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The -adic Numbers

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 9) 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
2012-09-24