What are the rules in writing roman numeral?

Answer 1

There are no rules, as such, but there is an accepted convention that has been in use for the past few hundred years.

The only real rules relate to the original system used by Romans themselves. Firstly, each symbol has a value that never changes: I=1, V=5, X=10, L=50, C=100, D=500 and M=1000. Numerals are created by writing these symbols in descending order (from left to right). To read a numeral, convert each symbol to its corresponding value and add the values together.

The Romans also used a shorthand for the values 4, 9, 40, 90, 400 and 900, denoted by symbolic pairings IV, IX, XL, XC, CD and CM respectively. Note that the symbols in each of these pairs are in ascending order (not descending order). This means that rather than adding the values together, the smaller of the two values is negated, such that IV = -1 + 5 = 4. These pairing are known as subtractive pairs.

The Romans themselves rarely used IV and IX, preferring the longhand notations IIII and VIIII, however either form is acceptable today. They did use the other 4 pairings more consistently, but there occasional instances of longhand, or a mixture of the two in the same numeral.

Regardless, the 6 subtractive pairs are each treated as if they were one symbol. Thus we now have 13 symbols at our disposal: I=1, IV=4, V=5, IX=9, X=10, XL=40, L=50, XC=90, C=100, CD=400, D=500, CM=900 and M=1000.

To convert a number to a Roman numeral, subtract the largest value symbol that is less than or equal to the number, write it down, and repeat with the remainder. For example, if our value is 1999, we would carry out the following steps:

1. M (1000) is the largest value that is less than or equal to 1999, leaving 999.
2. CM (900) is the largest value that is less than or equal to 999, leaving 99.
3. XC (90) is the largest value that is less than or equal to 99, leaving 9.
4. IX (9) is the largest value that is less than or equal to 9, leaving zero.

Thus the Roman numeral for 1999 is MCMXCIX. The Romans themselves would have more likely written MCMXCVIIII, but either form is acceptable.

Note that you will often find examples that bend the rules slightly. For instance, the number 1999 can also be represented by the Roman numeral MIM, even though IM (-1+1000=999) is not an acceptable subtractive pair.

The accepted convention states that no symbol may be repeated more than three times in succession (meaning IV is the preferred version of IIIII, etc), that V, L and D may never repeat (because VV can be simplified to X), and that subtractive pairs may never repeat (IXIV is not permitted). Also, although there is no rule that states IM is not permitted for the value 999, the accepted convention states that I may only be subtracted from V and X, X may only be subtracted from L and C, and C may only be subtracted from D and M. This is simply in keeping with the original Roman rules.

Under this convention, all the values in the range 1-3999 can be represented in only one unique way.

Numbers larger than 3999 can be represented using Middle Age notation. However, the notation is relatively unknown so few will understand its meaning, even if they understand the standard Roman numerals. Numbers such as 4000 are easier to understand as just MMMM, however the larger the number, the more difficult it becomes to notate. For instance, one million would require one thousand Ms, which is clearly unacceptable. But just as we'd write "1 million" today, the Romans would have written "M mille" (one thousand thousands), or even just "I million" (million being a Latin word for a thousand thousands).

Middle Age notation has two forms. The first uses overlines and vertical bars to multiply a number by 100, 1,000 or 100,000. An overline multiples by 1000, while vertical bars either side of a numeral multiply by 100. Combining the two multiplies by 100,000. To notate the least-significant portion of a number, we simply need to leave a space and notate it using the standard notation. Using this convention, we can cater for values up to one thousand million but, as previously stated, few people will understand the notation.

The second form of notation uses the original symbols for M and D, which were CIƆ and IƆ, and extends them such that CCIƆƆ means 10,000 and IƆƆ means 5000. The two forms can also be combined such that CCIƆƆƆ means 10,500 and CCIƆƆƆƆ means 15,000. Although this convention allows a greater range, there's still the problem of readability. Few will understand it, but extremely large numbers would still be extremely unwieldy. However, the symbols can be written using shorthand, such that CIƆ is written ↀ, CCIƆƆ is written ↂ, IƆ is written D (which is what it became anyway) and IƆƆ is written ↁ. Even using shorthand, extremely large numbers become so cryptic that it's really not worth the effort. The Romans themselves never wrote extremely large numbers out in full (it was implied by the context), so why should we?

Answer 2

There are no rules as such. While there are conventions that have been in use for several hundred years, there is no authority that governs whether a given Roman numeral is written rightly or wrongly. The number 1999 can therefore be written as MCMXCIX, MIM or MDCCCCLXXXXVIIII, or even a mixed form such as MCMXCVIIII. All are intrinsically correct, however the accepted convention would consider the first to be the only valid form.

In general, all Roman numerals are written in descending order, with the largest value to the left. If one small number value precedes one larger number value, the smaller value is subtracted from the larger value.

There's at least one known instance where the number IIXX is intended to mean 18, however this is deemed unacceptable in the accepted convention because a) the values are not in descending sequence and b) it implies II must be subtracted from XX, when subtractions should only be applied to 2 individual values, not combined values. For similar reasons, IVX would also be deemed invalid as it is ambiguous. Does it mean 10-5-1=4? Or 10-4=6? Or is it really 4+10=14?

The accepted convention states that subtractive pairs are limited to just 6 pairs: IV, IX, XL, XC, CD and CM, where each pair represents a single value (4, 9, 40, 90 400 and 900 respectively). This then gives 13 possible symbols in all; I, IV, V, IX, X, XL, L, XC, C, CD, D, CM and M. The accepted convention also states that no individual value (I, V, X, L, C, D or M) may repeat more than 3 times in succession, and that no subtractive pairs (IV, IX, XL, XC, CD or CM) may ever be repeated. Furthermore, all values, including subtractive pairs, must be written in descending order of value.

As such, the accepted convention caters for values in the range 1-3999, because 4000 would require 4 Ms. Larger numbers such as 1,000,000 would also be impossible to notate as you'd need 1,000 Ms, which is impractical at best and completely useless in terms of readability.

Middle Age notation can extend the range beyond 3999 in a more practical manner, however you will rarely see this form in use today (Hindu-Arabic notation made it obsolete). However, the notation allowed a horizontal bar over a value to indicate the value be multiplied by 1,000 while vertical bars either side multiply it by 100. Combining the two would multiply the value by 100,000. However, this form of notation only applies to a complete value. The least-significant portion of the number would need to be notated separately, using the accepted convention.

Another form of notation, also from the Middle Ages, replaces the D and M with their earlier symbols: IƆ and CIƆ. These notations can be extended to multiply the value by a factor of 10, such that IƆƆ is 5,000 and CCIƆƆ is 10,000, and so on. They can also be combined such that CCIƆƆƆ is 10,500. The symbols can also be reduced hyperbolically, such that CIƆ can be written with the symbol ↀ, CCIƆƆ with ↂ, IƆ with D (the normal symbol for 500) and IƆƆ as ↁ.