Roman numerals
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As Roman numerals , the number of characters in the are Roman antiquity incurred and even for numbers in use and special-purpose number font called. In today's usual form, the Latin letters I (1), V (5), X (10), L (50), C (100), D (500) and M (1000) are used as numerals for the spelling of the used natural numbers .
It is an additive number font, with a supplementary rule for the subtractive writing of certain numbers, but without a place value system and without characters for zero . It is based on a combined quinary- decimal or biquinary number system with the base numbers 5 and 10.
presentation
General
The characters used in a Roman numeral have a fixed value. There are powers of ten as base values (the "ones") and the fivefold auxiliary base values (the "fives"). Apart from the subtractive writing , the value is independent of the position.
character | I. | V | X | L. | C. | D. | M. | ↁ | ↂ |
---|---|---|---|---|---|---|---|---|---|
value | 1 | 5 | 10 | 50 | 100 | 500 | 1000 | 5000 | 10,000 |
In modern times the representation with capital letters ( capitals ) is common. Spellings with lower case letters have been used since the Middle Ages and do not mean any difference for the numerical value, but i and l can be mixed up. However , they are still used for page numbering (for example in the foreword, to distinguish it from the actual main part of a book, which is numbered with Indian digits ) and alphanumeric structures .
Occasionally, especially in handwritten records, Roman numerals are marked with an overline or an overline and an underline to distinguish them from normal letters (as with IX = 9, not [ɪks]).
As in the Middle Ages and the early modern period of the letters I and V , the variations J and U developed, they were often used for the same respective numerical value. Especially with minuscules , the closing i was represented by a j: j = 1; ij = 2; iij = 3 etc. This notation is no longer common.
In countries with non-Latin script, the Roman numerals are sometimes written with other, graphically appropriate, native characters. In Russia , for example, the number 1, which was usually in the form of an I on typewriters, is still used for the Roman one, the letters П for II, Ш for III and У for V in typewritten texts :
character | 1 | П | Ш | 1У | У | У1 | УП | УШ | 1Х | Х |
---|---|---|---|---|---|---|---|---|---|---|
value | 1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th | 9 | 10 |
In the Middle Ages, letters other than the usual were used to represent numbers:
character | A. | B. | E. | F. | G | H | K | N | O | P | Q | R. | S. | T | Y | Z | B. | A. |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
value | 500 | 300 | 250 | 40 | 400 | 200 | 151 | 90 | 11 | 400 | 500 | 80 | 70 | 160 | 150 | 2000 | 3000 | 5000 |
conversion
Easy conversion
To convert to a Roman number without the subtraction rule described below, it is sufficient, starting with the large Roman numerals, to subtract their value from the number to be converted as often as possible and to note the Roman numerals. This will automatically sort the digits according to size:
1 × 1000 | + | 1 × 500 | + | 4 × 100 | + | 1 × 50 | + | 3 × 10 | + | 4 × 1 | = | 1984 |
M. | + | D. | + | Cci | + | L. | + | XXX | + | IIII | = | MDCCCCLXXXIIII |
To calculate back such a Roman number, just add the values of the individual numerals.
Subtraction rule
The subtraction rule is a common, shortened form of writing that avoids writing four identical numerals in direct succession. It was occasionally used as early as Roman times, but its consistent application has only appeared since the later Middle Ages, often in mixed use with the writing of individual numbers without the subtraction principle, and since then it has merely remained a largely predominant convention, of which especially in epigraphy many times no use is made.
The subtraction rule in its normal form states that the numerals I, X and C may be placed in front of one of their two next larger numerals and then their numerical value must be subtracted from its value:
- I before V or X: IV (4), IX (9)
- X before L or C: XL (40), XC (90)
- C before D or M: CD (400), CM (900)
Numbers in a bundle of five (V, L, D) are generally not placed in front of a larger character in a subtractive position.
