Roman Numerals

Roman numerals 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. Roman numerals, as used today, are based on seven symbols:

Symbol	I	V	X	L	C	D	M
Value	1	5	10	50	100	500	1,000

The numbers 1 to 10 are usually expressed in Roman numerals as follows:

I, II, III, IV, V, VI, VII, VIII, IX, X.

Numbers are formed by combining symbols and adding the values, so II is two (two ones) and XIII is thirteen (a ten and three ones). Because each numeral has a fixed value rather than representing multiples of ten, one hundred and so on, according to position, there is no need for "place keeping" zeros, as in numbers like 207 or 1066; those numbers are written as CCVII (two hundreds, a five and two ones) and MLXVI (a thousand, a fifty, a ten, a five and a one).

Symbols are placed from left to right in order of value, starting with the largest. However, in a few specific cases, to avoid four characters being repeated in succession (such as IIII or XXXX), subtractive notation is used: as in this table:

Number	4	9	40	90	400	900
Notation	IV	IX	XL	XC	CD	CM

• I placed before V or X indicates one less, so four is IV (one less than five) and nine is IX (one less than ten)
• X placed before L or C indicates ten less, so forty is XL (ten less than fifty) and ninety is XC (ten less than a hundred)
• C placed before D or M indicates a hundred less, so four hundred is CD (a hundred less than five hundred) and nine hundred is CM (a hundred less than a thousand)

For example, MCMIV is one thousand nine hundred and four, 1904 (M is a thousand, CM is nine hundred and IV is four).

Some examples of the modern use of Roman numerals include:

• 1954 as MCMLIV, as in the trailer for the movie The Last Time I Saw Paris^[
• 1990 as MCMXC, used as the title of musical project Enigma's debut album MCMXC a.D., named after the year of its release.
• 2014 as MMXIV, the year of the games of the XXII (22nd) Olympic Winter Games (in Sochi).

 Roman numerals are still used today in certain contexts. A few examples of their current use are:

Spanish Real using "IIII" instead of IV

• Names of monarchs and popes, e.g. Elizabeth II of the United Kingdom, Pope Benedict XVI. These are referred to as regnal numbers; e.g. II is pronounced "the second". This tradition began in Europe sporadically in the Middle Ages, gaining widespread use in England only during the reign of Henry VIII. Previously, the monarch was not known by numeral but by an epithet such as Edward the Confessor. Some monarchs (e.g. Charles IV of Spain and Louis XIV of France) seem to have preferred the use of IIII instead of IV on their coinage (see illustration).
• Generational suffixes, particularly in the US, for people sharing the same name across generations, for example William Howard Taft IV.
• In the French Republican Calendar, initiated during the French Revolution, years were numbered by Roman numerals - from the year I (1792) when this calendar was introduced to the year XIV (1805) when it was abandoned.
• The year of production of films, television shows and other works of art within the work itself. It has been suggested – by BBC News, perhaps facetiously – that this was originally done "in an attempt to disguise the age of films or television programmes." Outside reference to the work will use regular Hindu–Arabic numerals.
• Hour marks on timepieces. In this context, 4 is usually written IIII.
• The year of construction on building faces and cornerstones.
• Page numbering of prefaces and introductions of books, and sometimes of annexes, too.
• Book volume and chapter numbers, as well as the several acts within a play (e.g. Act iii, Scene 2).
• Sequels of some movies, video games, and other works (as in Rocky II).
• Outlines that use numbers to show hierarchical relationships.
• Occurrences of a recurring grand event, for instance:
• The Summer and Winter Olympic Games (e.g. the XXI Olympic Winter Games; the Games of the XXX Olympiad)
• The Super Bowl, the annual championship game of the National Football League (e.g. Super Bowl XXXVII; Super Bowl 50 is a one-time exception^[24])
• WrestleMania, the annual professional wrestling event for the WWE (e.g. WrestleMania XXX). This usage has also been inconsistent.

Specific disciplines

Entrance to section LII (52) of the Colosseum, with numerals still visible

In astronomy, the natural satellites or "moons" of the planets are traditionally designated by capital Roman numerals appended to the planet’s name. For example, Titan's designation is Saturn VI.

