# The third rule - subtractive notation of the Roman numerals (placing lower value symbols in front of larger value ones)

## Subtractive notation of Roman numerals:

• A numeral of a lesser value placed in front (next to the left) of a larger value numeral is subtracted from the larger one. This group of two numerals is called a group in subtractive notation.
• Except for symbol I (1), any other symbol may be used in a subtractive notation group as the larger value numeral.
• But only these symbols are allowed to be used in subtractive notation as numerals of lesser value: I (1), X (10), C (100), M (1,000), (X) (10,000), (C) (100,000), (M) (1,000,000).
• Numerals V (5), L (50), D (500), (V) (5,000), (L) (50,000), (D) (500,000) are NOT allowed to be used as numerals of lesser value in subtractive notation groups of Roman numerals.

## The only possible groups of numerals in subtractive notation are:

• IV, IX, XL, XC, CD, CM, M(V), M(X), (X)(L), (X)(C), (C)(D), (C)(M).
• IV: calculated as V - I = 5 - 1 = 4;
• IX: calculated as X - I = 10 - 1 = 9;
• XL: calculated as L - X = 50 - 10 = 40;
• XC: calculated as C - X = 100 - 10 = 90;
• CD: calculated as D - C = 500 - 100 = 400;
• CM: calculated as M - C = 1,000 - 100 = 900;
• M(V): (V) - M = 5,000 - 1,000 = 4,000;
• M(X): (X) - M = 10,000 - 1,000 = 9,000;
• (X)(L): (L) - (X) = 50,000 - 10,000 = 40,000;
• (X)(C): (C) - (X) = 100,000 - 10,000 = 90,000;
• (C)(D): (D) - (C) = 500,000 - 100,000 = 400,000;
• (C)(M): (M) - (C) = 1,000,000 - 100,000 = 900,000;

## How to use the lesser value numerals in subtractive notation groups

• The symbol I (1) may precede only a V (5) and a X (10) - the next two larger symbols up in the basic set of Roman numerals; the groups IL, IC, ID, IM, ... are not correct;
• The symbol X (10) may precede only an L (50) and a C (100) - also the next two larger symbols up in the basic set of Roman numerals; the groups XD, XM, X(V), X(X)... are not correct;
• The symbol C (100) may precede only a D (500) and an M (1,000) - again, also the next two larger symbols up in the basic set of Roman numerals; the groups C(V), C(X), C(L), C(C)... are not correct;
• As a rule, when used in subtractive notation as the lesser value numerals, the symbols I, X, C, M, ... may only precede their correspondent two larger symbols up in the basic set of Roman numerals.

## Examples:

• 99 is written as: 99 = 90 + 9 = (100 - 10) + (10 - 1) = (C - X) + (X - I) = XC + IX = XCIX and not IC (we are not allowed to subtract I from C to get 99).
• 95 is written as: 95 = 90 + 5 = (100 - 10) + 5 = (C - X) + V = XC + V = XCV, and not VC (we are not allowed to subtract V from C to get 95).