Example:
1 × 1000 | + | (−1 × 100 + 1 × 1000) | + | 1 × 50 | + | 3 × 10 | + | (−1 × 1 + 1 × 5) | = | 1984 |
M. | + | CM | + | L. | + | XXX | + | IV | = | MCMLXXXIV |
There are two deviations from this normal form, which have been found sporadically since ancient times and which have also occurred more recently:
- The character in the subtractive position is doubled and then the value is subtracted twice, e.g. B. IIX instead of VIII for 8, XXC instead of LXXX for 80
- I or X are used in the subtractive position not only in front of the next two characters in each case, but also in front of even higher characters, e.g. B. IL instead of XLIX for 49, IC instead of XCIX for 99 or XM instead of CMXC for 990
Both deviations sometimes occur in combination, i.e. IIL instead of XLVIII for 48, IIC instead of XCVIII for 98.
The subtractive spelling is sometimes associated with the subtractive Latin numerals , but does not match them. With the Latin numerals, the words for 1 and 2, but not those for 10 and 100, are used subtractively and then only the multiple of the 10 from 20 ( duodeviginti = 18, undeviginti = 19) and occasionally the 100 ( undecentum = 99) prefixed.
The zero
An additive or combined additive-subtractive number font like the Roman one does not need a character for the zero , as it plays a fundamental role as a placeholder in a place value system like the decimal system and its usual Indo-Arabic spelling. The Romans knew linguistic expressions for “not something” (nullum) and “nothing” (nihil) , but no number sign and no separate mathematical term for a number value “zero”. When numbers are displayed on the abacus , the absence of a place value is indicated by leaving the corresponding column blank; In tables, the missing number is sometimes marked by a horizontal line, sometimes combined with a small circle. Beda Venerabilis used the character N to denote the zero around 725 AD.
Big numbers
For large numbers (from 1000) there are several possibilities of representation:
Comparison of the spellings of Roman numerals | value | ||||
---|---|---|---|---|---|
with apostrophe | with frame | with Vinculum | Multiplication notation | with cifrão or calderón | |
I. | 1 | ||||
V | 5 | ||||
X | 10 | ||||
L. | 50 | ||||
C. | 100 | ||||
IↃ | D. | 500 | |||
ↀ, CIↃ | M. | I $ | 1000 | ||
ↁ, IↃↃ | V | V • M | V $ | 5000 | |
ↂ, CCIↃↃ | X | X • M | X $ | 10,000 | |
ↇ, IↃↃↃ | L. | L • M | L $ | 50,000 | |
ↈ, CCCIↃↃↃ | I. | C. | CM | C $ | 100,000 |
IↃↃↃↃ | V | D. | DM | D $ | 500,000 |
ProzIↃↃↃↃ | X | I , M | M • M | I $ $ | 1,000,000 |
GegIↃↃↃↃ ↈↈↂↂↂ ↀↁ IↃ LXVII | XII ↂↂↂMↁDLXVII | I CCXXXIV DLXVII | MCCXXXIV • M DLXVII | I $ CCXXXIV $ DLXVII | 1,234,567 |
The spellings were also mixed, like the spelling with apostrophes and the multiplication spelling.
Spelling with apostrophes
Roman numeral | value |
---|---|
Ⅾ, ⅠↃ | 500 |
ↀ, ⅭⅠↃ | 1,000 |
ↁ, ⅠↃↃ | 5,000 |
ↂ, ⅭⅭⅠↃↃ | 10,000 |
ↇ, ⅠↃↃↃ | 50,000 |
ↈ, ⅭⅭⅭⅠↃↃↃ | 100,000 |
The Roman apostrophus, a character that looks like a closing bracket or a C (Ↄ) mirrored on the vertical, is derived, like other Roman numerals, from Chalcidian-Greek numerals.
The original character for 1000, the Phi (Φ, also written ↀ or CIↃ) can be seen as a combination of a C, an I and an apostrophe: CIↃ. By adding more arcs, or C and apostrophes, the value was increased tenfold: ↂ or CCIↃↃ for 10,000, CCCIↃↃↃ or ↈ for 100,000 for short.
The Roman 500, half of 1000, is also created by halving the sign: ↀ → D. The formation of 5000, 50,000 and the following is analogous: ↁ or IↃↃ and ↇ or IↃↃↃ.
Notation with frame
Since the apostrophus notation for very large numbers was unwieldy, a frame was drawn around a digit or group of digits to multiply their value by 100,000. The frame was usually open at the bottom: X , but there are also completely closed: X and spellings that frame the number of characters left and right only with vertical lines: X ago.