In chemistry, Roman numerals are often used to denote the groups of the periodic table. They are also used in the IUPAC nomenclature of inorganic chemistry, for the oxidation number of cations which can take on several different positive charges. They are also used for naming phases of polymorphic crystals, such as ice.

In computing, Roman numerals may be used in identifiers which are limited to alphabetic characters by syntactic constraints of the programming language. In LaTeX, for instance, \labelitemiii refers to the label of an item in the third level iii of a nested list environment.

In military unit designation, Roman numerals are often used to distinguish between units at different levels. This reduces possible confusion, especially when viewing operational or strategic level maps. In particular, army corps are often numbered using Roman numerals (for example the American XVIII Airborne Corps or the WW2-era German III Panzerkorps) with Hindu-Arabic numerals being used for divisions and armies.

In music, Roman numerals are used in several contexts:

• Movements are often numbered using Roman numerals.
• In music theory, the diatonic functions are identified using Roman numerals. 
• Individual strings of stringed instruments, such as the violin, are often denoted by Roman numerals, with higher numbers denoting lower strings.

In pharmacy, Roman numerals are used in some contexts, including S to denote "one half" and N to mean "nothing".

In photography, Roman numerals (with zero) are used to denote varying levels of brightness when using the Zone System.

In seismology, Roman numerals are used to designate degrees of the Mercalli intensity scale of earthquakes.

In tarot, Roman numerals (with zero) are used to denote the cards of the Major Arcana.

In theology and biblical scholarship, the Septuagint is often referred to as LXX, as this translation of the Old Testament into Greek is named for the legendary number of its translators (septuaginta being Latin for "seventy").

In entomology, the broods of the thirteen and seventeen year periodical cicadas are identified by Roman numerals.

In advanced mathematics (including trigonometry, statistics, and calculus), when a graph includes negative numbers, its quadrants are named using I, II, III, and IV. These quadrant names signify positive numbers, negative numbers on the X axis, negative numbers on both axes, and negative numbers on the Y axis, respectively. The use of Roman numerals to designate quadrants avoids confusion, since Hindu-Arabic numerals are used for the actual data represented in the graph.

Modern non-English use

Capital or small capital Roman numerals are widely used in Romance languages to denote centuries, e.g. the French xviii^e siècle and the Spanish siglo XVIII mean "18th century". Slavic languages in and adjacent to Russia similarly favour Roman numerals (XVIII век). On the other hand, in Slavic languages in Central Europe, like most Germanic languages, one writes "18." (with a period) before the local word for "century".

Boris Yeltsin's signature, dated 10 November 1988. The month is specified by "XI" rather than "11".

In many European countries, mixed Roman and Hindu-Arabic numerals are used to record dates (especially in formal letters and official documents, but also on tombstones). The month is written in Roman numerals, while the day is in Hindu-Arabic numerals: 14.VI.1789 is 14 June 1789.

Sample hours-of-operation sign

I	9:00–17:00
II	10:00–19:00
III	9:00–17:00
IV	9:00–17:00
V	10:00–19:00
VI	9:00–13:00
VII	—

In parts of Europe it is conventional to employ Roman numerals to represent the days of the week in hours-of-operation signs displayed in windows or on doors of businesses, and also sometimes in railway and bus timetables. Monday, taken as the first day of the week, is represented by I. Sunday is represented by VII. The hours of operation signs are tables composed of two columns where the left column is the day of the week in Roman numerals and the right column is a range of hours of operation from starting time to closing time. In the sample chart (left), the business opens from 9 AM to 5 PM on Mondays, Wednesdays, and Thursdays; 10 AM to 7 PM on Tuesdays and Fridays; and to 1 PM on Saturdays; and is closed on Sundays.

Sign at km. 17·9 on route SS4 Salaria, north of Rome

In several European countries Roman numerals are used for floor numbering. For instance, apartments in central Amsterdam are indicated as 138-III, with both a Hindu-Arabic numeral (number of the block or house) and a Roman numeral (floor number). The apartment on the ground floor is indicated as '138-huis'.

In Italy, where roads outside built-up areas have kilometre signs, major roads and motorways also mark 100-metre subdivisionals, using Roman numerals from I to IX for the smaller intervals. The sign "IX| 17" thus marks kilometre 17.9.

