Roman Numerals

Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Modern style uses seven symbols, each with a fixed integer value:

Roman Numerals

1	5	10	50	100	500	1000
I	V	X	L	C	D	M

In the Imperivm Romanvm we use a few more Roman Numerals that are attested in classical and medieval texts:

Roman Numerals in the Imperivm Romanvm

1/24	1/12	1/6	1/4	1/3	5/12	0.5	1	5	10	50	100	500	1000	5000	10,000	50,000	100,000	500,000	1,000,000	5,000,000	10,000,000	50,000,000	100,000,000
Σ	·	:	∴	∷	⁙	S	I	V	X	L	C	D	M	ↁ	ↂ	ↇ	ↈ	D	M	ↁ	ↂ	ↇ	ↈ

The use of Roman numerals continued long after the decline of the Roman Empire. From the 14th century on, Roman numerals began to be replaced by Arabic numerals; however, this process was gradual, and the use of Roman numerals persists in some applications to this day.

One place they are often seen is on clock faces. For instance, on the clock of Big Ben (designed in 1852), the hours from 1 to 12 are written as:

I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII

The notations IV and IX can be read as "one less than five" (4) and "one less than ten" (9), although there is a tradition favouring representation of "4" as "IIII" on Roman numeral clocks. Other common uses include year numbers on monuments and buildings and copyright dates on the title screens of movies and television programs. MCM, signifying "a thousand, and a hundred less than another thousand", means 1900, so 1912 is written MCMXII. For the years of this millenium, MM indicates 2000. The current year is MMXXIV (2024). In Ab Urbe Condita it is MMDCCLXXVII (2777).

Description

Roman numerals for integers use a base-ten system (each number represents a power of ten) though unlike the Arabic Numeral system, wherein each number is placed either before or after a zero to represent its order, the numbers in the Roman Numeral system are assigned set values and use additive counting (unless subtractive counting is used in the case of before a 5 or 10, though even this isn't a set rule). Thus when using Roman numerals we use tally-like counting to create numbers. This structure allows for significant flexibility in notation, and many variant forms are attested.

In fact, there has never been an officially binding, or universally accepted standard for Roman numerals. Usage in ancient Rome varied greatly and became thoroughly chaotic in medieval times.

Standard Form

The following table displays how Roman Numerals are usually written:

Individual Decimal Places

	Thousands	Hundreds	Tens	Units
1	M	C	X	I
2	MM	CC	XX	II
3	MMM	CCC	XXX	III
4		CD	XL	IV
5		D	L	V
6		DC	LX	VI
7		DCC	LXX	VII
8		DCCC	LXXX	VIII
9		CM	XC	IX

The numerals for 4 (IV) and 9 (IX) are written using "subtractive notation", where the first symbol (I) is subtracted from the larger one (V, or X), thus avoiding the clumsier (IIII, and VIIII). Subtractive notation is also used for 40 (XL), 90 (XC), 400 (CD) and 900 (CM). These are the only subtractive forms in standard use.

The largest number that can be represented in this notation is 3,999 (MMMCMXCIX). Prior to the introduction of Arabic numerals in the West, ancient and medieval users of the system used various means to write larger numbers.

Fractions

The base "Roman fraction" is S, indicating 1⁄2. The use of S (as in VIIS to indicate 7 1⁄2) is attested in some ancient inscriptions, but while Roman numerals for whole numbers are decimal S does not correspond to 5⁄10, as one might expect, but 6⁄12. This is because Roman Numeral system for numbers less than 0 are duodecimal in base (base-12).

The Romans used a duodecimal rather than a decimal system for fractions, as the divisibility of twelve (12 = 2 × 2 × 3) makes it easier to handle the common fractions of 1⁄3 and 1⁄4 than does a system based on ten (10 = 2 × 5).

Notation for fractions other than 1⁄2 is mainly found on surviving Roman coins, many of which had values that were duodecimal fractions of the unit as. Fractions less than 1⁄2 are indicated by a dot (·) for each uncia "twelfth", the source of the English words inch and ounce; dots are repeated for fractions up to five twelfths. Six twelfths (one half), is S for semis "half". Uncia dots were added to S for fractions from seven to eleven twelfths, just as tallies were added to V for whole numbers from six to nine.

