How Did Anyone Do Math in Roman Numerals?

Subtraction is as easy as addition (unless you go into negative numbers). The stategy is to walk the subtrahend left-to-right, eliminating from both as you go. If you run into a value that you don't have, then you expand the minuend from the right until you do.

For your "difficult" example of 42 - 13:

XXXXII - XIII

1 Eliminate an X from both.

Now you have XXXII - III.

2 Eliminate I from both

Now you have XXXI - II. Repeating (2) gives you XXX - I.

3 Since there is no I to eliminate, expand the minuend from the right to create one.

Now you have XXVV - I, repeat since you still don't have a I, repeat (3) which gives you XXVIIIII - I

Eliminate the from I both, giving the final result: XXVIIII.

Optional, you can "reduce" this to XXVIV, like you would if you had a fraction, like 3/9, but as stated in the article, this notation was not common place in ancient Rome.

If you spent several decades working with a counting board and had only occasionally seen Hindu–Arabic arithmetic, you would likely feel the opposite (as, indeed, people did for the first few centuries after written arithmetic was introduced to Europe – for example our word cipher, meaning secret code, comes from the word for the Arabic 0, reflecting people’s early confusion about pen and paper arithmetic).

The Romans (and others in the Roman empire) were the premier engineers, merchants, bureaucrats, astronomers, etc. of their era and region. They didn’t have any problem doing extremely complex computations.

As for your specific concerns: the easy pattern is that the letter for a group of five literally looks like half of the letter for a group of ten.

V = X/2, L = C/2, D = ↀ/2, ↁ = ↂ/2, etc.

So you need to remember the meanings and relations for the symbols for I, X, C, ↀ, ↂ and then just count them. The patterns that: IIIII = V; XXXXX = L; CCCCC = D, ↀↀↀↀↀ = ↁ are really not that hard to remember. The groups of five are mostly there as a shorthand, because writing nine of the same symbol in a row is harder to count and takes up more space.

These patterns are certainly no harder than remembering the English words ten, hundred, thousand (or Latin words unus, decem, centum, mille), which are also arbitrary symbols.

Since the numbers are always written in order, you can learn to separate them by digits. People would have “chunked” a long string of these symbols into the word for each each digit, and pronounced them using words pretty much like modern languages. Just like in our natural languages, the system is not strictly positional – you just skip writing/pronouncing any lines on the counting board with no pebbles.

So if you see DCCCXXXXVII you think of it / read it as “eight hundred forty seven”, first split into the groups DCCC XXXX VII, with the digit represented by the pattern and the order of magnitude represented by the symbols used. Or alternately, you would have seen them as the visual patterns “three pebbles on the hundred line and one in the space above; four pebbles on the tens line; two pebbles on the one line and one in the space above”.

When you’re thinking of the meaning of these symbols, you’re going to be fluently translating them in your head back and forth between three representations: verbal, counting board, and written. Once these have all been worked with extensively, there’s not much friction. It’s just like learning to read, or learning music notation. Someone experienced can read a musical phrase on a score and hear the sound of the whole expression in their head, rather than trying to count which line each note is on, count out the tempo, etc.

Objectively, the Roman system is easier to teach up to a basic level, especially to someone illiterate. Basic calculations on a counting board are straight forward and easy to explain and motivate. Multi-digit multiplication gets a bit annoying in both cases because it involves the summation of many partial partial products. Long division and square roots get nasty in both systems.

Where the Hindu–Arabic system really shines is when people need to frequently work with very large numbers, very precise numbers (though remember there were no decimal fractions per se in Europe until >1600), or numbers of different orders of magnitude, have access to cheap and abundant paper, and can spend years training to do basic arithmetic. The biggest advantages of pen and paper methods for basic arithmetic are that it’s easy to see the whole work process, and therefore more easily check for mistakes, and that writing the final answer doesn’t take as much space. It’s also much easier to write down and explain pen and paper arithmetic methods in a printed book. The counting board methods are often faster to perform.

But more importantly still, pen and paper arithmetic is easy to generalize to more sophisticated mathematical notation for fractions, algebraic equations, etc.