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Roman numerals
Roman numerals are a numeral system that developed in Ancient Rome and was used throughout the Roman Empire. The system remained in use after the Empire's disappearance and is still used in some fields. It uses certain capital letters as symbols to represent specific values, and numbers are written as combinations of these letters. For example, the year 2020 is written as MMXX, where each M represents one thousand units and each X represents ten more units.
The system is based on Etruscan numeration. Unlike decimal numeration, which is based on a positional system, Etruscan numeration is based on an additive system, in which each sign represents a value that is added to the previous one. Roman numeration later evolved into a subtractive system, in which some signs subtract instead of adding. For example, 4 was represented in Etruscan numeration as IIII (1+1+1+1), while in modern Roman numeration it is represented as IV (1 subtracted from 5).
Symbols
The following table shows the valid symbols in the Roman numeral system and their equivalents in the decimal system:
Sign Value Name Origin I 1 VNVS (ūnus) From Etruscan numeration: I V 5 QVINQVE (quinque) From Etruscan numeration: Λ, which was inverted in Roman numeration X 10 DECEM (decem) From Etruscan numeration: X L 50 QVINQVAGINTA (quinquaginta) Evolution in Etruscan: Ψ → ᗐ → ⊥ → L C 100 CENTVM (centum) First letter of CENTVM D 500 QVINGENTI (Quingenti) D is half of Φ (evolution in Etruscan of the symbol for five hundred: ⊕ → Φ) M 1000 MILLE (Mille) First letter of MILLE
Capital letters are used in Roman numeration because the Latin alphabet originally had only capital letters.
Origins
Roman numerals are written with letters of the Roman alphabet, but they originally came from the Etruscans, who used I, Λ, X, Ψ, 8 and ⊕ to represent I, V, X, L, C and M, respectively. The Romans took letters resembling the Etruscan symbols to represent these values. Thus, for I and X they used the letters I and X; they inverted Λ and used V; the symbol Ψ was not uniform in Etruscan and evolved into various forms: Ψ → ᗐ → ⊥. From the last of these, the Romans took half of the symbol, which became L because it was the most similar letter. For 8 and ⊕ they used the initials of the Latin names corresponding to those values, C and M, since there were no letters similar to those symbols. The number 500 initially had no symbol, but the symbol ⊕ for 1000 was also sometimes represented as Φ, and from half of that symbol they took D to represent half of 1000.
Nevertheless, it appears that the Etrusco-Roman numerals actually derive from notches, marks or lines carved into rods, sticks and bones to keep counts (such as the Ishango bone), which were used by both Dalmatian and Italian shepherds until the 19th century.[1] Thus, the numeral 'I' descends from a notch carved into the rod. In Etruscan numeration, every fifth notch was a double notch (e.g., ⋀, ⋁, ⋋, ⋌, etc.), and every tenth notch was a cross (X), IIIIΛIIIIXIIIIΛIIIIXII…, much in the style of European tally marks to this day. This gave rise to an additive system: eight on a tally rod was eight marks—IIIIΛIII—which could be abbreviated as ΛIII (or VIII), since the presence of Λ implies four preceding notches. By extension, eighteen was the eighth notch after the first ten, which could be abbreviated with X, and so it was XΛIII.
A particular feature of this system is that symbols of greater value are written before those of lesser value, since they occur earlier in the sequence of marks. For this reason, the system was able to evolve into a subtractive system in which a sign of lesser value placed before one of greater value subtracted instead of adding, which made it possible to shorten the writing of large numbers. Thus the number 1999 went from M·DCCCC·LXXXX·VIIII to M·CM·XC·IX. This also made reading easier, since reading more than 3 identical letters in a row led to errors. Thus it is easier to read IX than VIIII, which also avoids confusing the latter with VIII.
However, until the Middle Ages the additive method (up to 4 identical letters in a row) was combined with the subtractive method (symbols that also subtract). For example, it was quite common to represent 4 as IIII instead of IV, because these two letters are the first of the word IVPITER (Jupiter), the supreme god of the Romans, so using the initials of his name was considered blasphemy.
Today, the same sign must not be repeated more than three times in a row. The exception is the representation of 4 on clock faces with Roman numerals, which may be written either as IV or as IIII.[2]
Modern notation
Although lowercase letters were sometimes used to represent Roman numerals in ancient texts, today Roman numerals are written only in capital form. The only exception is Roman numerals used to number sections or items in a list, which are frequently written in lowercase and are called romanitos.
