Roman numeration

The Roman numerals were used by the Romains antiquity for, starting from only seven letters, to write integers until approximately 4  999 (but not the zero, that they did not know or more exactly did not regard as a number). Classification was standardized in the current use and rests on four principes :

• Any letter placed at the right-hand side of another appear a higher value or equalizes à* Any letter unit placed immediately at the left of a letter stronger than it, indicates that the number which corresponds to him must be cut off with the number which follows;
• the values are grouped in decreasing order, except for the values being cut off according to the preceding rule.
• the same letter cannot be employed four times consecutively except Mr.

Origin

Contrary to an generally accepted idea, the Roman numerals are not acronymic  : for example, C is not, at the beginning, the abbreviation of centum (written CENTVM ). The figures, attested in other languages and writings of Italy, were at the beginning of the separated symbols, then confused with the letters. Thus, in Etruscan (of which the alphabet was borrowed and adapted by the Romans) one finds the symbols for I|1 , V|5 , X|10 , L|50 , C|100 and M|1  000 .

In fact, the modern critic recognizes that Roman numeration is a survival of an antiquated practice, former to the invention even of the writing (and thus, with strictly speaking, prehistoric), and that one finds in many civilizations.

These figures would be derived from the use of sticks with notches and the need for making there appear reference marks: The shepherd who wants to count his animals without knowing to enumerate takes simply a stick on which are reproduced of the notches, makes pass its herd in front of him, and shifts its nail of a notch to each time an animal passes in front of him: the final notch corresponds to the number of animals, and it is enough to locate its position to preserve the number. With this system, the first figures are always simple notches, transcribed later on by " I".

The location is not easy as soon as the number of notches exceeds a handle, because the eye clearly does not perceive the collections beyond three or four items: to see IIIIIIII is practically impossible (by comparison with VIII, much simpler). The shepherd is naturally led to intercalate notches of different form regularly, to be used as visual reference mark; and the natural regrouping (for a shepherd cash on its fingers) is by groups of five. Such a regrouping is always used nowadays on the rules to measure.

The reference mark " cinq" naturalness could be a longer notch (used on the rules), or in skew (used on the sizes), but these two marks are not different many simple notches when it is a question of transcribing them. The simple marks finally used are formed by a double notch (in the shape of V, or of when one reads it in the other direction). The following regrouping, with ten notches, is practically always a notch in cross X. the later reference marks have more worked out forms, with three notches: 50 corresponds to " V plus a encoche" , which initially gives forms in NR, Z or E; and hundred correspond to " X plus a encoche" , giving forms of the star type (*). These forms were less stable, and evolved/moved thereafter worms of the shapes with two features, in L for fifty, and C percent.

With a stick thus marked, the shepherd can rather easily locate the notch on which its calculation stopped. If it has thirteen animals, for example, its nail stops on the third notch after first ten, which retranscribes simply XIII. If it has twenty-nine of them, its nail is with a notch before third ten, which notes XXIX. If it has fifty-nine of them, its finger passed first around fifty, and is with a notch before following ten: LIX.

This primitive location can lead to very atypical writings: for example, a notch before ten before fifty would note IXL (for thirty-nine). It was regularized thereafter, to form the system known nowadays.

Basic traditional notation

}|2=5}} framed (and Ɔ|100 superimposed) become, confused then with D|500 . |----- |style=" make-size: 24px; " | {{Romanian|M|1  000}} |1  000 |style=" text-align: left; " |one X|10 surrounded or framed, near to X|1  000 , which, passing by several forms, of which, was written like a Greek phi , then became CIƆ|1  000 and (inter alia), which forms were finally confused with M|  000 , more especially as 1  000 says thousand Latin . |}

This system, which simplified old Greek numerations and phenicians, makes it possible to write all the numbers of 1 with 4  999, by using the letters of the alphabet Latin more resembling the old lunar systems. Nevertheless this system did not replace them completely, because it was simplified too much and insufficient to express all the numbers (in particular larger numbers, which gave place to all kinds of extensions).