A notable exception to the use of Roman numerals in Europe is in Greece, where Greek numerals (based on the Greek alphabet) are generally used in contexts where Roman numerals would be used elsewhere.

Special values

Zero

The number zero does not have its own Roman numeral, but the word nulla (the Latin word meaning "none") was used by medieval scholars in lieu of 0. Dionysius Exiguus was known to use nulla alongside Roman numerals in 525.^[30][31] About 725, Bede or one of his colleagues used the letter N, the initial of nulla or of nihil (the Latin word for "nothing"), in a table of epacts, all written in Roman numerals.

Vinculum

Another system is the vinculum, where a conventional Roman numeral is multiplied by 1,000 by adding an overline. For instance:

• V for 5,000
• XXV for 25,000

Adding further vertical lines before and after the numeral might also be used to raise the multiplier to (say) one hundred thousand, or a million. thus:

• |VIII| for 800,000
• |XX| for 2,000,000

This needs to be distinguished from the custom of adding both underline and overline to a Roman numeral, simply to make it clear that it is a number, e.g. MCMLXVII.

https://books.google.com/books?id=-uHqQSWSPdwC&pg=PA25#v=onepage&q&f=false

Roman Arithmetic

When in Rome, do as the Romans do!

Roman Numerals

Roman Numerals were used in Europe until the 18th century and even today in certain applications. Generally the only place we see them today is an alternative way of expressing a given Natural Number such as year, or hour on a clock, or Superbowl number.

Roman Numerals are not a positional system and contain no symbol of zero. Rather, they are an additive system with each symbol taking on a definite numeric value regardless of where it appears. Actually there a few subtractive rules that makes writing a value somewhat more compact that do depend upon position relative to another symbol.

The symbols and their values are:

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

The table can be extended to larger values by using the convention that a bar over a symbol indicates "1000 times:"

V	X	L	C	D	M
5000	10,000	50,000	100,000	500,000	1,000,000

For our purpose we will generally utilize values that are sufficiently small to be written without the "barred" symbols.

While a particular symbol takes on a given value regardless of its position, it is customary to write the number from left-to-right with the "largest" symbols first.

A Roman value of, as an example: DCCLXVII

would be:

500 + 100 + 100 + 50 + 10 + 5 + 1 + 1 = 767

Of course, the Romans would not have been able to think in terms of our numeric values. They could only manipulate the symbols directly. As an example, they would not have known that VV is an X because 5 + 5 = 10. They would have needed to memorize and use a grouping equivalence table:

IIIII	is equivalent to	V
VV	is equivalent to	X
XXXXX	is equivalent to	L
LL	is equivalent to	C
CCCCC	is equivalent to	D
DD	is equivalent to	M

Such an equivalence would have reduced the number of symbols in a number. As an example, 600 could be written as CCCCCC (or 100+100+100+100+100+100). This would have been "simplified" by the equivalence rules to the properly written DC (or 500+100). For that matter, 600 could be expressed as 12 L's or 60 X's or 120 V's or 600 I's or some combination as long as the sum of all the characters totaled 600. In all cases, by applying the equivalence rules to shorten the expression by replacing several symbols with one we would get DC.

Simply applying the rules so far could lead to a Roman value such as VIIII (or 9). To write this more compactly, we use the convention that we can "subtract" a symbol representing a 1, 10, or 100 from the next two higher symbols, respectively, by writing the smaller to the left. Therefore the only subtractive forms are:

write	instead of	value
IV	IIII	4
IX	VIIII	9
XL	XXXX	40
XC	LXXXX	90
CD	CCCC	400
CM	DCCCC	900

Therefore, as an example: MCMXCIV

would be:

1000 + 900 + 90 + 4 = 1994

The 900, of course, comes from the pair of symbols: CM.

The Roman value is equivalent to: MDCCCCLXXXXIIII. While this is less compact, we shall see that this equivalent form without any subtractives can be useful.

By the way, not every source agrees to some of the details of Roman Numerals. As an example, we could expand the definition of a subtractive and form 99 as IC rather than XCIX as we would according to the rules above. Indeed, it is not clear if the Romans actually used the subtractives to write the numbers compactly.

