Roman Numerals

Roman numerals are a system of numerical notations used by the Romans. They are an additive (and subtractive) system in which letters are used to denote certain "base" numbers, and arbitrary numbers are then denoted using combinations of symbols. Unfortunately, little is known about the origin of the Roman numeral system (Cajori 1993, p. 30).

The following table gives the Latin letters used in Roman numerals and the corresponding numerical values they represent.

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

For example, the number 1732 would be denoted MDCCXXXII in Roman numerals. However, Roman numerals are not a purely additive number system. In particular, instead of using four symbols to represent a 4, 40, 9, 90, etc. (i.e., IIII, XXXX, VIIII, LXXXX, etc.), such numbers are instead denoted by preceding the symbol for 5, 50, 10, 100, etc., with a symbol indicating subtraction. For example, 4 is denoted IV, 9 as IX, 40 as XL, etc. However, this rule is generally not followed on the faces of clocks, where IIII is usually encountered instead of IV. Furthermore, the practice of placing smaller digits before large ones to indicate subtraction of value was hardly ever used by Romans and came into popularity in Europe after the invention of the printing press (Wells 1986, p. 60; Cajori 1993, p. 31).

The following table gives the (Europeanized) Roman numerals for the first few positive integers.

# RN # RN # RN 1 I 11 XI 21 XXI 2 II 12 XII 22 XXII 3 III 13 XIII 23 XXIII 4 IV 14 XIV 24 XXIV 5 V 15 XV 25 XXV 6 VI 16 XVI 26 XXVI 7 VII 17 XVII 27 XXVII 8 VIII 18 XVIII 28 XXVIII 9 IX 19 XIX 29 XXIX 10 X 20 XX 30 XXX

For large numbers, the Romans placed a partial frame around numbers (open at the bottom), which indicated that the framed number was to be multiplied by , as illustrated above (Menninger 1992, p. 44; Cajori 1993, p. 32). In more recent practice, the strokes were sometimes written only on the sides, e.g., (Cajori 19993, p. 32). It should also be noted that the Romans themselves never wrote M for 1000, but instead wrote (I) for , (I)(I) for , etc., and also occasionally wrote IM, IIM, etc. (Menninger 1992, p. 281; Cajori 1993, p. 32). However, in the Middle Ages, the use of M became quite common. The Romans sometimes used multiple parentheses to denote nested multiplications by 10, so (I) for , ((I)) for , (((I))) for , etc. (Cajori 1993, p. 33).

The Romans also occasionally used a vinculum (called a titulus in the Middle Ages) over a Roman numeral to indicate multiplication by 1000, so , , etc. (Menninger 1992, p. 281; Cajori 1993, p. 32).

Roman numerals are encountered in the release year for movies and occasionally on the numerals on the faces of watches and clocks, but in few other modern instances. They do have the advantage that addition can be done "symbolically" (and without worrying about the "place" of a given digit) by simply combining all the symbols together, grouping, writing groups of five Is as V, groups of two Vs as X, etc.

The number of characters in the Roman numerals for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... (i.e., I, II, III, IV, V, VI, VII, VIII, IX, X, ...) are 1, 2, 3, 2, 1, 2, 3, 4, 2, 1, 2, 3, 4, ... (OEIS A006968). This leads to a scale-invariant fractal-like stairstep pattern which rises in steps then falls abruptly.