Each fraction from 1⁄12 to 12⁄12 had a name in Roman times; these corresponded to the names of the related coins:

Fraction	Roman numeral	Name (nominative and genitive)
1⁄24	Σ	Semuncia, Semunciae
1⁄12	·	Uncia, Unicae
2⁄12 = 1⁄6	·· or :	Sextans, Sextantis
3⁄12 = 1⁄4	··· or ∴	Quadrans, Quadrantis
4⁄12 = 1⁄3	···· or ∷	Triens, Trientis
5⁄12	····· or ⁙	Quincunx, Quincuncis
6⁄12 = 1⁄2	S	Semis, Semissis
7⁄12	S·	Septunx, Septuncis
8⁄12 = 2⁄3	S·· or S:	Bes, Bessis
9⁄12 = 3⁄4	S··· or S∴	Dodrans, Dodrantis or Nonuncium, Nonuncii
10⁄12 = 5⁄6	S···· or S∷	Dextans, Dextantis or Decunx, Decuncis
11⁄12	S····· or S⁙	Deunx, Deuncis
12⁄12 = 1	I	As, Assis

Large Numbers

During the centuries that Roman numerals remained the standard way of writing numbers throughout Europe, there were various extensions to the system designed to indicate larger numbers, none of which were ever standardised.

Apostrophus

One of these was the apostrophus, in which 500 was written as IↃ, while 1,000 was written as CIↃ. This is a system of encasing numbers to denote thousands (imagine the Cs and Ↄs as parentheses), which has its origins in Etruscan numeral usage. The IↃ and CIↃ used to represent 500 and 1,000 most likely preceded, and subsequently influenced, the adoption of "D" and "M" in conventional Roman numerals.

Each additional set of C and Ↄ surrounding CIↃ raises the value by a power of ten: CCIↃↃ represents 10,000 and CCCIↃↃↃ represents 100,000. Similarly, each additional Ↄ to the right of IↃ raises the value by a power of ten: IↃↃ represents 5,000 and IↃↃↃ represents 50,000. Numerals larger than CCCIↃↃↃ do not occur.

Sometimes CIↃ was reduced to ↀ for 1,000. John Wallis is often credited for introducing the symbol for infinity (modern ∞), and one conjecture is that he based it on this usage, since 1,000 was hyperbolically used to represent very large numbers. Similarly, IↃↃ for 5,000 was reduced to ↁ; CCIↃↃ for 10,000 to ↂ; IↃↃↃ for 50,000 to ↇ (ↇ); and CCCIↃↃↃ (ↈ) for 100,000 to ↈ.

Vinculum

Another system was the vinculum, in which conventional Roman numerals were multiplied by 1,000 by adding a "bar" or "overline". It was a common alternative to the apostrophic ↀ during the Imperial era: both systems were in simultaneous use around the Roman world (M for '1000' was not in use until the Medieval period). The use of vinculum for multiples of 1,000 can be observed, for example, on the milestones erected by Roman soldiers along the Antonine Wall in the mid-900s AUC (2nd century AD). The vinculum for marking 1,000s continued in use in the Middle Ages.

In The Imperivm Romanvm

The Imperivm Romanvm uses a custom set and combination of the above methods to be able to represent any number that can be represented as a whole fraction over 24. Thus the set of numbers that can be represented by the Imperivm Romanvm Roman Numeral System is:

IRRNS = {q ∈ Q⁺: q/24}

This is done through a few rules:

• The lowest possible number to be represented is 0.0416.
• All numbers can thus be represented by that number in additive form, and thus are able to be represented.
• When creating numbers additive form should be used, unless there is a number whose set value is larger and is 1 value away from being created, then the subtractive form takes presedence.

This results in all numbers up to 399,999,999 (ↈↈↈↂↈↂↈↂↈↂↈMↂCMXCIX) being able to be represented, however the last rule is:

• The symbol ↈ can be used additively, infinitely.

Resulting in an infinite set of numbers where all numbers that can be represented must be able to be represented as a fraction over 24.