The complexity of the basic Roman system (without the numbers higher than 4  999) appears already in the suivants  examples;:

Also its complex design, mixing additions and subtractions was also difficult to include/understand, even for the Romans themselves which continued to use purely additive systems from which result these forms “  simplificatrices  ” (in particular for calculations). It persisted about it of many alternatives not complying with the rules imposed above, and calling upon the true purely additive origins of this numbering system.

Certain numbers can be written in several ways: 99 can be written XCIX or IC.

It should be noted that the imposed subtractive rules always ceased their effect beyond the thousands, as testifies some the attested and persistent writing MM|4  000 .

Procedure

To know the value of a number written in Roman numerals, the number should be read from left to right. If a figure is larger or equal to its successor, one adds it to the sum. In the contrary case one withdraws it. But beyond 3  000, the rule change : the system of subtraction is not applied any more for the thousands.

• XVI| = 10 + 5 + 1 = 16  ;
• XIV| = 10 + (5 - 1) = 14, because I|1 is lower than V|5   ;
• TEN| = 500 + (10 - 1) = 509, because I|1 is lower than X|10   ;
• MM|4  999 = 1  000×4 + (1  000 - 100) + (100 - 10) + (10 - 1) = 4  999  ;
• MM|4  888 = 4888, the Roman number longest in quantity of symbols .

Examples of Roman numerals in the basic system

• 888 = DCCCLXXXVIII|888   ;
• 1  000 = M|1  000   ;
• 1  515 = MDXV|1  515 = 1  000 + 500 + 10 + 5.
• 1  975 = MCMLXXV|1  975 = 1  000 + (1  000 - 100) + 50 + 10×2 + 5.
• 2  002 = MMII|2  002 = 1  000×2 + 1×2.

Once these conventions of writing were posed, one can write entireties in Roman numerals. But the mathematicians of the time did not use this notation to make additions or multiplications, they had recourse to abacus S. They used of this fact a positional numeration, but without being aware that this positional numeration could have been used to write the larger numbers in a permanent way.

Extensions

Medieval alternatives

With the the Middle Ages, the writing of the Roman numerals is sometimes debased.

4 is written IIII|4 instead of IV|4 (what, in fact, is only one old alternative). This use was taken again in Horlogerie where 4 hours is written IIII|4 , primarily for reasons of Legibility on a round dial, especially when the beers (quantified engravings) there are tilted. One speaks about four of clock and watch maker .

From 60 to 400, one counts and one writes per score, the figure twenty ( xx|20 ) being placed in exposant : that is to say IIII|4 ^xx|20 for 80  :

the Hôpital of Quinze-Vingts to Paris owes its name with this way of compter : it could accommodate 300 (15×20) patient.

For the hundreds, one can indicate the number of hundreds followed by the marker of the hundreds ( C|100 , even in the plural ctz|100 for centz ) in exposant : thus 300 is written III|3 ^C|100 or III|3 ^ctz|100 .

These rules were not such as of the first certificates, especially epigraphic  : several possible C-Ws communication coexisted freely (like IIX|8 for VIII|8 in order to reduce the number of symbols by extension of the subtractive rule, or contrary VIIII|9 for IX|9 in order not to use the subtractive rule). It is only recently that the procedure was fixed.

In certain texts of 15th and 16th centuries one also uses (attention ! there are problems of returned characters Unicode with certain navigators. Signs for 1  000, 5  000 and 10  000 are indicated in the image opposite)  :

• ↀ , or the binding of , for 1  000  ;
• ↁ , or the binding of , for 5  000  ;
• ↂ , or the binding of for 10  000  ;
• but one does not use this notation in notation soustractive : 4  000 is written MM|4  000 and not M|- 1  000 .

