0 through 9. But as the picture at the bottom of the facing page shows, one can equally well use other bases. And in practical computers, for example, base 2 is almost always what is used.

So what this means is that in a computer numbers are represented by sequences of 0's and 1's, much like sequences of white and black cells in systems like cellular automata. And operations on numbers then correspond to ways of updating sequences of 0's and 1's.

In traditional mathematics, the details of how operations performed on numbers affect sequences of digits are usually considered quite irrelevant. But what we will find in this chapter is that precisely by looking at such details, we will be able to see more clearly how complexity develops in systems based on numbers.

In many cases, the behavior we find looks remarkably similar to what we saw in the previous chapter. Indeed, in the end, despite some confusing suggestions from traditional mathematics, we will discover that the general behavior of systems based on numbers is very similar to the general behavior of simple programs that we have already discussed.

Elementary Arithmetic

The operations of elementary arithmetic are so simple that it seems impossible that they could ever lead to behavior of any great complexity. But what we will find in this section is that in fact they can.

To begin, consider what is perhaps the simplest conceivable arithmetic process: start with the number 1 and then just progressively add 1 at each of a sequence of steps.

The result of this process is to generate the successive numbers 1, 2, 3, 4, 5, 6, 7, 8, ... The sizes of these numbers obviously form a very simple progression.

But if one looks not at these overall sizes, but rather at digit sequences, then what one sees is considerably more complicated. And in fact, as the picture below demonstrates, these successive digit sequences form a pattern that shows an intricate nested structure.

Digit sequences of successive numbers written in base 2. The overall pattern has an intricate nested form.