• Roman Numeration: The basics and the precise rules (including medieval ones).
• Larger Numbers, like 18034...
• Extending the Roman system.
• IIS (or HS) is for sesterce (originally, 2½ asses, "unus et unus et semis").

Related articles on this site:

• Roman numerals are awkward for larger numbers.
  • Roman numerals Unabridged discussion. (below)
• Calendars & Chronology.
• The names of operands in common numerical operations.
• The names of "lines": Vinculum, bar, solidus, virgule, slash.
• Zillion. Naming large numbers.
• Zillionplex. Naming huge numbers.
• Scientific Icons and Symbols.

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

1 I i 2 II ii 3 III iii 4 IV iv 5 V v 6 VI vi 7 VII vii 8 VIII viii 9 IX ix 10 X x

11 XI xi 12 XII xii 13 XIII xiii 14 XIV xiv 15 XV xv 16 XVI xvi 17 XVII xvii 18 XVIII xviii 19 XIX xix 20 XX xx

21 XXI xxi 22 XXII xxii 23 XXIII xxiii 24 XXIV xxiv 25 XXV xxv 26 XXVI xxvi 27 XXVII xxvii 28 XXVIII xxviii 29 XXIX xxix 30 XXX xxx

31 XXXI xxxi 32 XXXII xxxii 33 XXXIII xxxiii 34 XXXIV xxxiv 35 XXXV xxxv 36 XXXVI xxxvi 37 XXXVII xxxvii 38 XXXVIII xxxviii 39 XXXIX xxxix 40 XL xl

41 XLI xli 42 XLII xlii 43 XLIII xliii 44 XLIV xliv 45 XLV xlv 46 XLVI xlvi 47 XLVII xlvii 48 XLVIII xlviii 49 XLIX xlix 50 L l

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 restrictions about 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 usually done on clockfaces (to "balance" their left and right halves, so we're told).

The subtractive principle (a subtrahend preceding a minuend) may apply:

• Only to a numeral (the subtrahend) which is a power of ten (I, X or C).
  For example, "VL" is not a valid representation of 45 (XLV is correct).
• Only when the subtrahend preceeds a minuend no more than ten times larger.
  For example, "IL" is not a valid representation of 49 (XLIX is correct).
• Only if any numeral preceeding the subtrahend is at least ten times larger.
  For example, "VIX" is not a valid representation of 14 (XIV is correct), and "IIX" is not correct for 8 (VIII is correct).
• Only if any numeral following the minuend is smaller than the subtrahend.
  For example, "XCL" is not 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 number of thousands (or hundreds) which was to be added to the number located to the right of M (or C). When this convention is intended, it's best to write M (or C) as a superscript (as explained below). For example, CM means 900, but LLM may only mean 100000. You may not assume that everyone is an expert at medieval numeration, though, and it's more likely that someone writing MXMI intends 1991, rather than 1010001 = MX^MI... An ancient writer would probably have hesitated to use this multiplicative convention beyond XCIXM (99000) or XCIXMCMXCIX (99999). Our LLM example was a stretch (probably OK, but not based on an historical instance), whereas something like MIMIMI (1001001001) would definitely have sickened most medieval minds...

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kdomenick (2001-04-02)
What are the Roman numerals for 18 034?

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 sometimes also used by the ancient Romans for slightly larger numbers. The archaic symbol used for 10000 was a large "m" with 5 legs instead of 3, and it may be typed as "((I))". The symbol for the number 100000 had 7 legs and may be typed as "(((I)))". The obvious extension to 9 legs or more was apparently not 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 corresponding number. 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 represent 1000 (with the added advantage 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 of our "(" and an upside-down "C" instead of our ")" (which is called an apostrophus in this context), so that you will find "" appearing instead 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 detailed above, proper Roman numbers are subject to precise restrictions when the numerals do not appear in decreasing order.) Using the above Roman/Etruscan numerals for 10000 and 5000, the number 18034 may thus be expressed as:

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

This archaic system was replaced by one which used only the 7 basic symbols (I=1, V=5, X=10, L=50, C=100, D=500, M=1000), with the simple convention that putting an overbar (a "vinculum") over a basic number would denote a value 1000 times as large. It became customary to add little downward-pointing corner marks to such a multiplying 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 widow of Emperor Octavian (Livia Drusilla, 58 BC-AD 29) who willed either 500 000 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 that quia 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 does not show any instances of multiple overstrikes to indicate successive multiplications by 1000 and/or 100 000 (stay away from this dubious extension of the system).  When dealing with the very large amounts involved in public affairs, the Romans understood that the "basic" unit was centena milia (100 000, one hundred thousand [sesterces]).  As Emperor Vespasian took office in AD 69, the amount of money in the state treasury was reported to him as quadringenties milies (400 times 1000 times) namely: 40 000 000 000 sesterces.

Finally, as noted above, it's worth mentioning that the familiar subtractive 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 of Common 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^C or VI^M ), 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

Note that the (recommended) superscripting is not strictly necessary because the unsuperscripted XVIIIMXXXIV would not otherwise be a valid number...

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

Extensions of the Roman system like the one you suggest have been proposed.  However, the historical record does not seem to show that any such extended system was ever actually used.  Multiple overstriking, or combinations of overstriking and superscripting, are apparently 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 (tell me if you know better).

This is not to say that such extensions would be illogical.  They would probably even be interpreted correctly, but they are just not a proper part of the system.

This used to be a genuine problem when Roman numeration was dominant in the Western World.  It no longer is, though.  There would be little point now in advocating a new extension to this antiquated system, which is best reserved to undemanding tasks like the numbering of pages, chapters, or copyright years...

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Nicholas Stevenson (2002-10-07; e-mail)
I am currently translating the Chronographus Anni CCCLIIII.  It has a few strange numbers like CCCLXIIS and LXXXVIIS dealing 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 the abbreviation for sesterce "IIS" (this later became "HS", which is better and less ambiguous).  This symbol comes from the fact that a sesterce was originally two and a half asses  (when the as was still the primary Roman monetary unit).  If that's the case, CCCLXIIS (or LXXXVIIS) would be 360 (or 85) sesterces.

I do not know whether the abbreviation "IIS" was still commonly used when the "Chronography of 354" was written.  (Please tell me whatever you may know.)

On 2002-10-07, Nicholas Stevenson wrote: I thought about the money aspect of the symbol S, but it doesn't explain what it means in terms of measurement.  I dont think that it is a symbol for money, but perhaps it does mean half a denarius, or half a foot.  I'll give you both examples in full: Congiarium dedit d LXXIIS. [Where "d" replaces the original symbol for denarii.]  ... et oboliscum cum sua sibi base, qui est in circo maximo, altum edes LXXXVIIS. I just found something that probably explains it in one of my grammar books:  "As an abbreviation, S denotes sacrum, semis, sibi suis, etc."  I think S must [simply] mean a half. Nick Stevenson