I=1, V=5, X=10, L=50, C=100, D=500, M=1000,
archaic:=5000,=10000,=50000,=100000.

1 i 2 ii 3 iii 4 iv 5 v 6 vi 7 vii 8 viii 9 ix 10 x

11 xi 12 xii 13 xiii 14 xiv 15 xv 16 xvi 17 xvii 18 xviii 19 xix 20 xx

21 xxi 22 xxii 23 xxiii 24 xxiv 25 xxv 26 xxvi 27 xxvii 28 xxviii 29 xxix 30 xxx

31 xxxi 32 xxxii 33 xxxiii 34 xxxiv 35 xxxv 36 xxxvi 37 xxxvii 38 xxxviii 39 xxxix 40 xl

41 xli 42 xlii 43 xliii 44 xliv 45 xlv 46 xlvi 47 xlvii 48 xlviii 49 xlix 50 l

2012 = MMXII, 2013 = MMXIII, 2014 = MMXIV, 2015 = MMXV, 2016 = MMXVI, 2017 = MMXVII, 2018 = MMXVIII, 2019 = MMXIX,  etc.
 
Superbowl numbers :  XLVIII = 48 (Feb. 2014), XLIX = 49 (Feb. 2015), 
50 (not "L", Feb. 2016), LI = 51 (2017), LII = 52 (2018), LIII = 53 (2019)...

Subtractive Principle:

Any numeral is counted positively unless there's a larger numeral anywhere to its right,in which case it is counted negatively. However,proper Roman numbers are subject to the following restrictionsabout the applicability of the subtractive principle.

The use of the subtractive principle has always been optional.Its systematic use is fairly modern. For example, it's acceptable to use IIII instead of IV, as is usuallydone on clockfaces (to "balance" their left and right halves, so we're told).

The subtractive principle (asubtrahend preceding aminuend) may apply:

• Only to a numeral (thesubtrahend) that is a power of ten (I, X or C).
  For example, "VL" isnot a valid representation of 45 (XLV is correct).
• Only when thesubtrahend precedes aminuend no more than ten times larger.
  For example, "IL" isnot a valid representation of 49 (XLIX is correct).
• Only if any numeral preceding thesubtrahend is at least ten times larger.
  For example, "VIX" isnot a valid representation of 14 (XIV is correct),and "IIX" is not correct for 8 (VIII is correct).
• Only if any numeral following theminuend is smaller than thesubtrahend.
  For example, "XCL" isnot a valid representation of 140 (CXL is correct).

Multiplicative Principle (medieval numeration only):

When the second of the above conditions was not met in front of an M (or C) numeral,a medieval convention was that the number to the left of M (or C) was the numberof thousands (or hundreds) which was to be added to the number located tothe right of M (or C).  When this convention is intended, it's best towrite M (or C) as a superscript (as explained below).

For example, CM means 900, but LLM could only translate into 100000. (This is a rather dubious example which is not  based on an historical instance.)

Donot assume that everyone is an expert at medieval numeration... It's more likely that someone writing  MXMI  intends  1991, rather than  1010001 = MX^MI.

A medieval writer would definitely have hesitated to use thismultiplicative convention beyond  XCIXM (99000) or  XCIXMCMXCIX (99999). Something like  MIMIMI (1001001001) would have sickened most medieval minds!

(2013-04-15) 
A simple conversion table provides an easy, foolproof way.

All of the above rules for standard Roman numeration can be precisely summarized by the following conversion table  (for numbers up to 9999).

Matching multi-digit decimal numbers with
their Roman counterparts (highlighted example is 4096):

Digit	Thousands	Hundreds	Tens	Units
0
1	M	C	X	I
2	MM	CC	XX	II
3	MMM	CCC	XXX	III
4	MMMM	CD	XL	IV
5	MMMMM	D	L	V
6	MMMMMM	DC	LX	VI
7	MMMMMMM	DCC	LXX	VII
8	MMMMMMMM	DCCC	LXXX	VIII
9	MMMMMMMMM	CM	XC	IX

For example, the Roman representation of  4096 is the juxtaposition of  MMMM  for "4" in the thousands  column, nothing for the hundreds,  XC  for "9" in the tens  column, and  VI  for "6" in the units  column:

4096   =   MMMMXCVI

Likewise,  you convert  2020  by concatenating  MM  and  XX :

2020   =   MMXX

One last example:   1956   =   M + CM + L + VI   =   MCMLVI.