It will be noticed that in the symbols above, the number of circles or half-circles (called apostrophus in Latin) indicates a factor 10 applied to the figure médial I|1 whose origin could be in fact the binding of the vertical sérifs of the letters CƆ|1  000 coupled (with the result that it C|100 turned over in Ɔ|×10 show clearly in fact a factor 10, the I médial often being then omitted when the two figures are coupled one with the other).

It should be noted that at the origin the I médial was in fact longer than it I|1 indicating the unit, and our long vertical bar more resembled | , exceeding above the unit bars and under their base line, so that another approximate form that it D|500 the medieval '' thorn '' Þ still used today in the Scandinavian languages should more have resembled.

It will be also noticed that the forms in half-circle are worth half of the form full (in this case the notation with I|1 initial the diameter of the half-circle is necessary for to close)   ; the form D|500 thus appears well also half of , or like the binding of .

This theorization is in fact a contemporary adaptation of an old medieval Greek writing, where the capital letter Phi (Φ) was also used to indicate the 1  number; 000, and resulted from one adaptation to the Greek alphabet of the initial unary system using vertical bars | to frame the multiples of 100, and an additional horizontal bar as a chief to indicate a multiple of 1  000. Also 1  000 in the beginning resembled one more X|10 framed, which resembled itself the Greek phi . Also the Latin apostrophus would have had also a squarer appearance, before one confuses it with a C reversed, as that had already been made for the letter C|100 Latin symbolizing number 100.

Note : the separation of the symbol representing 1  000 with the apostrophus , combined with the I médial (of which the shortening continued then) would be at the origin of the character $\backslash \; infin$ used today in mathematics to symbolize the infinite one, like an evolution of the use of the Latin word thousand in the plural (and of its unknown value) to represent any arbitrarily large and unknown number (one will note the French expression still used aujourd' ui “  of the hundreds and milles  ” pointing out this use). This symbol would be simply the evolution of the tiny binding CIƆ|1  000 in uncial manuscript writing.

Thus the 5  number; 000 can also be represented by (500×10) instead of above. But like all “  C renversés  ” are also related to the I initial, one makes some can be unaware of the presence of this I and transform all the “  C renversés  ” simply in D|500 . Thus the 5  number; 000 is then written simply DD|5  000 , and the 10  number; 000 normally written CCIƆƆ is written today more simply CCDD|10  000 .

Ici  attention;! number 400 is written today normally CD|400 (500-100), it is distinct not very used historical form (not recommended today) (100×10) for 1  000 (at this time there, 400 was written rather CCCC|400 without using the subtractive mode, to avoid confusion). One can on the other hand use CID|1  000 to indicate without ambiguity the historical notation of 1  000, if one does not lay out of the character “  C|100 retourné  ” (replaced here by the letter D|500 , nonambiguous since one precedes it well by the I médial).

Other symbols using this principle of composition could be used to indicate the billion (3 circles) or half-milliard (3 half-circles). It will be noticed that the vertical diameter is always noted, and that the layout of another diameter or horizontal ray could also be used instead of a circle or additional half-circle.

However, the layout using the letter simply C|100 , turned over in and placed after the letter I|1 , was essential quickly (in particular in printing works), because that did not require additional cast iron and improved the legibility of the numbers while being easier to trace with the feather (badly adapted to the layout of small circles)   ; of this fact the form of the C|100 at the place or with back could take that of brackets ( and ) related to the I médial. One finds this use in the old accounts books, of the Moyen-âge until the Renaissance where the C-W communication did not cease becoming more and more complex.

Alternatives for the insertion of the Roman numbers in a text

With the Middle Ages, when the monumental Latin C-W communication was replaced by the Onciale, easier to trace with the feather, the figures were written in small letters as the remainder of the text. The use of the Majuscule S rare (not even at the beginning of sentence) and rather was reserved for the decorative reference letters at the beginning of paragraph (which were only increased alternatives of the letters of the alphabet).

