michael_nielsen

Now, I'm pretty certain this would have seemed a wild idea to most of your contemporaries. But from our modern vantage point we know that in fact a much better system is possible: arabic numerals.

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.

Stated in roman numerals: there are DIXXX possible words of length II in the roman alphabet.

What about words of length III? Well, we can just enumerate again: AAA, AAB, AAC, ...., AAZ, ABA, ABB, ... ABZ. Keep going, you get to BAA, BAB. And so on. It's not too difficult to see the answer is 23 x 23 x 23.

Unfortunately, 23 x 23 x 23 (= 12,167) was starting to stress the roman numeral system. They had ways of dealing with such numbers, but it got pretty convoluted. The system was mostly designed to work up to a few thousand.

The reason was that the basic strategy behind roman numerals is to keep introducing new symbols as you get to larger and larger numbers. That's okay for a while, but eventually breaks down.

So a curious thing about the word-counting problem is that you get large numbers of different possible words, without needing to introduce new symbols. That's interesting if you've been bugged by the need to introduce new symbols to describe large numbers.

It's also interesting that you get very large numbers of different possible words, even with very short strings.

Now, if you were playful, you might wonder a bit about using strings of letters to _represent_ numbers. One way would be to identify I with A, II with B, and so on, through XXIII -> Z.

You could just continue from there: XXIV -> AA. XXV -> AB, and so on. Symbolically: yz -> y*23 + z, where y and z are just single letters.

Of course, this isn't notation the romans would have used. But the ideas - multiplication, equality, addition, are all things the romans understood. It's not too far a stretch.

What about with three digits? Well, with two digits you get up to 23+23*23. So we have: xyz -> x*(23+23*23)+y*23+z. I must admit, that seems slightly ugly to me, and it gets uglier with longer words.

Let's try something slightly different - a tad less obvious, but also prettier. Let's imagine that instead of numbers of variable length, we have just a single universal counter, starting at: .....AAAAAA

We increment by 1 and get to: ...AAAAAB. And then by 1 more and get to: ...AAAAAC. And so on.

Of course, in practice we don't want to write out all the A's on the left. So we'll just adopt the convention of taking those as given. In other words, something like: BD really means ...AAAABD, but the A's on the left are implied.

Now there's a bit of a problem, which is this convention makes ...AAAA be written as just a blank. Obviously that would be unreadable. So we'll introduce an exception: for that, we'll write A, just as a kind of placeholder.

I don't claim, by the way, that this "universal counter" approach is necessarily _more_ obvious than what we were doing earlier. But it's at least plausible as an alternate. And it has the benefit that it's an attractively unified system.

Okay, so the way counting goes is: A, B, C, ..., Z, BA, BB, BC,...., and eventually on to BAA, BAB, and so on.

In this system, xy -> (x-1)*23+y. And xyz -> (x-1)*23*23+(y-1)*23+z. Those seem satisfyingly neat, and the pattern continues.

There is still a slightly ugly thing, which is the -1 terms, which show up over and over again. We could get rid of those if we decided to start counting at 1 -> B, 2 -> C, etc.

If we did that, then we'd have xy -> x*23+y, xyz -> x*23*23+y*23+z, and so on for longer strings. That's even simpler, though A is now playing pretty much the role of a placeholder.

At this point a clever designer might be bugged by the use of the alphabet in this number representation. After all, the alphabet is already being used for words! So it'd be better to switch to different symbols to prevent confusion.

The alternate symbols we use are pretty arbitrary. Let's go for A -> 0, B -> 1, C -> 2, and so on. Of course, we could introduce 23 new symbols. But the 23 is actually pretty arbitrary. So let's use just 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.