Take some

prime number `p`. The

p-adic norm |.|

_{p} gives us a

metric on the

rationals

**Q**; we can take the

completion of this

metric space to get the set of

`p`-adic rationals **Q**_{p}.

Which probably makes no sense unless you had 2 Banach spaces for breakfast (and a Frechet space chaser).
So let's do it another way.

Recall that the p-adic integers are a (commutative) ring with no zero divisors (an integral domain). Which means we can take their field of fractions; this is exactly **Q**_{p}. But what is it *really*?

Well, it is easy to see that in the p-adic integers, any element d=...d_{2}d_{1}d_{0} has an inverse (a p-adic integer, denoted 1/d, for which d*(1/d)=1) iff d_{0} != . The proof of this fact is by a simple explicit construction.

So the non-invertible d's are precisely those divisible by `p` (i.e. d=`p`*c). In other words, if we could divide by `p`, we'd have a field.

Now, in "normal" base `p` numbers, the way we introduce 1/`p` is by adding a decimal point (`p`-cimal point?). So let's do that. A p-adic rational has the form

...d_{2}d_{1}d_{0}**.**d_{-1}...d_{-k}.

(except the p-cimal point usually isn't big and

bold).
A

*finite* number of digits may follow the p-cimal point. In this respect a p-adic rational looks like a

reversed

real number written in

base `p`.

As before, all operations are defined; determining the `n`'th digit of the sum, product, difference, quotient or square root (if it exists) requires that we look only at finitely many digits. As everyone remembers (or should remember) from grade school, working a p-cimal point can be done by removing it, performing the desired operations, then adding it back at the correct location.

The obvious parallel to the relationship between p-adic integers and p-adic rationals is to the relationship between power series and Laurent series.