Also, to allow the insertion of numbers in a text, those were framed of points median in order to more easily distinguish them from the words. For example, ·xxvıı· represented number 27 in the medieval manuscripts (the small letter I did not comprise a Point yet superscribed, appeared well later in Gothic script to facilitate the reading of the text, in order to better distinguish the I from the m and N to which the jambs were very close).

The position of these points was variable according to the authors (the use of the punctuation, and in particular distinction of the point and the comma, having been well controlled only well later), and sometimes impossible to distinguish in the text from the point from normal punctuation (it is particularly true for the manuscripts in Catalan, old Occitan and Vieux French, but also the medieval manuscripts in England and of the Holy roman Empire). One also finds this use of the median point (which often took the form of small indents) on the monumental Latin inscriptions which mix the numbers with the text, for example the monuments and religious buildings.

The use of the median points was lost today because the Roman numbers are not employed more as determining numeral adjectives (to indicate quantities one uses today the decimal notation with Indo-European figures, often called improperly Arab numerals French ), but mainly as adjectival ordinal whose context poses less problems of interpretation (after a name of sovereign, or accompanied by an ordinal suffix) and normally in capital letters (or small capitals) within a sentence.

Later, when the letter J was different from the letter I , the official documents started to use the J instead of the I at the end of a number (this form marking the end of the number well which one cannot then lengthen any more). As at that time, there was no tiny/capital difference in the uncial writing, one thus wrote vııj instead of vııı or even ·vııj· (note : herebefore, the small letter J was also written without any superscribed point, appeared well later on the new consonant only by similarity with the vowel I with which it could still freely be confused in the orthography, the choice of the form used having remained a long time very often a question of style independent of the vocal value or consonantale of the lettre  ; for more details, to consult not superscribed ).

Traditional extensions

The various forms above often were diverted and sometimes were incompatible between them, also the accountants used a more logical and simpler marking system, coming from the provision of their abacuses of calculation, and in connection with the initial system, purely additive, where a bar is added to each unit. They took again this principe :

Beyond 4  999, one employs a macron (bars horizontal) superscribed above it number to indicate a factor 1  000 and two will macrons for a factor 1  000  000. For example  :

For the other multiples of 1  000, it M|1  000 superscribed is lengthened to cover the whole of the figures that it multiplie.
That remains true for the superscribed macron, for example CXLII|142 CCCLXVIII|368 DCCXCV|795 represents the 142  number; 368  795.

There however was many alternatives, as well in the layout as the procedure, before this model is not imposed. The notation M|1  000 thus was essential quickly instead of CIƆ (or CID|1  ) and I|1  000 , each time possible. Of the same M|1  000  000 is preferred with CCIƆƆ|1  000  000 (or CCIDD|1  000  000 or CCDD|1  000  000 ) each time possible. is used M|1  000 rather than CID|1  000 or CIƆ|1  000 , except if a group of thousands is higher than 4 in which case is preferred I|1 with macron if possible for the writing of the groupe  ; the writing must be consistent in the systematic use of the letters with will macrons if those are used (in which case the letter M|1  000 will not be used, the numbers being then written by group of 3 decimal digits transcribed using only the six letters D|500 , C|100 , L|50 , X|10 , V|5 , I|1 ).

Calculation

It can that the users of this numbering system were brought, to be carried out calculations, to know certain results by heart. If, for example, they knew the product of XII|12 by XII|12 , it was then easy to deduce the product from it from XII|12 by one of less or one moreover.

Modern extensions

The extension of the subtractive rule is sometimes used in a systematic way with symbols of lower row to shorten the numbers to the maximum, for example IIM|998 = 1  000 - 1×2 = 998, instead of CMXCVIII|998 = (1  000 - 100) + (100 - 10) + 5 + 1×3 according to the standard rule. Sometimes also, the identical unit symbols are gathered together after the subtractive symbols instead of being separated by these subtractive symbols. This shortened representation can be obtained with the algorithm suivant :

• Is the 3  number; 898 to represent,

one breaks up it figure by figure into 3  000 + 800 + 90 + 8 as in the standard rule.

