Imagine you're a designer or mathematician living in ancient Rome. Being a curious & imaginative sort, used to exploring wild ideas, you ask yourself: is there some way you can improve on the roman numeral system? Might it be possible to find a better way of representing number?

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.

(Of course, the numeric base we use is pretty uninteresting. Base 10 turns out to have some nice advantages, and also some slight problems. But I think this really is an accidental piece of history.)

With these choices, our number representation is: xyz -> x*10*10+y*10+z. And so on, for larger numbers.

We're very _used to_ this kind of representation. But there's nothing obvious about it at all. Every single thing in the representation can be questioned, and possibly changed. It's fun to try some experiments!

One particularly amusing thing: consider the numbers 132 and 123. The 2 actually has a very different meaning in those two numbers. The fact that _location matters_ is a deep idea.

(I've glossed over 0 in various ways in this account. And over the fact that place-number system pre-dated the romans. If I was more conscientious I'd have talked more about these - they're incredibly deep ideas - but this thread is already long, so I'll keep glossing.)

At this point we have a new numeral system. It's nice in a couple of ways when compared to roman numerals - it doesn't need new symbols to represent larger numbers, and it's extremely compact.

Still, those things perhaps don't seem that important. Certainly not worth replacing an entire piece of intellectual infrastructure with!

But where this number system really shines is in simplifying certain other things you might want to do. For instance, consider addition of the numbers wx and yz.

We have wx + yz = (w*10+x) + (y*10+z) = (w+y)*10 + (x+z). Fiddle with this for a while - it's more work than I want to go through here - and you eventually recover the grade school algorithm for adding two-digit numbers. Do some more work, & you get the n-digit algorithm.

There's a miracle going on here, one we don't notice b/c it's so familiar: I pointed out above that the numerals have very different meanings, depending on their location. But despite this, in the grade-school algorithm we use the _same rules_ for addition, regardless of place!

E.g., in computing 27+38 at some point in the computation we'll use 2+3 = 5; in computing 72+83 at some point we'll also use 2+3=5. That's despite the fact that the 2 and the 3 in the first sum have a very different meaning than in the second sum!

What's responsible for this astonishing fact? If you look back at the reasoning above, you see it's a consequence of associativity, commutativity, and distributivity. That's a pretty huge set of things! And it makes addition _really_ nice in this representation.

In particular, it means that although the numerals have different meanings in different locations, in many ways they continue to _behave_ as though they are in some sense "the same". They're _not_ the same. But they retain many of the same affordances, in modern interface lingo.

You can go through the same kind of thing, trying to figure out how to multiply two numbers: wx*yz = (w*10+x)*(y*10+z) = x*y*10*10 + ... I won't go through it, but you can guess what happens: you get the grade-school algorithm for multiplication.

And, again, even though the numerals have very different meanings in different positions, they have a lot of the same affordances, and so behave in some ways as the "same" object.

You can go through this with long division. Same story.

At this point this new numeral system looks vastly superior to roman numerals. It's exploiting all this structure to get compact representations, which don't require new symbols, & which exploit deep properties of numbers to simplify addition, multiplication, & division. Amazing!

I'm far from an expert on the history of mathematics or its representations. The story above is unrealistic in many ways. Still, I think it gives at least a hint of the incredible mathematical and design insight required to invent arabic numerals.

And, of course, this story requires many incredible earlier insights: words, the alphabet, various ideas about multiplication and addition, and so on.

To finish, a fun question: is there some way to improve still further on arabic numerals? I believe the answer is almost certainly yes! But that's a thread for another day.

A few addenda: (1) as lots of people have pointed out, it'd have been more accurate to name them Hindu-Arabic numerals; (2) the ancient Greeks seem to have known much of this (which I didn't know);

(3) the ancient Babylonians ditto (which I did know - it was meant as a thought experiment about discovery and lines of insight, not history); (4) @DavidDeutschOxf has a lovely discussion of number systems in chapter 6 of "The Beginning of Infinity".

(5) Via @CXGonzalez_, a paper arguing that for educated romans, the computational difficulty of working with roman numerals was comparable to us working with Hindu-Arabic: http://csjarchive.cogsci.rpi.edu/Proceedings/2008/pdfs/p2097.pdf

(6) I haven't published anything specifically on improving Hindu-Arabic numerals. But here's some related work inspired in part by that problem: on "Magic Paper" (new interfaces for mathematics) http://cognitivemedium.com/magic_paper/index.html

"Toward an Exploratory Medium for Mathematics" (on developing a logic of heuristic discovery, to underly creative exploration) http://cognitivemedium.com/emm/emm.html 

And "Thought as a Technology" (about the idea that we internalize the interfaces we use as part of our thinking; interface designers actually help us think new thoughts):
http://cognitivemedium.com/tat/index.html