Subtractive notation

Subtractive notation

Subtractive notation is an early form of positional notation used with Roman numerals as a shorthand to replace four or five characters in a numeral representing a number with usually just two characters. Using subtractive notation the numeral VIIII becomes simply IX. LXXXXVIII would become IIC, which is invalid in Roman numerals though.

Without subtractive notation, XIV represents the same number as XVI (16 in Arabic numerals). With the introduction of subtractive notation, XIV (14) no longer represents the same number as XVI but rather is an alternate way of writing XIIII.

By encoding information about the number into the order of the numerals, subtractive notation transformed the Roman numeral system from a variation of a unary numeral system which used alphabetic characters to represent groupings of tally marks (a position-independent counting system). This form of notation closely follows Latin language usage, in which the number 18 is pronounced as "duodeviginti", meaning "two [deducted] from twenty" ("duo-de-viginti"), and 19 is pronounced "undeviginti", meaning "one [deducted] from twenty" ("un-de-viginti").

While simplifying the presentation of numerals, subtractive notation removed from Roman arithmetic the advantages of a tally system for speedy addition and subtraction without adding the flexibility of later positional notations systems for mathematics.

The very positive advantage of subtractive notation is the reduction of counters needed on a abacus, the calculating devices used by the Romans, and those before them for thousands of years. They did not do arithmetic with their written numerals. They used their numerals only for recording the results of calculations on an abacus. Reduction of counters is also why they used intermediate 5 values, i.e. V, L, and D. The number of counters needed to represent IIIIIIIII, VIIII, and IX are 9, 5, and 2. Subtractive notation reflected the counter positions on an abacus with positive and negative counters on opposite sides of a median line.


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