Subtractive notation
Encyclopedia
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 With the introduction of subtractive notation, XIV (14) no longer represents the same number as XVI but rather is an alternate way of writing XIIII.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).
The very positive advantage of subtractive notation is the reduction of counters needed on an 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.
Positional notation
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude...
used with Roman numerals
Roman numerals
The numeral system of ancient Rome, or Roman numerals, uses combinations of letters from the Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as:...
as a shorthand to replace four or five characters in a numeral representing a number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
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
Arabic numerals
Arabic numerals or Hindu numerals or Hindu-Arabic numerals or Indo-Arabic numerals are the ten digits . They are descended from the Hindu-Arabic numeral system developed by Indian mathematicians, in which a sequence of digits such as "975" is read as a numeral...
). 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
Unary numeral system
The unary numeral system is the bijective base-1 numeral system. It is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol representing 1 is repeated N times. For example, using the symbol | , the number 6 is represented as ||||||...
which used alphabetic characters to represent groupings of tally marks
Tally marks
Tally marks, or hash marks, are a unary numeral system. They are a form of numeral used for counting. They allow updating written intermediate results without erasing or discarding anything written down...
(a position-independent counting
Counting
Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
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).
The very positive advantage of subtractive notation is the reduction of counters needed on an abacus
Roman abacus
The Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and presumably tax collectors...
, 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.