michael_nielsen

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.