It should be noted that Roman numeration, not being a positional system, does not require zero. The value zero (none, nothing), not really being a value, is not represented in an additive system such as Roman numeration. For this reason, the Romans did not know zero, which was introduced into Europe later with Arabic numeration.
The following rules are used for the modern notation of Roman numerals:
- Numbers are read from left to right, starting with the symbols of greatest value, or the group of symbols of greatest value.
- A symbol followed by another of equal or lesser value is added (e.g., X·X·I = 10+10+1 = 21), while if it is followed by one of greater value, both symbols form a group in which the value of the first must be subtracted from the value of the second (e.g., X·IX = 10+[10-1] = 19).
- The unit (I) and the base-10 numbers (X, C and M) may be repeated up to 3 times in a row as addends.
- The base-5 numbers (V, L and D) may not be repeated in a row, since the sum of two such symbols is represented by one of the preceding symbols.
- The unit and the base-10 symbols may also subtract when placed before a symbol of greater value, but according to the following rules:
- They may only appear subtracting from the base-5 and base-10 symbols of immediately higher value, but not from others of higher value (e.g., 'IV', 'IX' or 'XC', but not 'IL', 'IC' or 'XM').
- When subtracting, they may not be repeated.
- The base-5 symbols may not be used to subtract (e.g., 45 is written 'XLV' and not 'VL').
Examples of combinations:
Roman Name II two III three IV four VI six VII seven VIII eight IX nine XXXII thirty-two XLV forty-five
For numbers with values equal to or greater than 4000, a horizontal line is placed above the number to indicate that it is multiplied by 1000:
Roman
(thousands)Decimal Name V 5000 five thousand X 10 000 ten thousand L 50 000 fifty thousand C 100 000 one hundred thousand D 500 000 five hundred thousand M 1 000 000 one million
There is no format for numbers of greater magnitude, so a double bar or an underline bar is sometimes used to indicate that the multiplication is by one million. As an example, a value of ten million would be shown as follows: X
As a numeral system , the inventory of signs is and the set of rules could be specified as:
- As a general rule, symbols are written and read from left to right, from greatest to least value.
- The value of a number is obtained by adding the values of the symbols that make it up, except in the following case:
- If a symbol is immediately to the left of another of greater value, the value of the first is subtracted from the value of the second (e.g., IV=4, IX=9).
- Type-5 symbols always add, and may not be placed to the left of one of greater value.
- At most three consecutive repetitions of the same type-1 symbol are allowed.
- Repetition of the same type-5 letter is not allowed; its double is a type-10 letter.
- If a type-1 symbol appears subtracting, only a single symbol of greater value may appear to its right.
- If a type-1 symbol that appears subtracting is repeated, its repetition is only allowed to be placed to its right and not adjacent to the symbol it subtracts from.
- Subtraction of a type-1 symbol is only allowed from the next higher type-1 or type-5 symbol. Examples:
- the symbol I may only subtract from V and X.
- the symbol X only subtracts from L and C.
- the symbol C only subtracts from D and M.
- Two different symbols may appear subtracting if they are not adjacent.
Below are some examples of numbers that are not valid in the Roman numeral system, and the rule they break.