Conversion

Conversion to and from ordinary numbers is accomplished rather easily.

from Roman

Simply add up the values of the Roman symbols. Of course, if a subtractive appears with a I, X, or C to the left of a "larger" symbol, we need to substitute the pair for the correct numeric value.

to Roman

Start by removing the largest Roman values first and subtracting the removed value until we have converted the entire value. We keep trying to remove a given Roman value until we cannot, then we try the next smaller one. As an example:

value	symbol to try	result	value left	Roman value
2349	M or 1000	yes	1349	M
1349	M or 1000	yes	349	MM
349	M or 1000	no	349	MM
349	D or 500	no	349	MM
349	C or 100	yes	249	MMC
249	C or 100	yes	149	MMCC
149	C or 100	yes	49	MMCCC
49	C or 100	no	49	MMCCC
49	L or 50	no	49	MMCCC
49	X or 10	yes	39	MMCCCX
39	X or 10	yes	29	MMCCCXX
29	X or 10	yes	19	MMCCCXXX
19	X or 10	yes	9	MMCCCXXXX
9	X or 10	no	9	MMCCCXXXX
9	V or 5	yes	4	MMCCCXXXXV
4	V or 5	no	4	MMCCCXXXXV
4	I or 1	yes	3	MMCCCXXXXVI
3	I or 1	yes	2	MMCCCXXXXVII
2	I or 1	yes	1	MMCCCXXXXVIII
1	I or 1	yes	0	MMCCCXXXXVIIII

The table above would indicate the steps a computer would move through. A human would normally take some short cuts by lumping. As an example, we readily see that there are two "thousands" in the original value and would immediately write down: MM. There are three "hundreds" so we get MMCCC, etc until we get the entire number converted.

There is one more step we need to do. We need to substitute for any subtractive values. To do this we proceed right-to-left and look for four of the same symbol. Since there are four I's we check the next symbol on the left and substitute IX for the VIIII. (Note, if the Roman value were something like, MDIIII, we could not utilize the D so we would end up with MDIV.) After making this substitution we have: MMCCCXXXXIX.

In this value there are four X's with a preceding C. These X's become XL. Thus the final Roman numeral value is: MMCCCXLIX.

Addition

If you were asked to add two Roman valves such as CXXII + LXI, you would probably convert these two values to regular numbers, add them (122 + 61 = 183), and finally convert back to Roman Numerals: CLXXXIII. This is because it is relatively easy to convert and we know how to add ordinary decimal numbers.

The Romans could not do this! They needed a method of manipulating the Roman symbols directly to achieve the addition!

It turns out that an algorithm for adding Roman numbers directly is actually quite easy. This was fortunate for the Roman engineers and accountants.

The algorithm has just five steps:

1. Substitute for any subtractives in both values; that is; "uncompact" the Roman values.
2. Put the two values together—catenate them.
3. Sort the symbols in order from left-to-right with the "largest" symbols on the left.
4. Starting with the right end, combine groups of the same symbols that can make a "larger" one and substitute the single larger one.
5. Compact the result by substituting subtractives where possible.

As an example, perform CCCLXIX + DCCCXLV.

	1.	Substitute for any subtractives to obtain:	CCCLXVIIII + DCCCXXXXV
	2.	Catenate to obtain:	CCCLXVIIIIDCCCXXXXV
	3.	Sort to obtain:	DCCCCCCLXXXXXVVIIII
	4.	Combine groups to obtain:	DCCCCCCLXXXXXXIIII DCCCCCCLLXIIII DCCCCCCCXIIII DDCCXIIII MCCXIIII
	5.	Substitute any subtractives to obtain:	MCCXIV

You can verify that this is indeed the correct by converting the values to regular notation: 369 + 845 = 1214.

Subtraction

Subtraction directly is also reasonably easy, but there is a process akin to "borrowing" that needs to be included. If addition is accomplished by putting the two values together to form a result, subtraction is accomplished by "crossing out" symbols in the value that is being subtracted in the starting value.

Here is a simple example: LXVIII − XII. In the first value, LXVIII "cross out" or remove common symbols: LXVIII or, in final form, LVI.