The use of 100,000 as a multiplication number corresponds to the Roman numerals for large numbers, such as decies centena milia (literally “ten times per hundred thousand” = one million), quadringenties milies centena milia (“four hundred times a thousand times per hundred thousand” = 40 billion, Vespasian's national debt ). It was centena milia often omitted, by using the Multiplikativzahl instead of the cardinal number (decies instead decem) had realized that they are multiplied by 100,000.
Notation with Vinculum
A vinculum (also titulus) is a dash above the digits to indicate a multiplication by 1000: X , the dash could be drawn across several digits at the same time. Several slashes were possible for higher powers of thousands.
This notation should not be confused with the designation of Roman numerals with an overline (e.g. VI for 6) to distinguish them from normal letters.
Multiplication notation
With greater powers of ten from 1000, a place-value-related multiplication notation was sometimes used. A multiplication factor was written to the left of the symbol, for example V • M for five thousand (5 × 1000).
Spelling with Cifrão or Calderón
In the 16th century, spellings of numbers came up that used special thousand separators to separate large numbers. In Portugal the cifrão was used, a symbol similar to the $ , in Spain the calderón , a U-like symbol (⊍). These characters were used with both Indian and Roman numerals. The number 18,642 was written 18 $ 642 or XVIII $ DCXLII.
particularities
The occasional use of a larger, longer I instead of two consecutive i in Latin texts is rarely found in the representation of Roman numerals. When using this notation, MDCLXX I does not stand for 1671, but for 1672.
In the carpentry trade , the additive notation is generally used for joining symbols: 4 = IIII, 9 = VIIII, 14 = XIIII and so on. On the one hand, this is less complicated and, on the other hand, it prevents confusion between, for example, IX and XI. Another specialty is the frequently used notation X / for XV.
On watch dials , the number 9 is usually written as IX according to the subtraction rule, but the number 4 is often written as IIII.
Fractions
The Romans used fractions with the base 12 . The use of the 12 was obvious because the most frequently required fractions “a half”, “a third” and “a quarter” can be represented by multiples of . The Roman name for one twelfth is Uncia, a word that later became the " ounce " measure of weight . For fractions whose numerator is 1 less than the denominator, a subtractive designation was sometimes used, as in Dodrans (de quadrans, ). Fractions were written out or represented by widely varying characters. In some cases, they have been appended to a Roman number as a number of dots or small slashes equal to the twelfth of the eighth. As a symbol for (semis) or for (semuncia), S or Σ, for (sicilicus) a reversed C and for (duae sextulae) a symbol similar to the Z or the Indian 2 was used.
value | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Roman naming | Uncia ("twelfth") | Sextans ("sixths") | Quadrans ("quarter") | Triens ("third") | Quincunx (quinque unciae) | Semis ("half") | Septunx (septem unciae) | Bes, Bessis (bis triens; dues partes Asis) | Dodrans (de quadrans) | Dextans (de sextans) | Deunx (de uncia) | As |
Representation in Unicode
Capitals | Ⅰ | Ⅱ | Ⅲ | Ⅳ | Ⅴ | Ⅵ | Ⅶ | Ⅷ | Ⅸ | Ⅹ | Ⅺ | Ⅻ | Ⅼ | Ⅽ | Ⅾ | Ⅿ | ↁ | ↂ | ↇ | ↈ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Minuscule | Ⅰ | Ⅱ | Ⅲ | Ⅳ | Ⅴ | Ⅵ | Ⅶ | Ⅷ | Ⅸ | Ⅹ | Ⅺ | Ⅻ | Ⅼ | Ⅽ | Ⅾ | Ⅿ | ||||
value | 1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th | 9 | 10 | 11 | 12 | 50 | 100 | 500 | 1000 | 5000 | 10,000 | 50,000 | 100,000 |
The Unicode block Numbers contains its own codes for the Roman numerals 1–12, 50, 100, 500, 1000, 5000 and 10,000 as upper case and mostly lower case letters as well as some old Roman numerals that have become unusable at positions U + 2160 to U + 2188. These characters are usually represented with the same thickness by the fonts they contain, making them suitable for table digits as well as for East Asian (horizontal or vertical) typesetting. When written vertically, they are also displayed upright - in contrast to normal Latin letters. For most applications, however, the Unicode standard does not recommend the use of these characters; instead, Roman numbers should be represented with the usual Latin letters.
