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# What are subtractive roman numerals?

Subtractive roman numerals are used to simplify and shorten a numeral sequence.

Subtractive numerals are used to simplify certain number representations that would otherwise contain four or more repeated numeral characters. They also prevent confusion when visually calculating the numerals.

For example, the number four expressed using only base numerals would be:

IIII

To simplify this, we would use the subtractive numeral IV; or “five minus one”, hence the ‘subtractive’ description.

Similarly, the number ninety expressed using only base numerals would be:

LXXXX

Using the subtractive numeral this becomes:

XC

Much simpler and easier to translate.

Each ‘place’ (thousands, hundreds, tens, and units) in a number has exactly two additional subtractive numerals available. One just before the midpoint, and one before the upper limit.

For units, the midpoint is 5 and the upper limit is 10. So, in turn, subtractive numerals are used for 4 and 9.

'Unit' subtractive numerals
NumberNumeralWords
4IVfour
9IXnine

For tens, the midpoint is 50 and the upper limit is 100. So, in turn, subtractive numerals are used for 40 and 90.

'Tens' subtractive numerals
NumberNumeralWords
40XLforty
90XCninety

For hundreds, the midpoint is 500 and the upper limit is 1000. So, in turn, subtractive numerals are used for 400 and 900.

'Hundreds' subtractive numerals
NumberNumeralWords
400CDfour hundred
900CMnine hundred

For thousands, there are no subtractive numerals in the standard set; they are not required as the highest number of thousands that can be represented in the standard set is 3000, which is MMM.

However, if we extend into vinculum notation, we simply start over, this time using barred numerals.

'Thousands' subtractive numerals
NumberNumeralWords
4000IVfour thousand
9000IXnine thousand

And so on.