Talk:Roman numerals/Rules

The modern era has seen the emergence of a standardized orthography for Roman numerals, which permits only one permutation for any given value. This system may be described as a decimal pattern, as above, but also as a logical set of rules. While exceptions can be made (notably IIII instead of IV on clockfaces), the modern convention is widely recognized and adhered to, and may be prescribed by the following ruleset:^{[1][2][3][4][5][6][7][8][9][10][11]}

Basic Rules

1. Powers of ten are dealt with separately, ordered from greatest to smallest and from left to right, with numerals either added or subtracted.
2. Repeated 'tens' numerals (I, X, C, M) are added together, with up to three (3) permitted in sequence (subtraction allows an additional non-sequential repetition once per power).
3. 'Fives' numerals (V, L, D) may only be added once per power (not repeated nor subtracted).
4. Numerals placed to the right of a greater value are added, while those placed to the left are subtracted; if placed between two larger numerals, the 'tens' value is subtracted (i.e. subtraction takes precedence).
5. Only one (1) 'tens' numeral may be subtracted from a single numeral per power, and this must be by 1/5 or 1/10 (i.e. the next lower 'tens' numeral).
6. Addition must be less than subtraction for any given numeral (i.e. the added value to the right must be less than the subtractor).
7. Any subtracted 'tens' numeral must either be first, or be preceded by a numeral at least ten times (10x) greater.

The above describes the basic pattern of integers from 1 - 3,999.

Examples

Following the above rules:
90 must be XC. It cannot be LXXXX because that would break rule 2 (up to three in sequence), and it cannot be LXL because that would break rule 3 ('fives' not repeated).
45 must be XLV. It cannot be VL because that would break rule 3 ('fives' not subtracted).
99 must be XCIX. It cannot be IC because that would break rule 5 (subtract by one fifth or tenth).
18 must be XVIII. It cannot be IIXX or IXIX because these would break rule 5 (subtract once per power).
19 must be XIX. It cannot be IXX because that would break either rule 5 (subtract by one fifth or tenth) or rule 6 (subtract>add).
10 must be X. It cannot be IXI because that would break rule 6 (subtract>add).
14 must be XIV. It cannot be IXV because that would break rule 6 (subtract>add), and it cannot be VIX because that would break rule 7 (at least 10x).

1894 is MDCCCXCIV. Consider how this agrees with the rules: powers of ten are arranged properly (rule 1), C is used three times sequentially and a fourth time non-sequentially (rule 2), 'fives' numerals D and V are added once each (rule 3), X and I are subtracted by being placed to the left of larger values while the rest are added (rule 4), they are subtracted once per power by one tenth and one fifth respectively (rule 5), IV is less than X (rule 6), and C is 10x X and 100x I (rule 7).

Fractions & Vinculums

The following rules extend the system further:

• Fractions may only be added once to the right of all numerals (not subtracted), without redundancy, and are duodecimal.
• Beginning at 6/12, the S (semi) is used, followed by marks (unciae) for additional quantities up to 11/12.
• Placing a bar (vinculum) over a numeral multiplies it x1000.
• The vinculum is used for values of 4,000 and greater (it is not used up to 3,999), and may be iterated up to three (3) times.
• Starting at 4,000, and at the next two powers of a thousand thereafter, a new vinculum is added, with the pattern beginning at IV.
• Once vinculums are employed, the higher power is modified in preference to the lower (eg IV, not MV).
• Vinculums are always contiguous and are placed leftmost (ie no broken overlines or lower powers to the left per basic rule 1).

Thus, 2 2/3 would be rendered as IIS••, while 6,986 would be (VI)CMLXXXVI. Note that the symbol V is used more than once in that example, this is only permissible with the vinculum setting them apart. Also note that while M is used, 7,000 would be (VII).

(I), (II), and (III) are not used, with M, MM, MMM being preferred under 4,000. SS is never used in place of I (ss is sometimes seen as a variant of semi, but this is strictly pharmaceutical notation).

With fractions and vinculums employed, the lowest possible value is • or 1/12, while the highest is (((MMMCMXCIX)))((CMXCIX))(CMXCIX)CMXCIX or 3,999,999,999,999 (within standard usage). Four trillion, rendered as (((MMMM))), would be non-standard since it breaks basic rule 2, although the four M's can be used as a variant form, and it is more aesthetically pleasing as a maximum figure. Another alternative is ((((IV)))), but since the general principle is three sequential iterations, this should also apply to vinculums. The reason for the three-rule is that the eye can readily distinguish between one, two, or three marks, while four or greater become illegible. Thus, 4 M's are preferable as a variant form.