Unicode does not provide special characters for the representation of a frame or for characters with (multiple) Vinculum. This requires a so-called smart font technology such as OpenType in order to be able to select a variant of the characters U + 0305 "Combining Overline" and U + 033F "Combining Double Overline" that is adapted to the letter width and height, provided the font used allows this. A variant of U + 007C "Vertical Line" can be used to represent the frame. These solutions are currently very rare.
When using broken fonts and cursive fonts , Roman numerals are set in Antiqua . If available, this can be done using the Unicode characters mentioned instead of capital letters. As an alternative, an antiqua that matches the text font is used for Roman numerals.
history
Like most notch scripts and simple number systems, the Roman numerals were arranged additively according to the principle of the combined bundling of tens and fives, so that never more than four identical characters follow one another. Following the example of the Etruscan number script , a subtractive spelling was also practiced, in which the prefixing of a character in front of one of the two next higher in the bundle of tens indicates that its value must be deducted from this. In this case, never more than three identical characters follow one another.
The first three Roman numerals I (1), V or rounded U (5) and X (10) have essentially remained unchanged in their writing throughout history, apart from the fact that V and U in older Roman inscriptions are sometimes still in inverted writing - with an upward-pointing angle or arch - appears. You can find them in the same or rotated spelling (V and U regularly inverted Λ, instead of X sometimes an upright cross +) and with the same numerical values among the Etruscans. Largely similar - with V in some cases deviating as a single slash / or inclined backwards \ - they can be detected in older Italian cultures , there as the inscription of notch wood . According to the result of research by Lucien Gerschel from the 1960s, it can be considered certain that the Romans and Etruscans adopted these first three numerals from the cursive script of older Italian peoples.
The original Roman numerals for 50, in which the symbol V or U for 5 was divided by a vertical line and thus increased tenfold in value to 50 (approximately Ψ), can be found with the same spelling (only upside down) and with the same numerical value in the Etruscans and the like in inscriptions from other cultures. It also resembles - in Roman spelling - the letter Chi of the Chalcidian alphabet, a western Greek alphabet that was in use in the Greek colonies of Sicily , or the psi of the eastern Greek alphabets. In Greek, however, Chi stands as a number sign for the value 1000 (as the first letter of the numerical word for 1000: χιλιοι) or in the decimally structured Greek number for the value 600, and Psi there for the value 700. Their numerical use in Greek probably only came first later in use as the corresponding Roman and Etruscan numerals. Contrary to the assumption of older research, it can therefore be assumed, according to Gerschel, that the Romans and Etruscans did not take this number from the Chalcidian alphabet, but also from the cursive script of older Italian peoples. In the case of the Romans, it was then adjusted to the Latin letter L by flattening the angle or round arch to a horizontal line Str and shortening its left half. It is in this form for the first time in 44 BC. Occupied.
The Etruscans wrote the number 100 according to a similar principle as the 50, in that the character X for 10 was divided by a vertical line (approximately Ж) and thus increased tenfold to 100. According to the parallels proven by Gerschel, this sign was also adopted by the Etruscans from the older Italian script. The Romans and other peoples of Italy, on the other hand, wrote the 100 as a right or left-hand open C. In research this was traditionally interpreted as a derivation from the Greek letter theta (number 9). Gerschel and Georges Ifrah, on the other hand, suspect that there is also a modification of the notched and Etruscan symbol for 100, in the case of an Etruscan-based rounded variant of the Ж under the influence of the Latin numeral centum ("hundred") only the one round arch of this Character was retained.
The Romans originally wrote the number 500 as a kind of horizontally divided D, i.e. roughly D , and the number 1000 as a circle Φ or semicircle divided by a vertical line (i.e. as a kind of inverted Ψ) or as a kind of lying S or lying 8 ( ∞ ). In some cases this horizontal 8 ( ∞ ) was divided by a vertical smear. According to conventional belief, the Roman symbol for 1000 is derived from the Greek Phi (number value 500) and the Roman symbol for 500 by halving it. Gerschel and Ifrah, on the other hand, suspect that the Roman thousand symbol was originally a vertically divided circle or a circled X or cross ⊕ and that the D was created by halving it. The Roman sign of the thousand has been in use since around the first century BC. Chr. Increasingly replaced by the letter M (for mille : "thousand"). The numerals M and D are inscribed for the first time in 89 BC. Occupied.