• one converts each figure with the shortest representation (for the multiples of 8 one uses the subtractive rule on following ten, instead of the additive rule on the multiple of 5),

one obtains MM|3  000 DC|- 200 M|1  000 X|- 10 C|100 II|- 2 X|10   ;

• one reorders all the negative figures at the head, in inverse order (of smallest with largest),

one obtains II|- 2 X|- 10 DC|- 200 MM|4  000 C|100 X|10   ;

• one eliminates all the figures which are cancelled mutually while starting with the negative figures on the left,

one obtains II|- 2 C|- 100 MM|4  000   ;

• if there remain groups of 3 or 4 consecutive identical unit symbols, one reduces them with the subtractive rule,

one obtains II|- 2 C|- 100 M|- 1  000 , the final number is then , that one reads as 5000 - 1102 (one adds all the figures lower than the final, from which one will cut off this total).

According to these rules, one obtains also numbers sometimes easier to read and interpréter :

• IIC|98 (100-2) instead of XCVIII|99 (100-10 + 5 + 1×3) with the standard notation for 98  ;
• IC|99 (100-1) instead of XCIX|99 (100-10 + 10-1) with the standard notation for 99  ;
• XM|990 (1  000-10) instead of CMXC|990 (1  000-100 + 100-10) with the standard notation for 990  ;
• XMV|995 (1  000-10 + 5) instead of CMXCV|995 (1  000-100 + 100-10 + 5) with the standard notation for 995  ;
• IM|999 (1  000-1) instead of CMXCIX|99 (1  000-100 + 100-10 + 10-1) with the standard notation for 999  ; etc
• These C-Ws communication however are not always recognized by the readers.

Modern uses

The Roman numerals are still usually used nowadays to note the centuries and the millenia, like, for example, “  the 21e siècle  ” and “  .   ” One also finds the Roman numerals on the dials of the clocks and the watches. In this case, the figure four is often written IIII|4 instead of IV|4 for a purely esthetic question. Indeed, by doing this, the first four figures are made up only of I|1 ( I|1 , II|2 , III|3 and IIII|4 ), the four following is composed containing V|5 ( V|5 , VI|6 , VII|7 , VIII|8 ) and last four containing X|10 ( IX|9 , X|10 , XI|11 , XII|12 ). It should however be known that the C-W communication IIII|4 date from the Roman epoch and met a long time in the inscriptions, the medieval manuscripts then the traditional printed papers form in competition with IV|4 .

The Roman numerals were also formerly used to mark the date of construction of the houses. One still finds on the pediment of old masonries this date written thus.

Still currently, it is current to specify the date of production of a film in Roman numerals, at the end of the credits.

The Roman numerals were abandoned with the profit of the figures Indo-European, known as “  Arab   ”, which uses a decimal system making it possible to write the numbers shorter with hardly more letters (10 instead of 7), and which include the zero positional one (0). Moreover, the decimal system allows an alignment of the figures which enormously facilitates calculations on paper.

However, the extension of the notation with simple or double macron superscribed is still commonly used today above M|1  000 like abbreviation of the Million ( , 10 6 ) and of the Billion ( , 10 9 ).

In classifications of pages, one finds sometimes the Roman numerals in small letters, or more usually in small capitals (in printing works with a neat typography)  :

• I , II , III , iv , v , VI , vii , viii , ix , X , xi , xii , xiii ,…, xl ,…, L ,…, lx ,…, xc ,…, C ,…, Cd ,…, D ,…, cm ,…, m .
• 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 ,…, xl|40 ,…, L|50 ,…, lx|60 ,…, xc|90 ,…, C|100 ,…, Cd|400 ,…, D|500 ,…, cm|900 ,…, m|1  000 .

In Chemistry, one indicates by a Roman numeral the Step oxidation (such as for example copper (II)).

Bibliographical resource

• History compared of written numerations of Genevieve Guitel.
• universal History of the figures of Georges Ifrah

Reference

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