We will assume that the subtraction is possible; that is, the result is 1 or larger. (Note, Roman Numerals cannot express zero or negative numbers.)

The algorithm becomes:

1. Substitute for any subtractives in both values.
2. Any symbols occurring in the second value are "crossed out" in the first.
   1. If the symbol appears in the first, simply cross it out.
   2. If not, then convert a "larger" symbol into appropriate multiples of the needed one, then cross out.
3. Rewrite without the crossed out symbols.
4. Check for any groupings of the same symbol that needs to be replaced with a "larger" one.
5. Compact the result by substituting subtractives where possible.

As an example, perform CXXIX − XLIII.

	1.	Substitute for any subtractives to obtain:	CXXVIIII − XXXXIII
	2.	a.  cross out common symbols:	CXXVIIII and XXXXIII
		b.  need X's, convert C to LXXXXX	LXXXXXXXVIIII and XXXXIII LXXXXXXXVIIII and XXXXIII
	3.	Rewrite:	LXXXVI
	4.	Check for grouping:	LXXXVI
	5.	Substitute any subtractives to obtain:	LXXXVI

You can verify that this is indeed the correct result by converting the values to regular notation: 129 − 43 = 86.

The "borrowing" rule 2b above may need to take on a slightly more complicated approach. Consider D − X. The D must be replaced with a series of symbols that includes an X. It becomes: CCCCC which in turn becomes CCCCLL which finally becomes CCCCLXXXXX. After "crossing out" the common symbols, the result is CCCCLXXXX with a final compact answer of CDXC.

If the conversion in step 2 is done carefully, then there should not be anything for step 4 to do. That is, it should be possible always to introduce a minimum number of needed symbols during step 2.

Multiplication

Multiplication is rather obvious once you realize that the Roman symbols are additive; that is, CXI is really C + X + I. To multiply two multinomial expressions in algebra like (a+b)(x+y+z) we multiply each term in the first by every term in the second and add the results. This is the approach for multiplying Roman numbers.

First we need a multiplication table. Due to the nature of the values of the Roman symbols (every value involves only 5's and 10's) this is easy to form and learn:

times	I	V	X	L	C	D
I	I	V	X	L	C	D
V	V	XXV	L	CCL	D	MMD
X	X	L	C	D	M	V
L	L	CCL	D	MMD	V
C	C	D	M	V
D	D	MMD	V

The value V, of course, may be written as MMMMM.

The algorithm for multiplication is very similar to the one for addition with just five steps:

1. Substitute for any subtractives in both values; that is; "uncompact" the Roman values.
2. For each symbol in one value form the product with every symbol in the second and catenate them all together.
3. Sort the symbols in order from left-to-right with the "largest" symbols on the left.
4. Starting with the right end, combine groups of the same symbols that can make a "larger" one and substitute the single larger one.
5. Compact the result by substituting subtractives where possible.

As an example, perform XXI•XVII.

	1.	Substitute for any subtractives to obtain:	XXI•XVII
	2.	Form products and catenate to obtain:	CLXX CLXX XVII
	3.	Sort to obtain:	CCLLXXXXXVII
	4.	Combine groups to obtain:	CCLLLVII CCCLVII
	5.	Substitute any subtractives to obtain:	CCCLVII

In step two, we get X "times" XVII to obtain CLXX. The second X in the first value gives the same, and finally the I "times" gives XVII. It is these three values that are catenated together to form the result. (Note the catenation performs the addition of the intermediate partial products.)

You can verify that this is indeed the correct by converting the values to regular notation: 21•17 = 357.

Division

Division is somewhat more complicated. Any process should take the two values and produce a quotient and a remainder. Since we cannot represent a zero, a remainder of zero will be simply "no remainder."

Essentially, we can perform division by repeated subtraction. If we count the number of times we can subtract the divisor until the dividend becomes smaller than the divisor, the quotient is the count we get and the remainder is what is left. As an example, 39 divided by 8 gives us:

value	can we subtract 8?	result	count
39	yes	31	1
31	yes	23	2
23	yes	15	3
15	yes	7	4
7	no	7	4

The answer for 39 divided by 8 is a quotient of 4 and a reminder of 7.

