Roman numerals

In the sign of the Puerta de Alcalá two of the most common uses of Roman numerals are observed: real names (Rege Carolo III, "Rey Carlos Tercer") and years (Anno MDCCLXXVIII, "year 1778").

The Roman numerals is a numeral system that was developed in Ancient Rome and was used throughout the Roman Empire, remaining after its disappearance and still used in some areas. This system uses some capital letters as symbols to represent certain values. Numbers are written as combinations of letters. For example, the current year, 2023, is written numerically as MMXXIII, where each M represents a thousand units, each X represents ten units, and finally each I represents one unit more.

It is based on Etruscan numbering, which, unlike decimal numbering which is based on a positional system, is based on an additive system (each sign represents a value that is added to the previous one). Roman numerals later evolved into a subtractive system, in which some signs subtract instead of adding. For example, 4 in Etruscan numerals was represented as IIII (1+1+1+1), while in modern Roman numerals it is represented as IV (1 subtracted from 5).

Origin

The number 12 written from right to left as shown in the Cipo de Perugia, an Etruscan inscription of the centuryIIIa. C.

Roman numerals are written with letters of the Roman alphabet, but 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 similar to the Etruscan symbols to represent the values. So for I and X they used the letters I and X; for Λ they inverted it and used the V; the symbol Ψ was not uniform in Etruscan and evolved in several variants: Ψ → ᗐ → ⊥; from the latter, the Romans took half of the symbol which became L as 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 are no letters similar to those symbols. The 500 initially had no symbol, but the ⊕ symbol for the 1000 was also sometimes represented by Φ and from half of that symbol they took the D to represent the half. out of 1000.

This system has the particularity that the symbols of greater value are written before those of lesser value, as these are found earlier in the succession of marks. For this reason, this system was able to evolve into a subtractive system in which a sign of a smaller value in front of a larger one subtracted instead of added, 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 gave rise to errors. This makes it easier to read IX than VIIII, also avoiding the confusion of 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 with IIII instead of IV, due to that these two letters are the first of the word IVPPITER (Jupiter), the highest god of the Romans, so it was considered blasphemy to use the initials of his name.

Currently, the same sign should not appear more than three consecutive times. Exceptions are the representation of 4 on clock faces with Roman numerals, which can be done as IV or as IIII.

Comparison with Etruscan figures

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)	Of the Etruscan numeration: Ḥ, which in the Roman was invested (etrusco: ..A A makh "5"
X	10	DECEM (decem)	From Etruscan numeration: X (etrusco: XAP śar "10"
L	50	QVINQVAGINTA (quinquaginta)	Evolution in the Etruscan: → → → → → →
C	100	CENTVM (centum)	First letter CENTVM
D	500	QVINGENTI (Quingenti)	D is the half of Ω (evolution in the Etruscan symbol five hundred: → → Ω)
M	1000	MILLE (Mille)	First letter MILLE

Modern notation

Although lowercase letters were sometimes used in ancient texts to represent Roman numerals, today Roman numerals are written only in uppercase form. The only exception are the Roman numerals used to number sections or items in a list, which are often written in lower case and are called little romans.

Keep in mind that Roman numerals, not being a positional system, do not require a zero. The value zero (none, nothing), not really being a value, is not represented in an additive system such as Roman numerals. For this reason, the Romans were unaware of zero, which was later introduced to Europe with the Indo-Arabic numerals. Although the concept of 0 was known by the Romans, and for this reason N is currently used to refer to the Roman 0, since it comes from the Latin nullus, null, and not from 0.

For the modern notation of Roman numerals, the following rules are used:

• Numbers are read from left to right starting with symbols with greater value, or set of symbols of greater value.
• A symbol followed by another of equal or lower value, adds (e.g., X·X·I = 10+10+1 = 21), while if followed by another of greater value, both symbols form a set in which the value of the first must be subtracted to the value of the next (e.g., X·IX = 10+(10-1) = 19).
• Unity (I) and numbers with base 10 (X, C and M) can be repeated up to 3 consecutive times as adding.
• Numbers with base 5 (V, L and D), cannot be repeated in a row, as the sum of these two symbols has representation with one of the above symbols.
• The unit and base symbols 10 can also be subtracted before a symbol of greater value, but with the following rules:

1. They can only appear to subtract on symbols with base 5 and 10 of immediate superior value, but not others with higher values (e.g., 'IV','IX' or 'XC'but not 'IL'no 'IC'no 'XM'.
2. In the event of subtraction, they cannot be repeated.

