Roman numbers ↔ Latin alphabet counter

We can number lists using different counter styles, for example using Hindu–Arabic numerals and the decimal system 1, 2, 3, …, 9, 10, 11, … or using the Latin alphabet A, B, C, …, Z, AA, AB, … or using the Roman number system I, II, III, IV, V, VI, ….

When we ignore the values of these symbols for a moment and only look at the characters used, we see that the Roman numerals are a subset of the Latin alphabet. Furthermore the alphabet counter uses every possible combination of letters. Hence for every Roman number there is a number in the Latin alphabet counter that looks the same.

For example, the Roman numeral for one, I, also appears in the Latin alphabet counter where it represents the number nine, as it is the ninth letter in the alphabet. V is a Roman five and at the same time the twenty-second letter in the alphabet. DC is a Roman six hundred and at the same time one hundred and seven in the Latin alphabet counter.

I’ve compiled a table that lists all Roman numbers from one (I) to 3999 (MMMCMXCIX) and the corresponding values which the same representations have when interpreted as Latin alphabetic numbers:

How do you do this?

There is no universally accepted standard of writing Roman numbers. For example, analog clock faces often display four as IIII instead of IV like I did above. So let’s settle on a convention first. I chose the one used for counter styles in cascading style sheets (CSS). It corresponds perfectly to what I learned at school about Roman numbers, although the explanation differs a little.

Step 1

The CSS specification defines thirteen Roman symbols, each consisting of one or two letters, and their corresponding values displayed as decimal numbers here. When you want to write a natural number with Roman numerals, you start at the left side of the table, at the first symbol M.

If your number is at least as big as the value of the current symbol, you write down the current symbol and reduce your number by the symbol’s value, otherwise you move right to the next symbol. Repeat this until your number is zero.

You’ll write done your initial number in terms of Roman numerals in the process. Easy, isn’t it? That’s the first step. The second step is to interpret that string of symbols as a Latin alphabetic representation and compute its value.

Step 2

The alphabetic number system is a positional system like our decimal system, but without a number zero. The letters A to Z are its digits and correspond to the values 1 to 26. You will only need a small subset of these when reinterpreting Roman numbers:

You compute the value of an alphabetic number by multiplying each digit’s value with the positional value of its place in the number and adding these products up. The positional value of the rightmost place is 26⁰ = 1, the value of the second place from right is 26¹ = 26, the value of the third place from right is 26² = 676. In general, the value of the n-th place from right is 26ⁿ⁻¹.