What are the rules of transforming a Hindu Arabic into a roman numeral?

Answer 1

There are no rules as such, just conventions. The accepted convention is to place symbols in descending order of value, such that no value is repeated more than 3 times in succession. There are 7 individual symbols available:

I, V, X, L, C, D and M.

In order, they represent the values:

1, 5, 10, 50, 100, 500 and 1000

In addition, there are intermediate symbols known as subtractive pairs that may never be repeated:

IV, IX, XL, XC, CD and CM.

In order, they represent the values:

4, 9, 40, 90, 400 and 900.

They are called subtractive pairs because, while individual values are generally added together, these values subtract the smaller value from the larger value. Thus IV means 5-1, which is 4. By contrast, VI would be 6 (5+1).

To translate a Hindu-Arabic numeral, begin by breaking the number down into thousands, hundreds, tens and units. Let's suppose the value is 1,234.

We can break this value down as follows: 1000 + 200 + 30 + 4.

Now we translate each of these into Roman numerals:

1000 is M, so we place M to the far left of the numeral (it is the largest value).

There is no symbol for 200, so we must break this down further. We can make 200 by adding 100 + 100, and since C is 100, CC must therefore be 200. Thus we have MCC (1200).

Similarly, there is no symbol for 30, but we can make it with 10 + 10 + 10. Thus 30 is XXX. Now we have MCCXXX (1230).

Finally, the 4 can be represented as IV. So the final numeral is MCCXXXIV.

Although the accepted convention does not permit an individual symbol to repeat more than 3 times, the Romans themselves preferred to use IIII to IV, thus they would have written MCCXXXIIII instead. Similalrly with VIIII rather than IX. Occasionally they'd do the same with the other subtractive pairs, however generally it was just the 4 and 9 they preferred to write in full.

Note that because no value can repeat more than 3 times, 3999 is the largest possible value using the accepted convention. That would be represented by MMMCMXCIX.

Middle Age notation permits larger numbers to be represented by place a horizontal bar over a numeral to multiply it by 1000. The the following would mean 4,000:

__

IV

Vertical bars on either side of a numeral multiplied it by 100. Thus |M| would mean 100,000.

Combining the two would multiply by 100,000, thus the following would mean 10 million:

___

|M|

Using this convention, combined with the accepted convention, allows a greater range of numbers to be notated. When using both conventions, the numbers are separated with a space. Thus the following is 10,000,000 + 3,999:

___

|M| MMMCMXCIX

Another convention uses the original symbol for M which was CIƆ. By itself this simply mean "many", but the notation could be extended such that the value is multiplied by a factor of 10. Thus CIƆ is 1000, but CCIƆƆ is 10,000 and CCCIƆƆƆ is 100,000, and so on.

Similarly, the original symbol for D was IƆ, and this could also be raised by a factor of 10, such that IƆ was 500, IƆƆ was 5,000 and IƆƆƆ was 50,000, and so on.

Moreover, the two could be combined such that CIƆƆ was 1,500, CCIƆƆƆ was 10,500 and CCIƆƆƆƆ was 15,000.