• Symbols with base 5 cannot be used to subtract (e.g., 45 is written 'XLV'and not 'VL'.

Entrance to the section LII of the Colosseum, with the still visible numbers

Examples of combinations:

Roman	Nomination
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 1000, a horizontal line is placed above the number, to indicate that the multiplication base is by 1000:

Roman (thousands)	Decimal	Nomination
V	5000	5,000
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	One million

There is a format for numbers with a larger value, in this case a double bar is used to indicate that the multiplication is performed by one million. As an example, to display a value of ten million you would do the following, but with a double underline: X . Three dashes multiply the million by a thousand, making a thousand million, 4 dashes, a billion, 6 dashes, a trillion, etc.

As a numbering system N=(S,R){displaystyle scriptstyle {mathcal {N}}=(S,{mathcal {R}}}}}}, the sign inventory is S={I,V,X,L,C,D,M,! ! !{displaystyle scriptstyle {mathcal {S}}={mathrm {I,V,X,L,C,D,M,} {bar { }}{mathrm {I,V,X,L,C,D,M,} {bar { }}}}}} and the set of rules R{displaystyle scriptstyle {mathcal {R}}}} could be specified as:

• As a general rule, symbols are written and read from left to right, of greater to lower value.
• The value of a number is obtained by adding the values of the symbols that compose it, except in the following exception:
• If a symbol is on the immediate left of another of greater value, the value of the first is subtracted to the value of the second (e.g., IV== ============================================================================================================================================================================================================================================================== IX=9).
• Type 5 symbols always add up, and cannot be on the left of one of greater value.
• Maximum three consecutive repetitions of the same type 1 symbol are allowed.
• The repetition of the same type 5 font is not allowed; its duplicate is a type 10.
• If a type 1 symbol appears subtracting, only a single symbol of greater value can appear on your right.
• If a subtracting type 1 symbol is repeated, it is only allowed that its replay is placed on your right and not adjacent to the remaining symbol.
• Only the subtraction of a type 1 symbol on the immediate major type 1 or type 5. Examples:
  • symbol I can only subtract V and X.
  • symbol X only subtracts L and C.
  • symbol C only subtracts D and M.
• Two different symbols are allowed to appear by subtracting if they are not adjacent.

Here are some examples of invalid numbers in the Roman numeral system, and the rule they break.

Fractions

A coin triens (1/3-4/12) a). The four points •••• indicate their value.

A coin semis (1/2-6/12) a). The letter S indicates its value.

Although the Romans used a decimal numbering system for whole numbers that mirrored the Latin way of counting, they used a duodecimal system for fractions. A system based on twelfths (12 = 3 × 2 × 2) allows you to handle common fractions like 1/3 and 1/4 more easily than a system based on tenths (10 = 2 × 5). Many Roman coins, whose value was a duodecimal fraction of a unit, displayed a notation based on halves and twelfths. A dot • indicated a uncia "twelfth", the etymological origin of the word ounce; and the dots were concatenated to represent fractions up to five twelfths. Six twelfths (a half) were abbreviated with the letter S for semis "half". For fractions between seven and eleven twelfths, uncia points were added in the same way that vertical strokes are added to the V to indicate whole numbers between six and nine.

Each of these fractions had a name that was the same as the corresponding coin for example:

The arrangement of the points was variable and not necessarily linear. The figure formed by five dots arranged as on the face of a dice (:·:) is called quincunx from the name of the fraction and Roman currency. The Latin words sextans and quadrans are the origin of the words sextant and quadrant.