This process should work in any notation that we can perform subtraction including Roman values.

However, consider the work involved if we divide 2417 by 17! (Quotient is 142 with a remainder of 3.)

If we really did this with regular numbers we could save considerable work by "shifting" the divisor by multiplying by a power of 10. We see that if we start with 100•17 = 1700. When we count as we subtract, we will count by 100's. When this value is too large, we divide by 10 and count by 10's, then finally divide again by 10's and count by 1's.

starting value	trial subtractor	can we subtract it?	result	count
2417	1700	yes	717	100
717	1700	no	717	100
717	170	yes	547	100+10=110
547	170	yes	377	110+10=120
377	170	yes	207	120+10=130
207	170	yes	37	130+10=140
37	170	no	37	140
37	17	yes	20	140+1=141
20	17	yes	3	141+1=142
3	17	no	3	142

The quotient is 142 and the remainder is 3. This involves 10 steps rather than the 142 steps needed if we simply subtracted 17 over and over!

We took advantage of the fact that with decimal numbers we can multiply by 10 quite easily by simply catenating a zero to the right.

With Roman numerals we can easily multiply a Roman value by C or X or even L or V. Each symbol in the original is replaced a single or set of symbols according to the multiplication table above. We can subtract by "crossing out." Counting is performed by catenation.

We do need one additional process—we need to be able to determine if one Roman number is larger than another.

This "subprocess" is not hard. It is based upon: "If a number contains more of a given symbol, then it is larger." This will be true if we proceed from the "largest" to the "smallest" symbols ignoring those that contain the same number of a symbol until we find the first occurence where there is more of one.

As an example, MMCXXXVI is smaller than MMCCCCXVI because the second contains more C's.
And MMMDXXXVI is larger than MMMCCCCLVII because the first contains more D's.
In both the cases the M's cannot decide the answer because there are the same number in each of the pairs.

Let us try dividing MMMDCCIV by XIV.

The first step is to rewrite these without subtractives: MMMDCCIIII and XIIII

Next form multiples of the divisor:

I	XIIII
V	LVVVV
X	CXXXX
L	DLLLL
C	MCCCC
D	MMMMMDDDD

The last one is clearly too large—contains more M's than the dividend. Therefore we start out subtraction process with the "C times."

starting value	trial subtractor	starting value      after "borrowing"	after "crossing out"	count
MMMDCCIIII	MCCCC	MMMCCCCCCCIIII	MMCCCIIII	C
MMCCCIIII	MCCCC	MDCCCCCCCCIIII	DCCCCIIII	CC
DCCCCIIII	MCCCC too large		DCCCCIIII	CC
DCCCCIIII	DLLLL	DCCLLLLIIII	CCIIII	CCL
CCIIII	DLLLL too large		CCIIII	CCL
CCIIII	CXXXX	CLXXXXXIIII	LXIIII	CCLX
LXIIII	CXXXX too large		LXIIII	CCLX
LXIIII	LVVVV too large		LXIIII	CCLX
LXIIII	XIIII	LXIIII	L	CCLXI
L	XIIII	XXXXVIIIII	XXXVI	CCLXII
XXXVI	XIIII	XXXIIIIII	XXII	CCLXIII
XXII	XIIII	XVIIIIIII	VIII	CCLXIIII
VIII	XIIII too large		VIII	CCLXIIII

We are done! The quotient is CCLXIIII or CCLXIV and the remainder is VIII.

If the divisor divided the dividend exactly then the last "crossing out" step would be complete with nothing left.

Conclusion

Did the Romans actually calculate using these exact procedures? Probably not. They may have utilized shortcuts and other schemes. However, the above processes do perform the arithmetic operations by manipulating the symbols of the Roman written values directly without first converting them to our decimal representation. In many respects these Roman procedures are easier than the corresponding ones for ordinary numbers since they involve only processes such as catenation, arranging the symbols in order, grouping symbols, "borrowing", and "crossing out."

It is important to remember that the Romans did not think in terms of our familiar numbers—they thought only in terms of the Roman Numbers. They were familiar with numbers such as XIII and MCMLXVIII directly without thinking these were 13 and 1968. They had a lifetime of experience with numbers represented only as Roman Numerals.