It's a familiar fact that every real number can be written in the form

where each digit is between 0 and , and the sequence can continue indefinitely to the right.

The -adic numbers also have decimal expansions, but everything is backward! To get a feeling for why this might be the case, we consider Euler's nonsensical series

One can prove (see Exercise 55) that this series converges in to some element .

What is ? How can we write it down? First note that for all , the terms of the sum are divisible by , so the difference between  and is divisible by . Thus we can compute modulo  by computing modulo . Likewise, we can compute  modulo  by compute , etc. We obtain the following table:

Continuing we see that

in $Q_10$ !

Here's another example. Reducing modulo larger and larger powers of  we see that

in $Q_10$

Here's another example, but with a decimal point.

We have

which illustrates that addition with carrying works as usual.

William Stein 2004-05-06