Take some prime number p
. The p-adic norm
gives us a metric
on the rational
; we can take the completion
of this metric space
to get the set of p-adic rationals Qp
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 Qp. But what is it really?
Well, it is easy to see that in the p-adic integers, any element d=...d2d1d0 has an inverse (a p-adic integer, denoted 1/d, for which d*(1/d)=1) iff d0 != . 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
(except the p-cimal point usually isn't big and bold
number of digits may follow the p-cimal point. In this respect a p-adic rational looks like a reverse
d 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.