, Latin writers used a horizontal line 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:

• I for one thousand
• V for five thousand
• |I| for one hundred thousand
• |V| for five hundred thousand

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 font-faces, particularly Times New Roman, the capital letters when used without spaces simulates the appearance of the under/over bar, eg. MCMLXVII, which is often exaggerated when written by hand.

Sometimes 500, usually D, was written as I followed by an apostrophus, resembling a backwards C (Ɔ), while 1,000, usually M, was written as CIƆ. This is believed to be a system of encasing numbers to denote thousands (imagine the Cs as parentheses). This system has its origins from Etruscan numeral usage. The D and M symbols to represent 500 and 1,000 were most likely derived from IƆ and CIƆ, respectively.

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

Sometimes CIƆ was reduced to an lemniscate symbol (ↀ) for denoting 1,000. John Wallis is often credited for introducing this symbol to represent infinity (), and one conjecture is that he based it on this usage, since 1,000 was hyperbolically used to represent very large numbers. Similarly, 5,000 (IƆƆ) was reduced to ↁ; and 10,000 (CCIƆƆ) was reduced to ↂ

In medieval times, before the letter j emerged as a distinct letter, a series of letters i in Roman numerals was commonly ended with a flourish; hence they actually looked like ij, iij, iiij, etc. This proved useful in preventing fraud, as it was impossible, for example, to add another i to vij to get viij. This practice is now merely an antiquarian's note; it is never used. (It did, however, lead to the Dutch diphthong IJ.)

Table of Roman numerals

The "modern" Roman numerals, post-Victorian era, are shown below:

An accurate way to write large numbers in Roman numerals is to handle first the thousands, then hundreds, then tens, then units.
Example: the number 1988.
One thousand is M, nine hundred is CM, eighty is LXXX, eight is VIII.
Put it together: MCMLXXXVIII (ⅯⅭⅯⅬⅩⅩⅩⅤⅠⅠⅠ).

Unicode has a number of characters specifically designated as Roman numerals, as part of the Number Forms range from U+2160 to U+2183. For example, MCMLXXXVIII could alternatively be written as ⅯⅭⅯⅬⅩⅩⅩⅧ. This range includes both upper- and lowercase numerals, as well as pre-combined glyphs for numbers up to 12 (Ⅻ or XII), mainly intended for the clock faces for compatibility with non–West-European encodings. The pre-combined glyphs should only be used to represent the individual numbers where the use of individual glyphs is not wanted, and not to replace compounded numbers. Similarly precombined glyphs for 5000 and 10000 exist.

The Unicode characters are present only for compatibility with other character standards which provide these characters; for ordinary uses, the regular Latin letters are preferred. Displaying these characters requires a user agent that can handle Unicode and a font that contains appropriate glyphs for them.

Games

After the Renaissance, the Roman system could also be used to write chronograms. It was common to put in the first page of a book some phrase, so that when adding the I, V, X, L, C, D, M present in the phrase, the reader would obtain a number, usually the year of publication. The phrase was often (but not always) in Latin, as chronograms can be rendered in any language that utilises the Roman alphabet.

Mnemonic devices

There are several mnemonics that can be useful in remembering the Roman numeral system.

The following mnemonics recall the order of Roman numeral values above ten, with L being 50, C being 100, D being 500, and M being 1000.

• Lucky Cows Drink Milk
• Lucy Can't Drink Milk

References

• Menninger, Karl (1992). Number Words and Number Symbols: A Cultural History of Numbers. Dover Publications. ISBN 0-486-27096-3.

See also

• Roman arithmetic

External links

• Roman Numeral Conversion Calculator on MAS with open code
• Roman Numeral Conversion Calculator and Self Test
• FAQ: Roman IIII vs. IV on Clock Dials
• Why do clocks with Roman numerals use "IIII" instead of "IV"? (from The Straight Dope)
• Roman Numerals listing from Archive.org
• Roman numbers in Unicode
• Roman numerals from sizes.com
• Roman Numeral Conversion Exercises