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Roman numerals

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Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. The system used in classical antiquity was slightly modified in the Middle Ages to produce the system we use today. It is based on certain letters which are given values as numerals.

Roman numerals are rarely 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 and video game 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.



[edit] Symbols

Roman numerals use a basic set of seven symbols:

I1 (one) (unus)
V5 (five) (quinque)
X10 (ten) (decem)
L50 (fifty) (quinquaginta)
C100 (one hundred) (centum)
D500 (five hundred) (quingenti)
M1000 (one thousand) (mille)

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, rather than iiii. Again, for the numbers not assigned a specific symbol, the above given symbols are combined:

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

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.

[edit] 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 19th 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 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 , , etc.); in the other, became and , eventually changing to M under the influence of the word mille ('thousand').

[edit] 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.

[edit] Fractions

A triens coin (1/3 or 4/12 of an as). Note the four dots •••• indicating its value.
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.
A semis coin (1/2 or 6/12 of an as). Note the S indicating its value.

Even though the Romans used a decimal system for whole numbers, presumably because of the number of human fingers, they commonly used duodecimal for fractions, 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:

Fraction1/122/12 = 1/63/12 = 1/44/12 = 1/35/126/12 = 1/27/128/12 = 2/39/12 = 3/410/12 = 5/611/1212/12 = 1
Roman numeral••••••••••••••SS•S••S•••S••••S•••••I

[edit] IIII or IV?

An inscription on Admiralty Arch, London.
An inscription on Admiralty Arch, London.

The notation of Roman numerals has varied through the centuries. Originally, it was common to use IIII to represent "four", because IV represented the pagan god Jupiter. 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.

[edit] 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.
  • With IIII, 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 a V, five 'I's, 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 IX's are rotated 180° to form XI and XII. The alternative with IV 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]

[edit] Chemistry

As it relates to the nomenclature of inorganic compounds, only IV should be used. For example MnO2 should be named manganese (IV) oxide; manganese (IIII) oxide is unacceptable.

[edit] XCIX or IC?

Rules regarding Roman numerals often state that a symbol representing 10x may not precede any symbol larger than 10x+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 universal MCMXCIX.


"Out Of My Mind"

Judging by the look on the organ-grinder,
He'll judge me by the fact that my face don't fit.
It's touching that the monkey sits on my shoulder.
He's waiting for the day when he gets me,
But I don't need no alibi - I'm a puppet on a string.
I just need this stage to be seen.
We all need a pantomime to remind us what is real.
Hold my eye and know what it means.

I'm out of my mind.

Judging by the look on the organ-grinder,
He'll judge me by the fact that my face don't fit.
It's touching that the monkey sits on my shoulder.
He's waiting for the day when he gets me,
But I won't be your concubine - I'm a puppet not a whore.
I just need this stage to be seen.
Will you be a friend of mine to remind me what is real?
Hold my heart and see that it bleeds.

I'm out of my mind.

Tags: wisdom

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