Roman numerals are based on an additive concept of denominations, like physical currency. For example, to pay $19, one can give a $10 bill, a $5 bill and four $1 bills. With the 'denominations' of Roman numerals being letters of the Latin alphabet valued thus
Ⅰ 1 Ⅴ 5 Ⅹ 10 Ⅼ 50 Ⅽ 100 Ⅾ 500 Ⅿ 1000
and the 'denominations' being sorted from greatest to smallest, then 19 is ⅩⅤⅠⅠⅠⅠ. But just as people realized one could also give two $10 bills and receive a $1 bill, the Romans evolved a concept of "giving change" on a limited basis. By placing a smaller numeral to the left of a larger numeral, it is then understood that the value of the smaller numeral is subtracted from the larger numeral. From Ⅴ and Ⅹ we may subtract Ⅰ; from Ⅼ and Ⅽ we subtract Ⅹ; and from Ⅾ and Ⅿ we may subtract Ⅽ. Thus 19 is ⅩⅠⅩ, somewhat more manageable than ⅩⅤⅠⅠⅠⅠ. A006968 gives the number of Roman "digits" of in the "new" style while A092196 does so for the "old" style. A092197 compares the efficiency of "new" style Roman numerals to "old" style Roman numerals.
Still, Roman numerals are exceedingly clunky for the study of number theory. Maybe Ramanujan would still have figured out that ⅩⅠⅠ ⅠⅠⅠ + Ⅰ ⅠⅠⅠ = Ⅹ ⅠⅠⅠ + ⅠⅩ ⅠⅠⅠ = ⅯⅮⅭⅭⅩⅩⅠⅩ. Given the frivolous theorem of arithmetic, Roman numerals are clearly much less adequate than decimal numbers for modern number theoretic pursuits such as the hunt for large prime numbers. Under the "new" style, the maximum number that may be represented is 3999.
The idea of placing bars over numerals to multiply them by thousands did not have a chance to become standard prior to place value numeration becoming widely accepted. This was because "in an effort to overcome this difficulty [of counting beyond 3999], certain graphic conventions were adopted, but their number was increased so rashly that the Roman system lost its cohesion." Thus we find different methods in use well into medieval times.
The Romans did indeed have a need to beyond 3999: consider, for example, that ancient Roman warships transported about a hundred soldiers each, and that at the Battle of Actium, Octavian's fleet numbered about 700 ships. Counting just the total number of soldiers would have run well past 5000, not to mention other personnel, such as perhaps prisoners of war.
Nowadays, Roman numerals are mostly only used for ceremonial purposes (e.g. year of inauguration) or when there are not many items to count, such as for indexing the chapters of a novel.
However, some researchers have studied to what extent concepts studied in standard place value representation (such as binary and decimal) can be applied to Roman numerals. So, there are palindromic Roman numerals (A078715), "pandigital" Roman numerals (A105417), etc., and even Roman numeral Friedman numbers, such as ⅬⅩⅩⅤ = Ⅼ × ⅩⅤ / Ⅹ.
- Georges Ifrah, From One to Zero: A Universal History of Numbers translated from French by Lowell Bair. New York: Penguin (1981): 313.
- Brian Williams, Ancient Roman War and Weapons. Chicago: Heinemann-Raintree Classroom, 2003, p. 36.
- Williams, ibid. 37.