| Incorrect | Correct | Value | Reason |
|---|---|---|---|
| VL | XLV | 45 | Type-5 letter subtracting |
| VD | CDXCV | 495 | Type-5 letter subtracting |
| LD | CDL | 450 | Type-5 letter subtracting |
| IIII | IV | 4 | More than three repetitions of a type-1 letter |
| VIV | IX | 9 | Repetition of a type-5 letter |
| XXXX | XL | 40 | More than three repetitions of a type-1 letter |
| LXL | XC | 90 | Repetition of a type-5 letter |
| CCCC | CD | 400 | More than three repetitions of a type-1 letter |
| DCD | CM | 900 | Repetition of a type-5 letter |
| IXX | XIX | 19 | Type-1 letter to the left of two of greater value |
| XCC | CXC | 190 | Type-1 letter to the left of two of greater value |
| CMM | MCM | 1900 | Type-1 letter to the left of two of greater value |
| IXVI | XV | 15 | Type-1 letter to the left of two of greater value |
| XCLX | CL | 150 | Type-1 letter to the left of two of greater value |
| CMDC | MD | 1500 | Type-1 letter to the left of two of greater value |
| IVI | V | 5 | Subtracting letter and its repetition adjacent to the symbol it subtracts from |
| XLX | L | 50 | Subtracting letter and its repetition adjacent to the symbol it subtracts from |
| CDC | D | 500 | Subtracting letter and its repetition adjacent to the symbol it subtracts from |
| IXI | X | 10 | Subtracting letter and its repetition adjacent to the symbol it subtracts from |
| XCX | C | 100 | Subtracting letter and its repetition adjacent to the symbol it subtracts from |
| CMC | M | 1000 | Subtracting letter and its repetition adjacent to the symbol it subtracts from |
| IIV | III | 3 | Type-1 letter subtracting and repeated to its left |
| XXL | XXX | 30 | Type-1 letter subtracting and repeated to its left |
| CCD | CCC | 300 | Type-1 letter subtracting and repeated to its left |
| IIX | VIII | 8 | Type-1 letter subtracting and repeated to its left |
| XXC | LXXX | 80 | Type-1 letter subtracting and repeated to its left |
| CCM | DCCC | 800 | Type-1 letter subtracting and repeated to its left |
| IL | XLIX | 49 | Letter I subtracting from L |
| IC | XCIX | 99 | Letter I subtracting from C |
| ID | CDXCIX | 499 | Letter I subtracting from D |
| IM | CMXCIX | 999 | Letter I subtracting from M |
| XD | CDXC | 490 | Letter X subtracting from D |
| XM | CMXC | 990 | Letter X subtracting from M |
| XIL | XLI | 41 | Letters I and X adjacent and subtracting |
| IXL | XXXIX | 39 | Letters I and X adjacent and subtracting |
| CXD | CDX | 410 | Letters X and C adjacent and subtracting |
| XCD | CCCXC | 390 | Letters X and C adjacent and subtracting |
Fractions
Although the Romans used a decimal system of numeration for whole numbers that reflected the way of counting in Latin, for fractions they used a duodecimal system. A system based on twelfths (12 = 3 × 2 × 2) makes it easier to handle common fractions such as 1/3 and 1/4 than a system based on tenths (10 = 2 × 5). Many Roman coins, whose value was a duodecimal fraction of the unit, displayed a notation based on halves and twelfths. A dot • indicated an [uncia] Error: {{Lang}}: text has italic markup (help) "twelfth", the etymological origin of the word ounce; and dots were combined to represent fractions of up to five twelfths. Six twelfths (one half) were abbreviated with the letter S for [semis] Error: {{Lang}}: text has italic markup (help) "half". For fractions between seven and eleven twelfths, uncia dots were added in the same way that vertical strokes are added to V to indicate whole numbers between six and nine.
Each of these fractions had a name that was the same as that of the corresponding coin, for example:
| Fraction | Roman numeral | Name (nominative and genitive) | Meaning |
|---|---|---|---|
| 1/12 | • | [uncia, unciae] Error: {{Lang}}: text has italic markup (help) | "ounce" |
| 2/12 = 1/6 | •• or : | [sextans, sextantis] Error: {{Lang}}: text has italic markup (help) | "sixth" |
| 3/12 = 1/4 | ••• or ∴ | [quadrans, quadrantis] Error: {{Lang}}: text has italic markup (help) | "quarter" |
| 4/12 = 1/3 | •••• or :: | [triens, trientis] Error: {{Lang}}: text has italic markup (help) | "third" |
| 5/12 | ••••• or :•: | [quincunx, quincuncis] Error: {{Lang}}: text has italic markup (help) | "five ounces" (quinque unciae → quincunx) |
| 6/12 = 1/2 | S | [semis, semissis] Error: {{Lang}}: text has italic markup (help) | "half" |
| 7/12 | S• | [septunx, septuncis] Error: {{Lang}}: text has italic markup (help) | "seven ounces" (septem unciae → septunx) |
| 8/12 = 2/3 | S•• or S: | [bes, bessis] Error: {{Lang}}: text has italic markup (help) | "double" (understood as "twice a third") |
| 9/12 = 3/4 | S••• or S:• | [dodrans, dodrantis] Error: {{Lang}}: text has italic markup (help) or [nonuncium, nonuncii] Error: {{Lang}}: text has italic markup (help) |
"less a quarter" (de-quadrans → dodrans) or "ninth ounce" (nona uncia → nonuncium) |
| 10/12 = 5/6 | S•••• or S:: | [dextans, dextantis] Error: {{Lang}}: text has italic markup (help) or [decunx, decuncis] Error: {{Lang}}: text has italic markup (help) |
"less a sixth" (de-sextans → dextans) or "ten ounces" (decem unciae → decunx) |
| 11/12 | S••••• or S:•: | [deunx, deuncis] Error: {{Lang}}: text has italic markup (help) | "less an ounce" (de-uncia → deunx) |
| 12/12 = 1 | I | [as, assis] Error: {{Lang}}: text has italic markup (help) | "unit" |
The arrangement of the dots was variable and not necessarily linear. The figure formed by five dots arranged as on the face of a die (:·:) is called a quincunx, after the name of the Roman fraction and coin. The Latin words sextans and quadrans are the origin of the words sextant and quadrant.