The Roman numerals are used for epigraphic or decorative writing of numbers (especially dates), for counting rulers, popes and other bearers of the same name, for volume, book, chapter and section counting in texts and for figuring measuring instruments like the dial of the clock still in use. Roman numerals are also used in musical notation, for example to denote the position or fret of stringed instruments.
Calculating with Roman numerals
The Roman numerals played a role mainly in the writing of numerals, but hardly in written arithmetic operations. For this purpose tools such as were finger numbers , the abacus and abacus used. Here, values (number of fingers, calculating coins, balls) are assigned to the Roman numerals in a place value system and the arithmetic operation is carried out with these.
In 493 Victorius of Aquitaine put together a table with 98 columns in which he gave the products of the numbers from the fractions up to the value 1000 with the numbers from 2 to 50 in Roman numerals to facilitate multiplication and division, the so-called Calculus Victorii .
Mathematicians who had learned about the Indian numerals recognized their superiority. As early as 1202, the Italian mathematician Leonardo Fibonacci published his book Liber abaci , with which he wanted to make known in Europe the Indian numbers that he had got to know in Bejaja in North Africa.
After examining the existing number systems, the German arithmetic master Adam Ries also gave preference to the Indian numerals. Ries recognized that adding and subtracting tables in tabular form compared to Roman numerals was considerably simplified by the zero. With the establishment of modern arithmetic based on Indian numerals, he heralded the end of the use of Roman numerals in mathematical contexts at the beginning of the 16th century.
literature
- Adriano Cappelli (Ed.): Lexicon Abbreviaturarum. Dizionario Di Abbreviature Latine Ed Italiane (= Manuali Hoepli ). 6. edition, corredata con 9 tavole fuori testo. Ristampa. Ulrico Hoepli Editore, Milan 1999, ISBN 88-203-1100-3 .
Web links
Individual evidence
- ^ Adriano Cappelli : Lexicon Abbreviaturarum . Dictionary of Latin and Italian Abbreviations. JJ Weber, Leipzig 1928, Roman numerals, p. 413 ff . ( online ).
- ^ Peter Gallmann: Graphical elements of the written language: Foundations for a reform of the orthography . Niemeyer, Tübingen 1985, ISBN 3-484-31060-X , p. 282 .
- ↑ ^{a } ^{b } ^{c } ^{d} Georges Ifrah: Universal history of numbers. With tables and drawings by the author . Parkland-Verl., Cologne 1998, ISBN 3-88059-956-4 (French, original title: Histoire universelle des chiffres . Translated by Alexander von Platen).
- ^ Franz Krämer: Basic knowledge of the carpenter. Bruderverlag, Karlsruhe 1982, ISBN 3-87104-052-5 , page 276.
- ↑ John H. Conway , Richard K. Guy : Number magic: from natural, imaginary and other numbers. Birkhäuser, Basel 1997, ISBN 3-7643-5244-2 , Chapter 1 Romance of Numbers; Types of numbers.
- ↑ Julie D. Allen et al. (Ed.): The Unicode Standard, Version 8.0.0 . The Unicode Consortium, Mountain View, CA 2015, ISBN 978-1-936213-10-8 , Symbols, pp. 766 f . ( online [PDF]).
- ^ David W. Maher, John F. Makowski: Literary Evidence for Roman Arithmetic with Fractions . In: The University of Chicago (Ed.): Classical Philology . No. 96 , 2001, p. 376–399 (English, PDF file, 1.18 MB [accessed January 8, 2013]).
- ↑ Priya Hemenway: The Secret Code: The Enigmatic Formula That Determines Art, Nature and Science . Taschen Verlag, Cologne 2008, ISBN 978-3-8365-0708-0 , pp. 80 f . (English, original title: Divine Proportion: Φ In Art, Nature, and Science . Translated by Anita Weinberger).