For (much) larger numbers, read on...

kdomenick(2001-04-02) 

There are several correct answers for 18034, including the awkward:

MMMMMMMMMMMMMMMMMMXXXIV

The Roman system of numeration is based on an earlier Etruscan system which was sometimesalso used by the ancient Romans for slightly larger numbers. The archaic symbolused for 10000 was a large "m" with 5 legs instead of 3, and it may be typed as "((I))". The symbolfor the number 100000 had 7 legs and may be typed as "(((I)))". The obvious extension to 9 legs or more was apparentlynot used,so the Roman representation of a million would consist of 10 times a 7-legged "m":

Incidentally, the right half of such symbols was used to represent half the correspondingnumber.For example, the numeral for 5000 was, which may be typed as "I))". The numeral for 500 was, it could be typed as "I)",but it got transliterated into "D", the same way became "M" to represent1000 (amnemotechnicalbonus was that M is the initial of "mille", the Latin word for 1000). In print or engravings, such archaic numbers often appear with a regular "C" instead ofour "(" and an upside-down "C" instead of our ")" (which is called anapostrophusin this context), so that you will findinstead of "M" or ""in the publication dates of some early books. There are about a dozen (!) similar graphical variations on this archaic theme...

The basic rules of Roman numeration apply to such symbols:Any numeral is counted positively unless there's a larger numeral anywhere to its right,in which case it is counted negatively. (As explainedabove,proper Roman numbers are subject toprecise restrictions when the numerals do not appear in decreasing order.)Using the above Roman/Etruscan numerals for 10000 and 5000,the number  18034  translates into:

MMMXXXIV   which could be typed   ((I)) I)) MMMXXXIV

This archaic system was replaced by one which uses only7 basic symbols(I=1, V=5, X=10, L=50, C=100, D=500, M=1000),with the convention that putting an overbar (avinculum)  over a basic numberwould denote a value 1000 times as large. It became customary to add little downward-pointing corner marks to such amultiplying vinculum because the straight vinculum was also used(following the Greek custom) simply to distinguish numerals from regular letters,within ordinary text. Skipping that optional flourish, any Roman accountant would simply have expressed 18034 as:

XVIII	XXXIV

The convention about corner marks on the vinculum caused another problem:If those marks were too large, the whole thing could be misread as an upper half-frame,which indicated multiplication by 100000 instead! This ambiguity is the source of a famous dispute about the testament of the widowof Emperor Octavian (Livia Drusilla, 58 BC-AD 29) who willed either500 000 or 50 000 000 sesterces(most probably the latter) to the futur Emperor Galba,whereas her son, the reigning Emperor Tiberius, was the residual heir. The historian Suetonius reports thatquia notata non praescripta erat summa(because the intended sum had not been written out in words),Tiberius could rule that only the lesser amount was owed. Suetonius adds that Galba did not even receive that! The inscription in Livia's testament resembled the middle one below (she did use"CCCCC" instead of the more compact "D" numeral):

	CCCCC			CCCCC				CCCCC
500or500 000		500 000or50 000 000				50 000 000

Apparently, the historical record doesnotshow any instances ofmultiple overstrikesto indicate successive multiplications by 1000and/or 100 000 (stay away from thisdubiousextension of the system). When dealing with the very large amounts involved in public affairs,the Romans understood that the "basic" unit wascentena milia(100 000, one hundred thousand [sesterces]). As Emperor Vespasian took office in AD 69, the amount of money in the statetreasury was reported to him asquadringenties milies (400 times 1000 times)namely: 40 000 000 000 sesterces.

Finally, as notedabove,it's worth mentioning that the familiarsubtractive principle(according to which a numeral appearing before a higher one is to be counted negatively)was not always strictly respected in medieval or ancient times. Instead, smaller numerals appearing before M or C may have meant multiplication(by 1000 or 100) instead, so that VIIC would mean 700 and VIM would mean 6000(this is especially true in the context ofCommon Era dates;CE = Common Era = Christian Era = AD =Anno Domini). The unambiguous typography for this multiplicative convention is to put C or M as superscripts(VII or VI),which is consistent with ancient usage. The use of a dot has also been advocated (VII.C or VI.M), but it is less than satisfying.This gives yet another way to represent 18034, namely:

XVIII^MXXXIV

The  (recommended)  superscripting is not strictly necessary because theunsuperscripted  XVIIIMXXXIV  wouldn't otherwise be avalid number...

psudo(2002-03-01; e-mail)
Your discussion of roman numerals left me wondering if combinationsof superscripting and overstriking, say,would be an acceptable stretch of the Roman system to represent larger numbers...