These are other Roman fractions:
- 1/8 [sescuncia, sescunciae] Error: {{Lang}}: text has italic markup (help) (from sesqui- + uncia, that is, 1½ unciae), represented by the sequence of the symbol for the semuncia and that for the uncia.
- 1/24 [semuncia, semunciae] Error: {{Lang}}: text has italic markup (help) (from semi- + uncia, that is, ½ uncia), represented by a variety of glyphs derived from the Greek letter sigma Σ. One variant resembles the pound sign £ but without the horizontal bar, and another resembles the Cyrillic letter Є.
- 1/36 [binae sextulae, binarum sextularum] Error: {{Lang}}: text has italic markup (help) ("two sextulae") or [duella, duellae] Error: {{Lang}}: text has italic markup (help), represented by ƧƧ, that is, two reversed letters S.
- 1/48 [sicilicus, sicilici] Error: {{Lang}}: text has italic markup (help), represented by Ɔ, a reversed C.
- 1/72 [sextula, sextulae] Error: {{Lang}}: text has italic markup (help) (1/6 of an uncia), represented by Ƨ, a reversed S.
- 1/144 [dimidia sextula, dimidiae sextulae] Error: {{Lang}}: text has italic markup (help) ("half sextula"), represented by ƻ, a reversed S struck through by a horizontal line.
- 1/288 [scripulum, scripuli] Error: {{Lang}}: text has italic markup (help) (a scruple), represented by the symbol ℈.
- 1/1728 [siliqua, siliquae] Error: {{Lang}}: text has italic markup (help), represented by a symbol similar to a closing Latin quotation mark, ».
Examples
Below are several examples of Roman numerals and their decimal equivalents:
| Roman | Decimal |
|---|---|
| I | 1 |
| II | 2 |
| III | 3 |
| IV | 4 |
| V | 5 |
| VI | 6 |
| VII | 7 |
| VIII | 8 |
| IX | 9 |
| X | 10 |
| XI | 11 |
| XII | 12 |
| XIII | 13 |
| XIV | 14 |
| XV | 15 |
| XVI | 16 |
| XVII | 17 |
| XVIII | 18 |
| XIX | 19 |
| XX | 20 |
| XXI | 21 |
| XXII | 22 |
| XXIII | 23 |
| XXIV | 24 |
| XXV | 25 |
| XXVI | 26 |
| XXVII | 27 |
| XXVIII | 28 |
| XXIX | 29 |
| XXX | 30 |
| XL | 40 |
| L | 50 |
| LX | 60 |
| LXX | 70 |
| LXXX | 80 |
| XC | 90 |
| C | 100 |
| CDL | 450 |
| DCLXVI | 666 |
| CMXCIX | 999 |
| MCDXLIV | 1444 |
| MMMDCCCLXXXVIII | 3888 |
Arithmetic with Roman numerals
All arithmetic operations performed with Roman numerals, being a particular case of integer numeration, can be broken down into additions and subtractions.
