michael_nielsen

So a modern phrasing of the question might be: how to invent arabic numerals, assuming you only know roman numerals?

To be clear, I'm not talking here about delving into the actual history of arabic numerals. That's long and complex and fascinating, but this is a different kind of challenge.

What I'm talking about is a kind of discovery fiction, a plausible line of reasoning that might have led you to the discovery of arabic numerals, with roughly the set of raw materials on hand in ancient rome.

I spent quite a bit of time trying to find such a discovery fiction. Originally, I started from the question "If we changed notation [from roman numerals] might there be an easier way to multiply?"

But there's a different starting point, another rather natural question, that is more promising. It's to ask: how many possible distinct _words_ are there of any given length?

Now, the roman alphabet was much like our modern alphabet, but with a few differences. It had 23 characters, mostly the same as ours today: ABC.... Z (but with a few differences, eg no J).

How many words are there of length 2 in this alphabet? Well, the possible words are AA, AB, AC... AZ, followed by BA, BB, BC, ..., BZ. Then CA, CB,..., CZ.

Obviously, that means there are 23 times 23 = 529 possible "words" of length 2. Of course, multiplication was actually pretty tricky for the romans (mostly because they didn't have arabic numerals!) But figuring this out was well within their ken.