Extensions of Roman numeration have been toyed with, butthe historical record doesn't reveal any actual usage of an extended system.

Multiple overstriking, or combinations of overstriking and superscripting,are nowhere to be found, neither are multiple-legged symbols beyond,or equivalent parentheses combinations beyond (((I))). I am not even sure whether overstriking was ever used with larger symbols like.

Such extensions would be unambiguous enough to be interpreted correctly,but they are just not a proper part of the system.

This used to be a genuine concern when Roman numeration was dominant in theWestern World. That's no longer the case, though,and there would be little point in devising a new extension to this antiquatedsystem, now best reserved to the mundane numbering of...

• Bearers of the same name.  (Popes, kings, family members, etc.)
• Preface pages.
• Book chapters.
• Copyright years.
• Superbowl events.  Except in 2016  (Superbowl 50, not  L).

Nothing beyond  MMMCMXCIX  (3999)  is needed anytime soon.

(2013-04-15)  
What year has required the longest string of Roman numerals, so far?

Answer:  MDCCCLXXXVIII  (1888).  The record will be broken in  2888.

Nicholas Stevenson(2002-10-07; e-mail)
I am currently translating theChronographus Anni CCCLIIII. It has a few strange numbers like CCCLXIIS and LXXXVIISdealing with money and measurement. I am not sure what the S represents [...]

"S" probably stands for "semis" (one half), however...

In both of your examples, we could also be dealing with theabbreviation forsesterce "IIS"(this later became "HS", which is better and less ambiguous). This symbol comes from the fact that a sesterce wasoriginallytwo and a halfasses (when theas was still the primary Roman monetary unit). If that's the case, CCCLX(or LXXXV)would be 360 (or 85) sesterces.

I think the abbreviation "IIS" was stillused when the "Chronography of 354" was written. (Pleasetell mewhatever you may know.)

On 2002-10-07,Nicholas Stevensonwrote: I thought about the money aspect of the symbolS, but it doesn't explainwhat it means in terms of measurement. I dont think that it is a symbol formoney, but perhaps it does meanhalf a denarius, orhalf a foot. Here are both examples in full: Congiarium dedit LXXIIS. [Where "" replacesthe original symbol for denarii.]  ... et oboliscum cum sua sibibase, qui est in circo maximo, altum edes LXXXVIIS. I just found something that probably explains it in one of my grammarbooks:  "As an abbreviation, S denotes sacrum, semis, sibi suis, etc." I thinkS must [simply] mean ahalf. Nick Stevenson

Right! See below for other fractions used in Roman or Medieval times.

(2014-03-11)  
Here's the rudimentary Roman representation of submultiples of unity.

One twelfth of a unit was an uncia ;  one sixth of an uncia  was a sextula.

The names of most other Roman fractionsare given in the first column of the following table, from page 283 of Mathématiques etmathématiciens  (Editions Magnard, 1959)  by Pierre Dedron (1887-1970; ENS 1908)  and Jean Itard (1902-1979). prefaced by Joseph Pérès (1890-1962), 

The second column gives the glyphs found in the works of Victorius of Aquitaine (fifth century AD). The third column corresponds to the Basel edition  (1558) of the Latin translation ofEuclid's Elements by JohannesCampanus of Novara  (1220-1296). The last two columns provide the ratio of the quantity to one whole unit andto one ounce  (1/12)  respectively.

The smallest quantity expressible with the above Roman fractions isthe difference between a siliqua  and a chalcus, which amounts to  1/576 of an ounce  (or  1/6912 of a whole). The cerates,siliqua andchalcus arerespectively equal to 6, 4 and 3 times that elementary unit,which has been called lentes  (symbol unknown). That forgotten unit was to a whole ( as  or libra ) nearly what a grain is to a modern pound  (1 lb is 7000 grains). The ancient Romans never considered anything smaller than that.

The names of some small Latin fractions are loosely  inspired by Greekcoins or weights.  In ancient Athens, the equivalences were:

•   6 oboloito thedrachma (= fistful  of  6  bronzebullion rods).
•   8  chalkoi to the obol  (instead of 4 in the Roman system).
•   7  lepta  (or mites)  to the chalcos. (That's 336 lepta to the drachma.)

In pre-euro Greek currency, the equivalence was 100 lepta to the drachma.