Addition
CXVI + XXIV = CXL
| Step | Description | Example |
|---|---|---|
| 1 | Remove the subtractive notation | IV → IIII |
| 2 | Concatenate the terms | CXVI + XXIIII → CXVIXXIIII |
| 3 | Sort the numerals from greatest to least | CXVIXXIIII → CXXXVIIIII |
| 4 | Simplify the result by reducing symbols | IIIII → V; VV → X; CXXXVIIIII → CXXXX |
| 5 | Add subtractive notation | XXXX → XL |
| 6 | Solution | CXL |
The first step decodes the positional data into a single notation, which makes the arithmetic task easier. With this, the second step can be carried out, since the notation is now purely additive. After that, reordering is necessary, because the two addends keep their respective orderings, which is not a problem since no subtractive notation is present. Once the symbols have been reordered, they are grouped and subtractive notation is introduced again, applying the rules of Roman numeration.
Subtraction
CXVI − XXIV = XCII
| Step | Description | Example |
|---|---|---|
| 1 | Remove the subtractive notation | IV → IIII |
| 2 | Remove the numerals common to both terms | CXVI − XXIIII → CV − XIII |
| 3 | Expand the numerals of the first term until elements of the second appear | CV − XIII → LLIIIII − XIII → LXXXXXIIIII − XIII |
| 4 | Repeat steps 2 and 3 until the second term is empty | LXXXXXIIIII − XIII → LXXXXII |
| 5 | Add subtractive notation | LXXXXII → XCII |
| 6 | Solution | XCII |
The 4 on clocks
It is common to see IIII used for the numeral 4 on many clocks, instead of the correct IV. The Roman numeral system, derived from the one used by the Etruscans, was initially based on the additive method (I plus I was II, V plus I was VI, and II plus II was IIII). Over time they decided to start using the subtractive method, in which the preceding number subtracts its quantity from the following one. In this way, instead of writing 4 as the sum of 2 plus 2 (IIII), it came to be written as the subtraction of 1 from 5 (IV).[3]
Despite the change, IIII continued to be used on many clocks. Some of the supposed reasons for this are:[3]
- In 1370, a Swiss clockmaker was commissioned to make a clock to be placed in the tower of the Royal Palace of France, and when he delivered it, King Charles V reproached him for having represented 4 as IV. The clockmaker pointed out that this was how it was written, but Charles V replied angrily: "The King is never wrong." The clockmaker had to change the representation of 4 to IIII, and from then on all clocks began to represent it that way.
- Another version of the story says that it was the clockmaker who made the mistake of representing 4 as IIII, and the King had him executed for the error. From then on, as a protest against this and as a tribute, all his fellow clockmakers decided to use IIII instead of IV.
- It is also said that IIII is kept out of superstition. IV corresponds to the first two letters of the Roman god Jupiter [IVPITER in Latin], and therefore its use to denote a number could be considered inappropriate and blasphemous.
- The group IIII creates visual symmetry on the dial, since the symbol I is the only one that appears in the first four hours, V appears in the next four hours and X in the last four, providing a symmetry that would be disrupted if IV were used.
- Also for convenience, since IV is harder to read given its position on the clock face, being almost upside down (the number IV could be confused with VI in that position).
See also
- Numeral system
- Binary number system
- Decimal system
- Octal system
- Hexadecimal system
- Number theory
- Arabic numerals
- Numbers
References
- ↑ Ifrah 2000
- ↑ "Ortografía de los números romanos". Ortografía de la lengua española (in español). RAE and ASALE. 2010.
- ↑ 3.0 3.1 "¿Por qué en algunos relojes el 4 aparece escrito IIII y no IV?" (in español). 20 minutos. March 15, 2012.
Bibliography
- Asimov, Issac (1966, 1977). Asimov On Numbers. Pocket Books, Simon & Schuster, Inc.
{{cite book}}: Check date values in:|year=(help)CS1 maint: year (link) - Hooper, Alfred (1945). The River Mathematics. New York: H. Holt and Company.
- 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.
- Milham, W. I. (1947). Time & Timekeepers. New York: The Macmillan Company. Archived from the original on April 17, 2009. Retrieved April 16, 2009.
- Morterero y Simón, Conrado (1979). Apuntes de iniciación a la paleografía española de los siglos XII a XVII: lecciones pronunciadas (in español). Ediciones Hidalguia. ISBN 9788400044091.
External links
- Arabic-to-Roman and Roman-to-Arabic numeral converter (in Spanish)
- Números Romanos (in Spanish)
