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
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:
Here's another example. Reducing modulo larger and larger powers of we see that
Here's another example, but with a decimal point.
William Stein 2012-09-24