These are other Roman fractions:

• 1/8 'sescuncia, sescunciae' (by sesqui- + Uncia, i.e., 11⁄2 uncias), represented by the sequence of the symbol of semoncy and that of unice.
• 1/24 'Semoncy, semunciae' (by semi- + Uncia, that is, 1⁄2 uncia), represented by a variety of glyphs derived from the Greek letter sigma ・. There is a variant that looks like the symbol of the £ pound but without the horizontal bar, and another one that looks like the letter Cyrillic.
• 1/36 'binae sextulae, binarum sextularum' ("two sextules") or 'duella, sleep', represented by., that is, two letters S invertidas.
• 1/48 'sicilicus, sicilici'represented by., an inverted C.
• 1/72 'sextula, sextulae' (1/6 de uncia), represented by., an inverted S.
• 1/144 'dimidia sextula, dimidiae sextulae' (“media sextula”), represented by., an S inverted and tagged by a horizontal line.
• 1/288 'scripulum, scripuli' (a scruple), represented by the symbol..
• 1/1728 'siliqua, siliquae', represented by a symbol similar to a Latin closing quotes, ».

To make other fractions, simply put lines of underscores, and use the points 12 by 12.

Examples

Roman Numerals at Cutty Sark, Greenwich

Here are several examples of Roman numerals, and their decimal equivalents:

Arithmetic with Roman numerals

All the arithmetic operations carried out with Roman numerals, being a particular case of integer numeration, can be broken down into additions and subtractions.

Sum

Roman Numerals in a manuscript of the centuryXVI

CXVI + XXIV = CXL

The first step decodes the positional data into a single notation, which makes the arithmetic task easier. With this, the second step, having a solely additive notation, can come into operation. After that, a rearrangement is necessary, since the two addends maintain their respective orderings, which is not a problem since no subtractive annotation is present. Once the symbols have been rearranged, they are grouped and the subtractive notation is introduced again, applying the rules of Roman numerals.

Subtraction

CXVI − XXIV = XCII

Multiplication and division are performed in Romans, but they are very extensive, and are not shown here, but factoring and other operations are not performed as the Romans did not know powers despite having multiple knowledge of engineering and architecture. In algebra, Roman letters are used, but common to all operations.

The 4 on the clocks

Watch with Roman numeration, with IIII instead of IV

Numerical diagram in a book of 1560 in which the four is also represented as IIII

It is common to see in many clocks the use of IIII for the numeral 4, instead of the correct IV. The Roman numeral system, derived from that used by the Etruscans, was initially based on the additive method (I plus I were II, V plus I were VI, and II plus II were III). Over time they decided to start using the subtractive method in which the previous number subtracts its amount from the next. In this way, instead of writing 4 as the sum of 2 plus 2 (IIII) it came to be written as the subtraction of 5 minus 1 (IV).

Despite the change, many clocks continued to use the IIII. Some of the supposed reasons why this has been the case are:

• In 1370, a Swiss watchmaker was commissioned to make a watch that would be placed in the tower of the Royal Palace of France, and when King Charles V handed it, he was recriminated to have represented 4 as IV. The watchmaker pointed out that this was how it was written, but Carlos V responded angry: "The King is never wrong." The watchmaker had to change the representation from 4 to IIII and since then in all the watches it began to represent itself.
• In another version of the story it is said that it was the watchmaker who made the mistake of representing the 4 as IIIIAnd the king commanded him to execute for the wrong. Since then, as a protest for the fact and as a homage, all colleagues decided to use IIII instead of IV.
• It is also said that IIII is maintained by superstition. The IV corresponds to the first two letters of the Roman god Jupiter [IVPPITER in Latin], and therefore its use to name a number could be considered inappropriate and blasphemous.
• The whole IIII creates visual symmetry in the sphere, as the symbol I It's the only one that appears in the first four hours, V appears the following four hours and X in the last four, providing a symmetry that would be altered if the IV.
• Also for comfort, as IV It is more difficult to read given its position in the dial of the clock, when it is almost face down (number IV could be confused with the VI in that position).
• Because it's known that a number IV It is not used in watches but in arithmetic, and the watchmakers left it this way.